Chapter 20
Photons: Maxwell’s Equations in a
Nutshell
20.1
Introduction
Light has fascinated us for ages. And deservedly so. Everything we know about the
earth and the universe is because of light. Light from the sun sustains life on earth.
Learning to measure and understand the contents of light has enabled us to understand
the origins of the universe in the big bang, and talk about its future. And one cannot
forget the sheer visual pleasure of a beautiful sunset, a coral reef, or an iridescent flower
in full blossom. Indeed, the beauty of light and color is a rare thing that scientists and
artists agree to share and appreciate.
Our fascination with light has led to three of the greatest revolutions in 19th and 20th
century physics. Sunlight used to be considered a ‘gift of the Gods’ and the purest indivisible substance, till Newton observed that passing it through a prism split it into multiple colors. Passing each of the colors through another prism could not split it further.
Newton surmised that light was composed of particles, but in the early 19th century,
Young proved that light was a wave because it exhibited interference and di↵raction.
Michael Faraday had a strong hunch that light was composed of a mixture of electric and
magnetic fields, but could not back it up mathematically. The race for understanding the
fabric of light reached a milestone when Maxwell gave Faraday’s hunch a rigorous mathematical grounding. Maxwell’s theory combined in one stroke electricity, magnetism,
and light into an eternal braid1 . The Maxwell equations predict the existence of light
1
J. R. Pierce famously wrote “To anyone who is motivated by anything beyond the most narrowly
practical, it is worthwhile to understand Maxwell’s equations simply for the good of his soul.”
132
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
133
as a propagating electromagnetic wave. With Maxwell’s electromagnetic theory, the
‘cat’ was out of the hat for light.
The second and third revolutions born out of light occurred in early 20th century in
parallel. Trying to understand blackbody radiation, photoelectric e↵ect, and the spectral
lines of hydrogen atoms lead to the uncovering of quantum mechanics. And Einstein’s
fascination with the interplay of light and matter, of space and time led to the theory
of relativity. Much of modern physics rests on these three pillars of light: that of
electromagnetism, quantum mechanics, and relativity. It would be foolhardy to think
that we know all there is to know about light. It will continue to amaze us and help
probe deeper into the fabric of nature through similar revolutions in the future. In
this chapter, we discuss Maxwell’s theory of electromagnetism in preparation for the
quantum picture, which is covered in the next chapter.
20.2
Maxwell’s equations
Maxwell’s equations connect the electric field E and the magnetic field intensity H to
source charges ⇢ and currents J via the four relations
r·D
r·B
r⇥E
= ⇢,
Gauss’s law
= 0,
=
r⇥H = J
Gauss’s law
@B
@t ,
+ @D
@t ,
Faraday’s law
(20.1)
Ampere’s law.
Here the source term ⇢ has units of charge per unit volume (C/m3 ), and current source
term J is in current per unit area A/m2 . H is related to the magnetic flux density B
via B = µ0 H, and the displacement vector is related to the electric field via D = ✏0 E.
The constant ✏0 is the permittivity of vacuum, and µ0 is the permeability of vacuum.
They are related by ✏0 µ0 = 1/c2 , where c is the speed of light in vacuum.
+
-
Figure 20.1: Electrostatic Fields.
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
134
Gauss’s law r · E = ⇢/✏0 says that electric field lines (vectors) due to static charges
originate at points in space where there are +ve charges, and terminate at negative
charges, as indicated in Figure 20.1. Vectors originating from a point in space have a
positive divergence. This relation is also called the Poisson equation in semiconductor
device physics, and if the charge is zero, it goes by the name of Laplace equation. Gauss’s
law for magnetic fields tells us that magnetic field lines B have no beginnings and no
ends: unlike static electric field lines, they close on themselves.
Note that for electrostatics and magnetostatics, we put @(...)/@t ! 0, to obtain the static
magnetic field relation r ⇥ H = J. The magnetic field lines curl around a wire carrying
a dc current, as shown in Figure 20.1. Electrostatic phenomena such as electric fields
in the presence of static charge such as p-n junctions, transistors, and optical devices in
equilibrium, and magnetostatic phenomena such as magnetic fields near wires carrying
dc currents are covered by the condition @(...)/@t ! 0, and electric and magnetic fields
are decoupled. This means a static charge produces just electric fields and no magnetic
fields. A static current (composed of charges moving at a constant velocity) produces a
magnetic field, but no electric field.
Since in electrostatics, r ⇥ E = 0, the static electric field vector can be expressed as
the gradient of a scalar potential E =
r
because r ⇥ (r ) = 0 is an identity.
is
then the scalar electric potential. However, the same cannot be done for the magnetic
field vector even in static conditions, because r ⇥ H = J 6= 0. However, the magnetic
field can be written as the curl of another vector field B = r ⇥ A, where A is called the
magnetic vector potential. Hence from the Maxwell equations, E =
dA/dt.
Faraday’s law says that a time-varying magnetic field creates an electric field. The
electric field lines thus produced ‘curl’ around the magnetic field lines. Ampere’s law
says that a magnetic field intensity H may be produced not just by a conductor carrying
current J, but also by a time-varying electric field in the form of the displacement current
@D/@t. The original Ampere’s law did not have the displacement current. Maxwell
realized that without it, the four constitutive equations would violate current continuity
relations. To illustrate, without the displacement current term, r ⇥ H = J, and taking
the divergence of both sides, we get r · r ⇥ H = r · J = 0 because the divergence of
curl of any vector field is zero. But the continuity equation requires
r·J =
(20.2)
@⇢/@t, Continuity Equation
which is necessary for the conservation of charge. With the introduction of the displacement current term, Maxwell resolved this conflict: r · J =
which connects to Gauss’s law.
r · @D
@t =
@
@t (r · D)
=
@⇢
@t ,
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
20.3
135
Light emerges from Maxwell’s equations
+
+
+
+
+
-
Figure 20.2: Antenna producing an electromagnetic wave.
The displacement current term is the crucial link between electricity and magnetism,
and leads to the existence of light as an electromagnetic wave. Let’s first look at this
feature qualitatively. Figure 20.2 shows a metal wire connected to an ac voltage source.
The battery sloshes electrons back and forth from the ground into the wire, causing
a charge-density wave as shown schematically. Note that the charge density in the
wire is changing continuously in time and space. The frequency is !0 . As a result
of charge pileups, electric field lines emerge from +ve charges and terminate on -ve
charges. This electric field is changing in space and time as well, leading to non-zero
r⇥E and @E/@t. The time-varying electric field creates a time-varying magnetic field H
because of displacement current. The time-varying magnetic field creates a time-varying
electric field by Faraday’s law. Far from the antenna, the fields detach from the source
antenna and become self-sustaining: the time-varying E creates H, and vice versa. An
electromagnetic wave is thus born; the oscillations of electric and magnetic fields move
at the speed of light c. For an antenna radiating at a frequency !0 , the wavelength is
= 2⇡c/!0 . That the wave is self-sustaining is the most fascinating feature of light. If
at some time the battery was switched o↵, the far field wave continues to propagate forever, unless it encounters charges again. That of course is how light from the most
distant galaxies and supernovae reach our antennas and telescopes, propagating through
‘light years’ in the vacuum of space, sustaining the oscillations2 .
Now let’s make this observation mathematically rigorous. Consider a region in space
with no charges (r · D = ⇢ = 0 = r · E) and no currents J = 0. Take the curl of
Faraday’s equation to obtain r ⇥ r ⇥ E = r(r · E)
r2 E =
@
@t (r ⇥ B)
=
1 @2
E,
c2 @t2
where we make use of Ampere’s law. Since in a source-free region r · E = 0, we get the
wave equations
2
Boltzmann wrote “... was it a God who wrote these lines ...” in connection to “Let there be light”.
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
(r2
(r2
1
c2
1
c2
@2
)E
@t2
@2
)B
@t2
136
= 0, Wave Equations
(20.3)
= 0.
Note that the wave equation states that the electric field and magnetic field oscillate both
in space and time. The ratio of oscillations in space (captured by r2 ) and oscillations in
p
@2
time (captured by @t
µ0 ✏0 . The
2 ) is the speed at which the wave moves, and it is c = 1/
number is exactly equal to the experimentally measured speed of light, which solidifies
the connection that light is an electromagnetic wave. We note that just like the solution
to Dirac’s equation in quantum mechanics is the electron, the solution of Maxwell’s
wave equation is light (or photons). Thus one can say that light has ‘emerged’ from the
solution of Maxwell equations.
However, we must be cautious in calling the wave equation above representing light
alone. Consider a generic wave equation (r2
1 @2
)f (r, t)
v 2 @t2
= 0. This wave moves at a
speed v. We can create a sound wave, and a water wave that moves at the same speed
v, and f (r, t) will represent distinct physical phenomena. If a cheetah runs as fast as a
car, they are not the same object!
Consider a generic vector field of the type V(r, t) = V0 ei(k·r
!t) ⌘
ˆ,
where ⌘ˆ is the direction
of the vector. This field will satisfy the wave equations 20.3 if ! = c|k|, as may be verified
by substitution. This requirement is the first constraint on the nature of electromagnetic
waves. The second stringent constraint is that the field must satisfy Gauss’s laws r·E =
0 and r · B = 0 for free space. In other words, electric and magnetic vector fields are a
special class of vector fields. Their special nature is elevated by the physical observation
that no other wave can move at the speed of light. Einstein’s theory of relativity proves
that the speed of light is absolute, and unique for electromagnetic waves: every other
kind of wave falls short of the speed of light. Thus, Maxwell’s wave equation uniquely
represents light, self-sustaining oscillating electric and magnetic fields.
20.4
Maxwell’s equations in (k, !) space
Consider an electromagnetic wave of a fixed frequency !. Since E, B / ei(k·r
make two observations. Time derivatives of Faraday and Ampere’s laws give
i!e
i!t ,
which means we can replace
@
@t
the vector operators div and curl act on the
!
eik·r
i!,
@2
@t2
!(
i!)2 ,
!t) ,
we
@
i!t
@t e
=
and so on. Similarly,
part only, giving r·(eik·r ⌘ˆ) = ik·(eik·r ⌘ˆ)
and r ⇥ (eik·r ⌘ˆ) = ik ⇥ (eik·r ⌘ˆ). These relations may be verified by straightforward
substitution. Thus, we can replace r ! ik. With these observations, Maxwell equations
in free-space become
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
k·E
k·B
137
= 0,
= 0,
(20.4)
k ⇥ E = !B,
k⇥B =
!
E.
c2
Note that we have converted Maxwell’s equations in real space and time (r, t) to ‘Fourier’
space (k, !) in this process. Just as in Fourier analysis where we decompose a function
into its spectral components, light of a particular k and corresponding frequency ! = c|k|
is spectrally pure, and forms the ‘sine’ and ‘cosine’ bases. Any mixture of light is a linear
combination of these spectrally pure components: for example white light is composed
of multiple wavelengths. Since B = r ⇥ A, we can write B = ik ⇥ A, and hence the
magnitudes are related by B 2 = k 2 A2 = ( !c )2 A2 . The energy content in a region in
space of volume ⌦ that houses electric and magnetic fields of frequency ! is given by
1
1
1
1
Hem (!) = ⌦ · [ ✏0 E 2 + µ0 H 2 ] = ⌦ · [ ✏0 E 2 + ✏0 ! 2 A2 ] .
2
2
2
2
(20.5)
If you have noticed a remarkable similarity between the expression for energy of an
electromagnetic field with that of a harmonic oscillator (from Chapter 3) Hosc =
1
2 2
2 m! x ,
p̂2
2m
+
you are in luck. In Chapter 21, this analogy will enable us to fully quantize
the electromagnetic field, resulting in a rich new insights.
Let us now investigate the properties of a spectrally pure, or ‘monochromatic’ component
of the electromagnetic wave. From equations 20.4, we note that k ? E ? B, and the
direction of k is along E ⇥ B. The simplest possibility is shown in Figure 20.3. If we align
the x axis along the electric field vector and the y axis along the magnetic field vector,
then the wave propagates along the +ve z axis, i.e., k = kẑ. The electric field vectors
lie in the x
z plane, and may be written as E(r, t) = E0 ei(kz
!t) x̂,
which is a plane
wave. For a plane wave, nothing changes along the planes perpendicular to the direction
of propagation, so the E field is the same at all x
y planes: E(x, y, z) = E(0, 0, z).
From Faraday’s law, B = k ⇥ E/!, and the magnetic field vectors B(r, t) =
lie in the y
E0 i(kz !t)
ŷ
c e
z plane. Note that here we use ! = ck and k = kz . The amplitudes of the
electric and magnetic fields are thus related by E0 = cB0 , and the relation to magnetic
q
field intensity H = B/µ0 is E0 = cµ0 H0 = µ✏00 H0 = ⌘0 H0 . Since E0 has units V/m
and H0 has units A/m, ⌘ has units of V/A or Ohms. ⌘0 is called the impedance of free
space; it has a value ⌘0 ⇡ 377⌦.
The direction of propagation of this wave is always perpendicular to the electric and
magnetic field vectors and given by the right hand rule. Since the field vectors lie on
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
138
Figure 20.3: Electromagnetic wave.
well-defined planes, this type of electromagnetic wave is called plane-polarized. In case
there was a phase di↵erence between the electric and magnetic fields, the electric and
magnetic field vectors will rotate in the x
y planes as the wave propagates, and the
wave would then be called circularly or elliptically polarized, depending upon the phase
di↵erence.
For the monochromatic wave, Maxwell’s wave equation becomes (|k|2
( !c )2 )E = 0.
For non-zero E, ! = c|k| = ck. The electromagnetic field carries energy in the +ve
z direction. The instantaneous power carried by the wave is given by the Poynting
vector S(r, t) = E ⇥ H =
E02 i(kz !t)
ẑ.
⌘0 e
The units are in Watts/m2 . Typically we are
interested in the time-averaged power density, which is given by
1
E2
⌘
S = hS(r, t)i = Re[E ⇥ H? ] = 0 ẑ = H02 ẑ,
2
2⌘
2
(20.6)
where ẑ is the direction of propagation of the wave. In later chapters, the energy carried
by a monochromatic wave will for the starting point to understand the interaction of
light with matter. In the next chapter, we will discuss how the energy carried by an
electromagnetic wave as described by Equation 20.6 actually appears not in continuous
quantities, but in quantum packets. Before we do that, we briefly discuss the classical
picture of light interacting with material media.
20.5
Maxwell’s equations in material media
How does light interact with a material medium? Running the video of the process of
the creation of light in Figure 20.2 backwards, we can say that when an electromagnetic
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
139
wave hits a metal wire, the electric field will slosh electrons in the wire back and forth
generating an ac current. That is the principle of operation of a receiving antenna. What
happens when the material does not have freely conducting electrons like a metal? For
example, in a dielectric some electrons are tightly bound to atomic nuclei (core electrons),
and others participate in forming chemical bonds with nearest neighbor atoms. The
electric field of the electromagnetic wave will deform the electron clouds that are most
‘flexible’ and ‘polarize’ them. Before the external field was applied, the centroid of the
negative charge from the electron clouds and the positive nuclei exactly coincided in
space. When the electron cloud is deformed, the centroids do not coincide any more,
and a net dipole is formed, as shown in Figure 20.4. The electric field of light primarily
interacts with electrons that are most loosely bound and deformable; protons in the
nucleus are far heavier, and held strongly in place in a solid medium. Let us give these
qualitative observations a quantitative basis.
Figure 20.4: Dielectric and Magnetic materials. Orientation of electric and magnetic
dipoles by external fields, leading to electric and magnetic susceptibilities.
The displacement vector in free space is D = ✏0 E. In the presence of a dielectric,
it has an additional contribution D = ✏0 E + P, where P is the polarization of the
dielectric. The classical picture of polarization is an electric dipole pi = qdi n̂ in every
unit cell of the solid. This dipole has zero magnitude in the absence of the external
field3 . The electric field of light stretches the electron cloud along it, forming dipoles
along itself. Thus, pi points along E. The net polarization4 is the volume-averaged
3
Except in materials that have spontaneous, piezoelectric, or ferroelectric polarization.
This classical picture of polarization is not consistent with quantum mechanics. The quantum theory
of polarization requires the concept of Berry phases, which is the subject of Chapter 54.
4
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
dipole density P =
1
V
P
V
140
pi . Based on the material properties of the dielectric, we
absorb all microscopic details into one parameter by writing
P = ✏0
where the parameter
e
e E,
(20.7)
is referred to as the electric susceptibility of the solid. With
this definition, the displacement vector becomes
D = ✏0 E + ✏0
eE
= ✏0 (1 + e )E = ✏E,
| {z }
(20.8)
✏r
where the dielectric property of the material is captured by the modified dielectric constant ✏ = ✏0 ✏r = ✏0 (1 +
e ).
The relative dielectric constant is 1 plus the electric
susceptibility of the material. Clearly the relative dielectric constant of vacuum is 1
since there are no atoms to polarize and nothing is ‘susceptible’.
In exactly the same way, if the material is magnetically polarizable, then B = µ0 (H+M),
where M is the magnetization vector. If there are tiny magnetic dipoles mi = IAn̂
formed by circular loops carrying current I in area A in the material medium (see
P
Figure 20.4), the macroscopic magnetization is given by M = V1 V mi = m H, which
leads to the relation
B = µ0 (H +
m H)
= µ0 (1 + m )H = µH,
| {z }
(20.9)
µr
With these changes, the original Maxwell equations remain the same, but now D = ✏E
and B = µH, so we make the corresponding changes ✏0 ! ✏ = ✏0 ✏r and µ0 ! µ = µ0 µr
everywhere. For example, the speed of light in a material medium then becomes v =
p1
µ✏
p
=
pc .
✏ r µr
If the material is non-magnetic, then µr = 1, and v =
pc
✏r
= nc , where n =
✏r is called the refractive index of the material. Thus light travels slower in aq
material
medium than in free space. Similarly, the wave impedance becomes ⌘0 ! ⌘ = µ✏ = ⌘n0
where the right equality holds for a non-magnetic medium.
If the material medium is conductive, or can absorb the light through electronic transitions, then the phenomena of absorption and corresponding attenuation of the light is
captured by introducing an imaginary component to the dielectric constant, ✏ ! ✏R +i✏I .
This leads to an imaginary component of the propagation vector k, which leads to attenuation. We will see in Chapters 26 and 27 how we can calculate the absorption
coefficients from quantum mechanics.
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
141
Electric and magnetic field lines may cross interfaces of di↵erent material media. Then,
the Maxwell equations provide rules for tracking the magnitudes of the tangential and
perpendicular components. These boundary conditions are given by
E1t
E2t
= 0,
H1t
H2t
D1n
= Js ⇥ n̂,
D2n = ⇢s ,
B1n
B2n
(20.10)
= 0.
In words, the boundary condition relations say that the tangential component of the
electric field Et is always continuous across an interface, but the normal component is
discontinuous if there are charges at the interface. If there are no free charges at the
interface (⇢s = 0), ✏1 E1n = ✏2 E2n , implying the normal component of the electric field
is larger in the material with a smaller dielectric constant. This feature is used in Si
MOSFETs, where much of the electric field drops across an oxide layer rather than in the
semiconductor which has a higher dielectric constant. Similarly, the normal component
of the magnetic field is always continuous across an interface, whereas the tangential
component can change if there is a surface current flowing at the interface of the two
media.
The force in Newtons on a particle of charge q in the presence of an electric and magnetic
field is given by the Lorentz equation
F = q(E + v ⇥ B).
Since the energy of the charged particle changes as W =
(20.11)
R
F · dr, the rate of change
of energy is F · v = qE · v, which is the power delivered to the charged particle by the
fields. Note that a static magnetic field cannot deliver power since v ⇥ B · v = 0. Thus
a time-independent magnetic field cannot change the energy of a charged particle. But
a time-dependent magnetic field creates an electric field, which can.
When a point charge is accelerated with acceleration a, it radiates electromagnetic waves.
Radiation travels at the speed of light. So the electric and magnetic fields at a point far
from the charge are determined by a retarded response. Using retarded potentials, or
more intuitive approaches5 , one obtains that the radiated electric field goes as
5
An intuitive picture for radiation by an accelerating charge was first given by J. J. Thomson, the
discoverer of the electron.
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
Er = (
qa
sin ✓ ˆ
)
✓,
2
4⇡✏0 c
r
142
(20.12)
expressed in spherical coordinates with the charge at the origin, and accelerating along
the x axis. The radiated magnetic field Hr curls in the ˆ direction and has a magnitude
|Er |/⌘0 . The radiated power is obtained by the Poynting vector S = E ⇥ H as
S=(
µ0 q 2 a2 sin ✓ 2
)(
) r̂,
16⇡ 2 c2
r
(20.13)
Note that unlike static charges or currents that fall as 1/r2 away from the source, the
radiated E and H fields fall as 1/r. If they didn’t, the net power radiated very far from
H
the source will go to zero since S·dA ⇠ S(r)4⇡r2 ! 0. Integrating the power over the
angular coordinates results in the famous Larmor Formula for the net electromagnetic
power in Watts radiated by an accelerating charge:
P =
20.6
µ0 q 2 a2
6⇡c
(20.14)
Need for a quantum theory of light
Classical electromagnetism contained in Maxwell’s equations can explain a remarkably
large number of experimentally observed phenomena, but not all. We discussed in the
beginning of this chapter that radiation of electromagnetic waves can be created in an
antenna, which in its most simple form is a conducting wire in which electrons are
sloshed back and forth. The collective acceleration, coupled with the Larmor formula
can explain radiation from a vast number of sources of electromagnetic radiation.
By the turn of the 20th century, improvements in spectroscopic equipment had helped
resolve what was originally thought as broadband (many frequencies !) radiation into the
purest spectral components. It was observed that di↵erent gases had di↵erent spectral
signatures. The most famous among them were the spectral features of the hydrogen
atom, then known as the hydrogen gas. There is nothing collective about hydrogen
gas, since it is not a conductor and there are not much electrons to slosh around as
a metal. The classical theory for radiation proved difficult to apply to explain the
spectral features. Classical electromagnetism could not explain the photoelectric e↵ect,
and the spectrum of blackbody radiation either. The search for an explanation led to
the quantum theory of light, which is the subject of the next chapter.
Chapter 20. Photons: Maxwell’s Equations in a Nutshell
143
Debdeep Jena: www.nd.edu/⇠djena
Chapter 21
Quantization of the Electromagnetic
Field
21.1
Introduction
In Chapter 20 we viewed the electromagnetic field as a classical object. We found that a
monochromatic wave of frequency ! = c|k| is composed of electric E = êE0 sin (k · r
and magnetic vector potential A = êA0 cos (k · r
!k t)
!k t) fields oscillating in the direction
@A
@t .
ê and linked by the dispersion !k = c|k|, and E =
The classical energy stored in
the field in a volume ⌦ for that mode is
1
1
Hem (!) = ⌦ · [ ✏0 E 2 + ✏0 ! 2 A2 ].
2
2
(21.1)
How can we represent the electromagnetic field in quantum mechanics? Is there a
‘Schrodinger equation’ for light? Let us go back to how we ‘quantize’ a classical problem.
If the classical energy is Hcl (x, p), i.e., a function of the dynamical variables (x, p), then
they satisfy Hamilton’s equations if the conditions
dx
dt
cl
= + @H
@p and
dp
dt
@Hcl
@x
=
are met.
We transition to quantum mechanics by postulating that [x̂, p̂] = i~ 6= 0, and promoting
the dynamical variables to operators. The Schrodinger equation Ĥcl (x̂, p̂)| i = E| i
then is a di↵erential equation that provides a recipe to find the quantum states | i and
the allowed energies E. This is highlighted in Figure 21.1.
Let us look at the 1D harmonic oscillator problem to verify this route of quantization.
A particle of mass m experiences a potential V (x) = 12 m! 2 x2 classically. The particle
then can have any positive energy E = Hcl (x, px ) =
p2x
2m
+ 12 m! 2 x2 , i.e., 0  E  1.
The dynamical variables x, px are seen to satisfy Hamilton’s equations
and
dpx
dt
=
m! 2 x =
@Hcl
@x ,
dx
dt
=
px
m
=
@Hcl
@px
and [x, px ] = 0. To transition to quantum mechanics, we
144
Chapter 21. Quantization of the Electromagnetic Field
145
Hamilton’s equations
Figure 21.1: Quantization of a classical Hamiltonian after Hamilton’s equations following Schrodinger and Dirac methods.
2
p̂x
postulate that [x̂, p̂x ] = i~. By solving the Schrodinger equation [ 2m
+ 12 m! 2 x̂2 ]| i =
E| i, which is a di↵erential equation, we find that the allowed energies E become
restricted to En = (n + 12 )~!, where n = 0, 1, 2, .... The particle is forbidden from being
at the bottom of the well because E0 =
~!
2 ,
the energy due to zero-point motion required
to satisfy the uncertainty principle.
But photons have no mass, so at first glance the di↵erential equation version of the
Schrodinger equation cannot be simply carried over for quantizing the electromagnetic
field! How can one then write a corresponding quantum equation for photons? The
crucial breakthrough was found by Dirac, who realized that the harmonic oscillator
problem itself need not be solved using the di↵erential equation version of Schrodinger,
but by an entirely di↵erent technique. The key to the technique is similar to taking
square roots! Note that if a, b are numbers, a2 + b2 = (a
ib)(a + ib). So for the classical
energy, we can write
p2x
1
+ m! 2 x2 = ~![
2m 2
r
m!
(x
2~
i
px )][
m!
r
m!
i
(x +
px )].
2~
m!
(21.2)
Note that the terms in the square brackets are dimensionless, and are complex conjugates. This relation is an identity as long as x and px are numbers. Does it remain so if
we transition to quantum mechanics and set [x̂, p̂x ] = i~? Let us define the dimensionless
operators
â =
r
m!
i
(x̂ +
p̂x ) and ↠=
2~
m!
r
m!
(x̂
2~
i
p̂x ) ,
m!
(21.3)
Chapter 21. Quantization of the Electromagnetic Field
146
and the corresponding space and momentum operators in terms of them:
x̂ =
r
~
(↠+ â) and p̂x = i
2m!
r
m!~ †
(â
2
â) .
(21.4)
The right hand side of the energy equation then reads ~!↠â. Using the commutator
[x̂, p̂x ] = i~, we find that
~!↠â =
m! 2 2
p̂2
i
[x̂ + 2x 2 +
(x̂p̂x
2
m !
m!
x̂p̂x )] = [
p̂2x
1
+ m! 2 x̂2 ]
2m 2
~!
,
2
(21.5)
i.e., we are o↵ by a constant factor 1/2 because of the transition from numbers to
operators. However, the order of the operators matter:
~!â↠=
m! 2 2
p̂2
[x̂ + 2x 2
2
m !
i
(x̂p̂x
m!
x̂p̂x )] = [
p̂2x
1
~!
+ m! 2 x̂2 ] +
,
2m 2
2
(21.6)
which means the commutator of the operators is (subtracting Equation 21.5 from 21.6)
ââ†
↠â = [â, ↠] = 1.
(21.7)
This also means that the harmonic oscillator Hamiltonian operator may be written
completely in terms of the operators â and ↠:
Ĥosc =
p̂2x
1
1
+ m! 2 x̂2 = ~!(↠â + ).
2m 2
2
(21.8)
Note that the mass m does not appear explicitly in the new form of the Hamiltonian:
it is of course hiding ‘under the hood’ of the operators â and ↠. Let us now investigate
a few properties of the new operators. Is there a function for which â = 0? Let’s
p
define l = ~/m! as the characteristic length scale of the harmonic oscillator problem.
By direct substitution, we find (x +
Ae
x2
2l2
~ d
m! dx )
. Normalizing the function over
d
= (x + l2 dx
) = 0 has the solution
1 < x < +1, we get A = (⇡l)
1/4 .
This is a
surprise, because the Schrodinger route had told us that the Gaussian hx|0i =
(⇡l)
1/4 e
x2 /2l2
(x) =
0 (x)
=
is exactly the eigenfunction of the ground state! So the operator â seems
to be destroying the ground state of the harmonic oscillator, because hx|â|0i = âhx|0i =
â
0 (x)
= 0. We also find that â†
0
=
1,
i.e., ↠|0i = |1i. It does not need much more
work (for example, this is done in Chapter 3) to show that in general the action of the
operators on state |ni yields
â|ni =
p
n|n
1i , ↠|ni =
p
n + 1|n + 1i , and ↠â|ni = n|ni .
(21.9)
We realize then that â is an annihilation operator, it lowers the state of the system one
Chapter 21. Quantization of the Electromagnetic Field
147
rung of the harmonic oscillator ladder. It truncates at the lower end because â|0i =
0. This is shown in Figure 21.2. Similarly, its Hermitian conjugate ↠raises it one
p
p
rung, and is a creation operator. For example, ↠|9i = 10|10i, and â|10i = 10|9i.
Together in the form ↠â = n̂ they form the particle number operator, meaning ↠â
p
counts the quantum number of the state. For example, ↠â|10i = ↠( 10|9i) = 10|10i.
Figure 21.2: Actions of the creation and annihilation operators.
Also note that we can create any state we want by starting from the ground state |0i
† n
)
and repeatedly acting on it by the creation operator: |ni = (âpn!
|0i. For example,
p
p
p
†
3
†
2
†
)
)
)
1
|3i = p(â3.2.1
|0i = p(â3.2.1
( 1)|1i = p(â
( 2.1)|2i = p3.2.1
( 3.2.1)|3i. Equation 21.8
3.2.1
then gives us the eigenvalues En when the Hamiltonian acts on eigenstates |ni
1
Ĥosc |ni = ~!(↠â + )|ni = ~!(n +
2
| {z
En
1
) |ni .
2}
(21.10)
Let us then recap Dirac’s alternate quantum theory of the harmonic oscillator problem.
If the dynamical variables (v, w) of a classical problem with energy Hcl (v, w) satisfy
Hamilton’s equations
dv
dt
=u=
@Hcl
@w
and
dw
dt
=
!2v =
@Hcl
@u ,
then we can quantize the
problem by postulating [v̂, ŵ] = i~ 6= 0. We can then find the corresponding creation and
annihilation operators ↠and â as linear combinations of v̂ and ŵ. The allowed states
| i in the quantum picture are then those that satisfy the Hamiltonian ~!(↠â+ 12 )| i =
E| i, subject to the operator relations in Equations 21.9. The potential 12 m! 2 x2 and
kinetic
p̂2x
2m
components of energy have ‘vanished’ into a natural energy scale ~! and
the dimensionless creation and annihilation operators. The problem is now solved in
its entirety without recourse to x̂ and p̂x with their classical interpretations, but to â
and ↠with the much simplified interpretation of raising or lowering. If we consider the
zero-point energy ~!/2 as a reference, then the new energy E 0 = ~!↠â = ~!n̂ scheme
has a delightful alternate interpretation. The allowed eigenenergies form an equidistant
energy ladder E 0 = n~!. One particle in state |ni has the same energy as n particles
Chapter 21. Quantization of the Electromagnetic Field
148
in state |1i. The action of a creation operator then indeed looks like creating a new
particle of the same energy, and the annihilation operator seems to remove them. This
form of quantizing the ‘field’ is called second quantization, a topic we will encounter in
greater detail later. It is the beginning of the canonical formulation of quantum field
theory. We are now ready to quantize the electromagnetic field using Dirac’s insight.
Going back to energy content of a monochromatic electromagnetic field in volume ⌦
1
1
Hem (!) = ⌦✏0 E 2 + ⌦✏0 ! 2 A2 ,
2
2
(21.11)
we note the uncanny similarity to the classical harmonic oscillator problem. If we associate the electric field to the momentum via px $
✏0 ⌦E, and the vector poten-
tial to space coordinate via x $ A, and the quantity ✏0 ⌦ to the mass of the oscil-
lator via m $ ✏0 ⌦, then the the electromagnetic field energy transforms exactly into
p2x
2m
+ 12 m! 2 x2 , the harmonic oscillator energy. They become mathematically identical.
We should now follow Dirac’s prescription and check if the classical vector field pair E
and A satisfy Hamilton’s equations, and then proceed to the quantization of the field
if they do. To do that, let us first generalize from a monochromatic wave to a general
broadband wave. If the mathematics appears involved at any point, it is useful to go
back to the monochromatic picture. But the broadband treatment of the electromagnetic
field will enable us to obtain far more insights to the problem.
21.2
Modes of the Broadband Electromagnetic Field
From Faraday’s law r ⇥ E =
we realize that r ⇥ (E +
we have in general E =
@A
@t )
r
@B
@t
and the definition of the vector potential B = r ⇥ A,
= 0. But since r ⇥ (r ) = 0 for any scalar function ,
@A
@t .
We consider an electromagnetic wave propagating
in free space with no free charge. Gauss’s law requires r · E = 0. Then,
r2 +
@
(r · A) = 0
@t
(21.12)
must be satisfied. This is where we need to choose a gauge. To see why, one can shift the
scalar potential background by a constant scalar
21.12. Furthermore, if we choose a scalar function
!
+
0
and still satisfy Equation
and form the vector field r , then
B = r ⇥ (A + r ) = r ⇥ A, because r ⇥ (r ) = 0 is a vector identity. It will prove
useful to choose the scalar field
the Coulomb
1
gauge1 .
to ensure that r · A = 0. This choice of gauge is called
From equation 21.12, we should then have r2
= 0. We also
Nobody ever reads a paper in which someone has done an experiment involving photons with the
footnote that says ‘This experiment was done in Coulomb gauge’. This quote from Sidney Coleman
highlights that the physics is independent of the gauge choice. It is chosen for mathematical convenience
Chapter 21. Quantization of the Electromagnetic Field
choose
149
= 0. This is the gauge we choose to work with. With this choice, Maxwell’s
@A
@t
equations give us B = µ0 H = r ⇥ A, and E =
we get r ⇥ (r ⇥ A) = r(r · A)
equation
r2 A
=
r2 A =
1 @2A
.
c2 @t2
uniquely. From Ampere’s law,
Since r · A = 0, we get the wave
1 @2A
.
c2 @t2
(21.13)
Solutions of this Maxwell wave equation yields allowed vector potentials A(r, t). From
there, we obtain the electric field E(r, t) =
@A(r,t)
@t
and H(r, t) =
1
µ0 r⇥A(r, t)
uniquely
because the gauge has been fixed. The classical problem is then completely solved if we
find the allowed solutions for A(r, t).
Recall the procedure for solving the time-dependent Schrodinger equation, which had
a similar nature (albeit with
@(...)
@t
on the RHS). We found the subset of solutions that
allowed a separation of the time and space variables were eigenstates: they formed a
complete set of mutually orthogonal modes. Let us follow the same prescription here,
and search for solutions of the vector potential of the form A(r, t) = A(r)·T (t). Realizing
that equation 21.13 is a compact version of three independent equations because A(r) =
x̂Ax + ŷAy + ẑAz , we find
T (t)r2 Ax =
2
1
d2 T (t)
1 d2 T (t)
2 r Ax
A
=)
c
=
= constant =
x
c2
dt2
Ax
T (t) dt2
!2
(21.14)
must hold for all three spatial components (x, y, z). Since the x dependent and the
t dependent LHS terms depend on separate variables, they can be equal only if they
are both equal to the same constant. Setting the constant to be
! 2 , we find solutions
of the time part are of the form T (t) = T (0)e±i!t . For the spatial part, we realize
that functions of the form Ax (x, y, z) = Ax (0, 0, 0)e±i(kx ·x+ky ·y+kz ·z) = Ax (0, 0, 0)e±ik·r
2
x
satisfy c2 rAA
=
x
! 2 if
! = !k = c
q
kx2 + ky2 + kz2 = c|k|.
(21.15)
Thus, from all the allowed solutions, the subset that allow the separation of time and
space variables are of the form A(r, t) = êAk e±ik·r e±i!k t where ê is the unit vector
along A(r, t), !k = c|k|, and Ak is a number characterizing the strength of the vector
potential, i.e., it is not a function of r. Since r · A = i(k · ê)Ak e±ik·r e±i!k t = 0,
ê · k = 0, implying we must have ê ? k, where k is the direction of propagation of the
wave. It represents a linearly polarized TEM wave. If k = kẑ, then ê = x̂ and ê = ŷ
are independent solutions that satisfy ê · k = 0. They represent physically di↵erent
waves. Thus, two polarizations are allowed. Let us label the polarization as s, with the
for a particular problem. For example, for electrons in a constant magnetic field, the Landau gauge
A = (0, Bx, 0) is often used. In electrodynamics, the Lorenz gauge r · A + c12 @@t = 0 is also popular.
Chapter 21. Quantization of the Electromagnetic Field
150
understanding that ês1 = x̂ and ês2 = ŷ for the wave here. The two polarization states
are orthogonal, because ês1 · ês2 = 0. The allowed solutions may then be written as
P
A(r, t) = s,k ês Ak e±ik·r e±i!k t . It is evident that this is a Fourier decomposition.
The sum over k should really be an integral if all values of k are allowed. But every
integral is the limit of a sum. Here, it is both more physical and convenient to work
with discrete k values, and then move to the continuum. To see why, imagine placing
a cubic box of side L with the origin at r = (0, 0, 0) as shown in Figure 21.3, and
requiring the vector potential to meet periodic boundary conditions2 . Then, A(x +
L, y, z, t) = A(x, y, z, t) for each allowed k requires eikx L = 1, restricting it to k =
(kx , ky , kz ) =
2⇡
L (nx , ny , nz ),
where nx = 0, ±1, ±2, ... can only take integer values. The
set of allowed k vectors then form a 3D discrete lattice as shown in Figure 21.3. Each
lattice point represents a mode of the wave. For each mode (nx , ny , nz ), there are
two allowed polarizations s1 , s2 . We can write the mode index compactly by defining
= (s, nx , ny , nz ), with the understanding
have k
=
k and !
= (s, nx , ny , nz ). In that case, we
= ! . This just means that the wave indexed by
moving in the opposite direction to that indexed by
is
with the same polarization and
same frequency.
Figure 21.3: Allowed modes for electromagnetic waves.
2
This is not a restriction, because one can take L ! 1 in the end. But it is physical, because if
indeed there was a cubic cavity such as in a laser, it would enforce the fields to go to zero which are
hard-wall boundary conditions to be discussed later.
Chapter 21. Quantization of the Electromagnetic Field
151
Now we can ask the following question: how many modes are available between frequencies ! and ! + d!, or between energies E and E + dE? This number is called the density
of states (DOS) of photons, and is easy to find. From Figure 21.3, since the dispersion
relation is ! = c|k|, the modes of constant energy lie on the surface of a sphere defined
by E = ~! = ~c|k|. Modes ! + d! and corresponding energies E + dE form a slightly
bigger sphere. The thin spherical shell between the two spheres has a volume 4⇡k 2 dk
in the 3D k space. How many modes fall inside this shell? Now because the volume
3
occupied by each mode in the k space is ( 2⇡
L) =
(2⇡)3
⌦ ,
and because each mode allows
two polarizations, the DOS is given by
4⇡k 2 dk
D! (!)d! = DE (~!)d(~!) = 2 ·
(2⇡)3
⌦
=) D! (!) =
!2⌦
!2⌦
and
D
(~!)
=
E
⇡ 2 c3
~⇡ 2 c3
(21.16)
The photon DOS in 3D increases as the square of the frequency (or energy) as shown in
Figure 21.3. What this means is as we increase the energy, there are more modes that
have the same energy. The DOS depends on the dimensionality of the problem, we have
considered 3-dimensions here. The photon DOS relations in Equation 21.16 are central
to many problems of light-matter interaction, as will be discussed later.
Now since the modes indexed by
form a complete set, the most general real solutions
of the wave equation 21.13 are
A(r, t) =
X
[ê A ei(k
·r ! t)
+ ê A? e
i(k ·r ! t)
],
(21.17)
where the second term is the complex conjugate of the first. Note that if ê = x̂ and A =
A0 /2 for a mode, the vector potential of that mode is A (r, t) = x̂A0 cos(k · r
! t).
Because we will need to find the energy content of the wave by finding E and H from
A, we choose to club together the terms in the expansion in the following fashion.
Merge the vector and spatial part by defining the mode vector M (r) = ê eik
·r .
Merge
the scalar field strength and the time dependence of the mode into the scalar function
Q (t) = A e
i! t .
With these definitions, Equation 21.17 is rewritten as
A(r, t) =
X
[Q (t)M (r) + Q? (t)M? (r)].
(21.18)
We now claim that in the above sum in Equation 21.18, the mode vectors M (r) corresponding to di↵erent
Z
are orthogonal. To see why this is true, evaluate
d3 rM (r) · M?⌫ (r) = (ê · ê⌫ )
Z
L
0
ei(kx
kx⌫ )x
dx
Z
L
0
ei(ky
ky⌫ )y
dy
Z
L
0
ei(kz
kz⌫ )z
dz.
(21.19)
Chapter 21. Quantization of the Electromagnetic Field
Because kx =
2⇡
L nx
..., the integrals are of the form
which leads to
Z
d3 rM (r) · M?⌫ (r) = L3
,⌫
152
RL
0
2⇡
dxei L (nx
=⌦
nx⌫ )x
=L
,⌫ ,
nx ,nx⌫
(21.20)
where ⌦ = L3 is the volume of the cube. Now let us find E(r, t) and H(r, t) from the
vector potential in Equation 21.18. Since
E(r, t) =
@Q (t)
@t
=
i! Q (t), we obtain
@A(r, t) X
=
i! [Q (t)M (r) Q? (t)M? (r)],
|
{z
}
@t
(21.21)
E
and since r ⇥ M (r) = ik ⇥ M (r), we get
H(r, t) =
1
i X
r ⇥ A(r, t) =
k ⇥ [Q (t)M (r)
|
{z
µ0
µ0
H
Q? (t)M? (r)] .
}
(21.22)
The total classical energy of the electromagnetic wave in the cube is given by
Hem =
Z
X
1
1
d3 r[ ✏0 E · E + µ0 H · H] = ✏0 ⌦
! 2 [Q (t)Q? (t) + Q? (t)Q (t)]. (21.23)
2
2
Showing this requires some algebra, but the fact that it must be independent of M (r)
could have been anticipated because this is the total energy in the cubic box. Further,
we have written 2Q (t)Q? (t) = Q (t)Q? (t) + Q? (t)Q (t) in a symmetric form in anticipation of it appearing in the quantized Hamiltonian, where it will necessary need to be
Hermitian. To make the connection to the harmonic oscillator problem more explicit,
motivated by Dirac’s method of taking ‘square roots’ in Equation 21.2, we define
x (t) =
p
✏0 ⌦[Q (t) + Q? (t)] and p (t) =
i!
p
✏0 ⌦[Q (t)
Q? (t)] ,
(21.24)
i
p (t)] .
!
(21.25)
and the inverse relations
Q (t) = p
1
i
1
[x (t) +
p (t)] and Q? (t) = p
[x (t)
!
4✏0 ⌦
4✏0 ⌦
With these definitions, the classical electromagnetic energy in the cubic box becomes
Hem =
X
H =
X1
1
[ p2 (t) + ! 2 x2 (t)],
2
2
where it is clear that the energy is distributed over various modes
(21.26)
and each term
in the sum in Equation 21.26 is the energy content of that mode. This is what we
mean by broadband. All energies including zero are allowed. Each term in the sum
is equivalent to the energy content of a harmonic oscillator with mass m = 1, for
Chapter 21. Quantization of the Electromagnetic Field
which x (t) = x (0) cos(! t) and p (t) =
1 2 2
2 ! x (0).
21.3
d
dt x
(t) =
153
x (0)! sin(! t), and H
,osc
=
Equation 21.26 is now ready for quantization.
Quantization of the Broadband Electromagnetic Field
Following Dirac’s prescription, we identify x (t) and p (t) as the dynamical variables
of mode
from the classical energy in Equation 21.26. Do they satisfy Hamilton’s
equations? Because Q (t) = A e
i! t ,
dx (t) p
d
= ✏0 ⌦ [Q (t)+Q? (t)] =
dt
dt
we find
i
q
@H
✏0 ⌦! 2 [Q (t) Q? (t)] = p (t) =
, (21.27)
@p
implying the first Hamilton’s equation is indeed satisfied. Similarly, we find
dp (t)
=
dt
i
q
d
✏0 ⌦! 2 [Q (t)
dt
Q? (t)] = ! 2
p
✏0 ⌦[Q (t) + Q? (t)] =
! 2 x (t) =
@H
@x
(21.28)
is also satisfied. Thus, x (t) and p (t) form a classically conjugate pair that commute. Dirac’s prescription is then to promote them from scalars to Hermitian operators, and enforce the quantization condition [x̂ (t), p̂ (t)] = x̂ (t)p̂ (t)
Because of the orthogonality of di↵erent modes
[x̂ (t), p̂⌫ (t)] = i~
,⌫ ,
p̂ (t)x̂ (t) = i~.
, we can write this compactly as
and also collect the ancillary relations [x̂ (t), x̂⌫ (t)] = 0 and
[p̂ (t), p̂⌫ (t)] = 0. Note that the operators born out of promoting the dynamical variables are explicitly time-dependent. They are in the Heisenberg picture of quantum
mechanics. In the Schrodinger picture of quantum mechanics, quantum state vectors
| (t)i are time-dependent and operators p̂ are time-independent. In the Heisenberg
picture the state vectors | i are time-independent, all the time-evolution is in the operators p̂ (t). We identify the creation and annihilation operators for mode
in analogy
to Equation 21.3. Using the classical definitions in Equations 21.24 and 21.25 we write
the quantum version of the annihilation operator as
â (t) =
r
!
i
[x̂ (t) +
p̂ (t)] =
2~
!
r
2✏0 ⌦!
Q̂ (t) =
~
r
2✏0 ⌦!
 e
~
i! t
.
(21.29)
The classically scalar quantity  has now become an operator. It had the physical
meaning of the strength of the vector potential in the classical version, and will retain this
meaning in the quantum version as the operator whose expectation value is the strength of
the vector potential. Also note the explicit time-dependence of the annihilation operator
in the Heisenberg representation is simply â (t) = â (0)e
i! t .
Let us denote â =
Chapter 21. Quantization of the Electromagnetic Field
â (0) =
q
†
2✏0 ⌦!
~
â (t) =
r
154
 . Following the same procedure, we write the creation operator as
!
[x̂ (t)
2~
i
p̂ (t)] =
!
r
2✏0 ⌦! †
Q̂ (t) =
~
r
2✏0 ⌦! † +i! t
 e
.
~
and denote q
the time-dependence of the operator by ↠(t) = ↠(0)e+i!
where ↠= 2✏0 ⌦!
† . We also have the relations
~
p
x̂ (t) = ✏0 ⌦[Q̂ (t) + Q̂† (t)] =
p
i! ✏0 ⌦[Q̂ (t)
p̂ (t) =
s
†
t
(21.30)
= ↠e+i! t ,
~ †
[â (t) + â (t)],
2!
(21.31)
r
(21.32)
Q̂ (t)] = i
~! †
[â (t)
2
â (t)].
Now since [x̂ (t), p̂ (t)] = i~, we get
[
s
~
(↠(t) + â (t)), i
2!
r
~!
(↠(t)
2
â (t))] = i~ =) [↠(t) + â (t), ↠(t)
â (t)] = 2
(21.33)
from where we obtain the commutator for the creation and annihilation operators for
each mode
to be
[â (t), ↠(t)] = 1.
Substituting Q̂ (t) =
q
~
2✏0 ⌦!
â (t) and Q̂† (t) =
Equation 21.23 we get
Ĥem = ✏0 ⌦
X
q
! 2 [Q̂ (t)Q̂† (t) + Q̂† (t)Q̂ (t)] =
(21.34)
~
2✏0 ⌦!
X1
2
↠(t) into the classical energy
~! [â (t)↠(t) + ↠(t)â (t)],
(21.35)
which with the use of the commutator in Equation 21.34 becomes
Ĥem =
X
X
1
1
~! [↠(t)â (t) + ] =
~! [↠â + ].
2
2
We have used ↠(t)â (t)) = ↠e+i! t â e
i! t
(21.36)
= ↠â , which makes it clear that the net
energy is indeed time-independent. Equation 21.36 completes the quantization of the
broadband electromagnetic field. It is telling us that each mode of the field indexed
by
acts as an independent harmonic oscillator. The total energy is the sum of the
energy of each mode. We collect the expressions for the relevant operators and fields
in this quantized version before discussing the rich physics that will emerge from the
Chapter 21. Quantization of the Electromagnetic Field
155
quantization. The vector potential from Equation 21.18 is now written as the operator
Â(r, t) =
X
s
~
ê [â ei(k
2✏0 ⌦!
·r ! t)
+ ↠e
i(k ·r ! t)
],
(21.37)
the electric field operator is
Ê(r, t) = i
X
r
~!
ê [â ei(k
2✏0 ⌦
·r ! t)
↠e
i(k ·r ! t)
],
(21.38)
and the magnetic field operator is
i X
Ĥ(r, t) =
µ0
s
~
(k ⇥ ê )[â ei(k
2✏0 ⌦!
·r ! t)
↠e
i(k ·r ! t)
].
(21.39)
If we want to revert form the Heisenberg to the Schrodinger picture for the timedependent fields Ĥ(r, t) or Ê(r, t), we simply set t = 0 in Equations 21.37 and 21.38.
One can verify that each of the four expressions in Equations 21.36, 21.37, 21.38, and
21.39 are dimensionally correct. The creation and annihilation operators are dimensionless. Note the nature of the ‘field operators’: the space and time dependence takes the
nature of a wave, whereas the creation and annihilation operators are independent of the
space and time and act on particular modes. Let us now investigate the repercussions.
21.4
Aftermath of Field Quantization
21.4.1
The physics of quantized field operators
We make a few observations and postpone a detailed discussion. Because of the noncommutation of the creation and annihilation operators, it is clear that Ê and Ĥ will
now fail to commute for a single mode. This means that one cannot simultaneously
measure the electric and magnetic fields of a mode of electromagnetic wave with arbitrary
accuracy. Even when there are absolutely no photons in the electromagnetic field, the
net energy is infinite, as is seen from Equation 21.36. This remains an unresolved
problem in quantum field theory - that is, if it is considered to be a problem. The
zero-point vibrations of electric and magnetic fields in the vacuum trigger spontaneous
emission. One school of thought is that the electromagnetic vacuum is composed of
particle-antiparticle pairs (such as electrons and positrons) and indeed stores infinite
energy. The Casimir e↵ect where two metal plates in close proximity attract each other
is explained from this picture of vacuum, though there exist alternate explanations.
Chapter 21. Quantization of the Electromagnetic Field
21.4.2
156
Occupation number formalism, Fock states
We have dutifully followed Dirac’s method for quantizing the broadband electromagnetic
field, culminating the the quantized Hamiltonian field operator in Equation 21.36. What
then are the corresponding allowed eigenstates and eigenvalues of the field? They must
satisfy the equation Ĥem | i = E| i. Recall the quantization of the single harmonic
oscillator in Equations 21.8 and 21.10, which said that if the oscillator was in the eigenstate |ni, the energy was En = ~!(n+ 12 ). The broadband field Hamiltonian in Equation
21.36 is clearly a sum of the Hamiltonians of independent modes, where each of them is
a oscillator. The energies simply add, meaning the eigenstate must be composed of a
product of the independent eigenstates of each mode. Let us label an allowed eigenstate
by |n i, and form the composite eigenstate | i = |n 1 i|n 2 i|n 2 i.... Since
P
the Hamiltonian has the number operator Ĥem =
~! (n̂ + 12 ), when it acts on the
of mode
constructed eigenstate, because n̂ 1 |n 1 i = n 1 |n 1 i it will give us the total energy Etot
Ĥem | i = [
X
X
1
1
~! (n̂ + )]|n 1 i|n 2 i|n 2 i... =
~! (n + )] |n 1 i|n 2 i|n 2 i...
2
2
|
{z
}
P
Etot =
En
(21.40)
This implies that each mode of the broadband field can only change its energy by
discrete amounts En
En = ~! . We call the ‘particle’ that carries this discrete
+1
packet of energy the photon. It is very important to note that the frequency of a mode
q
! = c|k| = 2⇡c
n2x + n2y + n2y is not really quantized, and could well be continuous for
L
an infinite volume as ⌦ = L3 ! 1. What really is quantized is the number of photons
in the mode n .
The importance of the number of photons in each mode motivates an alternate way
of writing the eigenstate: we can simply write the composite eigenstate as | i =
|n 1 , n 2 , n 3 , ...i, where n
ber of photons in mode
2,
1
is the number of photons in mode
1,
n
2
is the num-
and so on. This way of writing the state vector is called the
occupation number representation, and the state is called a Fock state after the Russian
physicist Vladimir Fock who introduced it. Since there are an infinite number of modes,
the Fock space is infinite. Each of the photon numbers can assume the integer values
n = 0, 1, 2, .... For example, an allowed state could be |01 , 52 , 303 , 04 , ...i indicating 0
photons in mode 1, 5 photons in mode 2, and so on. The state |0i = |01 , 02 , 03 , ...i is
a perfectly respectable state: it represents a quantum state of the electromagnetic field
when there are no photons in any mode. It is therefore called the vacuum state. The
action of the photon creation and annihilation operators are now
â |n1 , n2 , ..., n , ...i =
p
n |n1 , n2 , ..., n
1, ...i,
(21.41)
Chapter 21. Quantization of the Electromagnetic Field
↠|n1 , n2 , ..., n , ...i =
So we have ↠|.., 5 , ...i =
p
p
157
n + 1|n1 , n2 , ..., n + 1, ...i.
(21.42)
6|.., 6 , ...i. Similarly, we have â |0i = 0. This says that
the annihilation operator kills the vacuum state, and converts it to the number 0. The
number 0 is not the same as the vacuum state |0i! Because of the orthogonality of
di↵erent modes, we have the inner product between allowed eigenstates
hm1 , m2 , ..., m⌫ , ...|n1 , n2 , ..., n , ...i =
n1 ,m1 n2 ,m2 ... n ,m⌫ ...,
(21.43)
which suggests that any quantum state of the broadband electromagnetic field may be
written as the linear combination of the complete set of eigenstates
| i=[
1 X
1
X
n1 =0 n2 =0
...]Cn1 ,n2 ,... |n1 , n2 , ...i.
(21.44)
Here |Cn1 ,n2 ,... |2 is the probability of finding the field in a state with n1 photons in mode
1, n2 in mode 2, and so on. The allowed number of eigenstates is vast, and the Fock-state
is enormous, larger than the Hilbert space. Much of this space is unexplored. Let us
now look at a few states of interest.
21.4.3
‘Thermal’ Photons: The Bose-Einstein distribution
From Equation 21.44, consider a state | i = |11 , 32 , 03 , ...i. This is a correlated state
of the photon field, because it requires 1 photon in mode 1, and 3 photons in mode
2 simultanesouly. Consider the special case of a highly uncorrelated photon field state
P
P1
which looks like | ithermal = ( 1
n1 =0 Cn1 |n1 , 02 , 03 , ...i)+( n2 =0 Cn2 |01 , n2 , 03 , ...i)+....
Taking the inner product, we see the reason it is called uncorrelated: h | ithermal =
P1
P1
2
2
2
n1 =0 |Cn1 | +
n2 =0 |Cn2 | + ..., meaning there are specific probabilities |Cn | of
finding the state with n photons in mode
, and no photons in other modes. The
probability of finding photons in any one mode is completely independent of how many
photons there are in other modes. The probability of finding 10 photons in the 3rd mode
would be |Cn3 =10 |2 .
This particular uncorrelated state of the photon field is a very important one - because
it gave birth to quantum mechanics. It is physically realized for ‘thermal’ photons, when
the energy in the photon modes is in equilibrium with a thermal bath characterized by
a temperature T . Boltzmann statistics state that if photons could be in many energy
En
states Em , then the probability of occupation of state En is P (En ) = e kT /Z, where
P
Em
Z = m e kT is called the partition function, and k is the Boltzmann constant. Clearly
P
n P (En ) = 1 = h | i. Consider a mode of the field with energy spacing ~! . We learnt
from field quantization in section 21.3 that the allowed energy values corresponding to
Chapter 21. Quantization of the Electromagnetic Field
158
n photons occupying that mode is En = ~! (n + 12 ). The probability of n photons
being present in that mode is then
e
P (En ) = P
1
m
~!
kT
=0 e
(n + 12 )
~!
kT
(m
+ 12 )
=e
n
~!
kT
(1
e
~!
kT
),
(21.45)
And the average number of photons in that mode is
hn i =
1
X
n =0
n P (En ) =
1
X
n e
n
~!
kT
(1
e
~!
kT
n =0
) =) hn i =
1
e
~!
kT
. (21.46)
1
We have thus arrived at a very fundamental result: that the statistical thermal average
number of photons hn i = h↠â ithermal in mode ! in incoherent light in equilibrium
with a bath at temperature T is given by the boxed Equation 21.46. This is the BoseEinstein distribution function, the root of which can be tracked all the way back to the
fact that the particle (photon) creation and annihilation operators follow the relation
[â , ↠] = 1 which we got in Equation 21.34. Any bosonic particle that follows this
commutation relation will have a corresponding Bose-Einstein distribution.
21.4.4
Planck’s Law of Blackbody Radiation
Now, we can ask the question what is the energy content in the thermal photon field
between modes ! and ! + d! ? The mean energy in the mode is hn i~! , and the
number of such modes is given by the photon DOS, DE (~! ) from Equation 21.16. The
energy density per unit volume is then, skipping the subscript ,
U! (!)d! = DE (~!)d(~!) · hni~! =)
1
U! (!)
~! 3
= u! (!) = 2 3 · ~!
.
⌦
⇡ c e kT 1
(21.47)
What we did here is Max Planck’s derivation of the blackbody radiation spectrum. The
boxed part of Equation 21.47 is Planck’s law for the blackbody spectrum, plotted in
Figure 21.4 for various temperatures as a function of frequency and wavelength. Also
shown is the pre-quantum result of Rayleigh-Jeans, which is urj
! (!) ⇡
!2
kT ,
⇡ 2 c3
which
diverges for high frequencies (the ultraviolet catastrophe). The Rayleigh-Jeans law is
the high-temperature limit of Planck’s law, and is easily derived assuming kT >> ~! in
Equation 21.47. Planck’s constant does not appear in it. Planck’s hypothesis that light
came in quantized packets of energy ~! resolved the ultraviolet catastrophe and gave
birth to quantum mechanics.
What may not be clear now is: where does the temperature T come from? Maxwell’s
equations do not have temperature anywhere, so clearly the light is in equilibrium with
Chapter 21. Quantization of the Electromagnetic Field
Rayleigh-Jean
UV catastrophe
5800 K
1500
1500
Rayleigh-Jean
UV catastrophe
5800 K
5800 K
(Sun)
500
4500 K
uλ [λ] (eV.s/m3 )
uω [ω ] (eV.s/m3 )
Planck’s law
1000
5800 K
(Sun)
1000
0
1000 2000 3000 4000 5000 6000
ω (1012 rad/s)
Planck’s
law
4500 K
500
3000 K
3000 K
0
159
0
0
500 1000 1500 2000 2500 3000
λ (nm)
Figure 21.4: Planck’s law resolved the ultraviolet catastrophe
some other objects? Yes, those other objects are composed of matter - specifically atoms
and crystals that have electrons. Einstein re-derived Planck’s law of radiation using the
full picture of light-matter picture. In Chapter 25 we will consider this problem and
show that it also holds the secrets of spontaneous emission, something that is remarkably
difficult to explain without the quantization of the electromagnetic field we have achieved
in this chapter. In the next few chapters, we arm ourselves with tools for time-dependent
perturbation theory that will enable us to tackle the light-matter interaction problem
in its full glory in Chapter 25.
Chapter 25
Optical Transitions in Quantum
Mechanics
25.1
Introduction
The classical Hamiltonian for a particle of mass m and charge q in the presence of
electromagnetic fields is
Hcl =
(p
qA)2
+ V (r) +
2m
Z
1
1
d3 r[ ✏0 E · E + µ0 H · H],
2
2
(25.1)
where p is the kinetic momentum of the particle, A is the magnetic vector potential
of the electromagnetic field, V (r) is the potential in which the particle is moving, E is
the electric field and H is the magnetic field. We recognize the integral term as the
energy content of the electromagnetic wave alone. We will consider the interaction of
light primarily with electrons and neglect the interaction with protons in the nucleus.
One justification is their large mass, which makes their contribution to the Hamiltonian
small1 . Because we are considering electrons, we replace q =
e, where e = +1.6⇥10
19
C is the magnitude of the electron charge. The quantum version of the light-matter
Hamiltonian then takes the form
Ĥtot
(p̂ + eÂ)2
=
+ V (r) +
2m
Z
1
1
d3 r[ ✏0 Ê · Ê + µ0 Ĥ · Ĥ],
2
2
(25.2)
where the hats represent operators. We are now treating electrons and photons at equal
footing, by quantizing both. From our treatment of field quantization in Chapter 21, we
1
The proton’s e↵ect on the electron energy is of course captured in the V (r) term.
182
Chapter 25. Optical Transitions in Quantum Mechanics
183
use Equation 21.36 for the field energy to write
Ĥtot =
X
(p̂ + eÂ)2
1
+ V (r) +
~! (↠â + ).
2m
2
(25.3)
Square the first term taking care of the operator order, and rearrange to get
Ĥtot =
⇥ p̂2
⇤ ⇥X
1 ⇤ ⇥ e
e2 Â · Â ⇤
+ V (r) +
~! (↠â + ) +
(p̂ · Â + Â · p̂) +
.
2
2m }
|2m {z
}
|2m
{z
|
{z
}
Ĥ
Ŵ =Ĥ
matter
light-matter
Ĥlight
(25.4)
The Hamiltonian is in a suggestive form, showing the ‘matter’ part for the electron, the
‘light’ part for the electromagnetic radiation field, and the term Ŵ that couples the
two. The equation is the starting point for the study of light-matter interaction. For
certain cases, the entire Hamiltonian can be diagonalized to obtain hybrid eigenstates
of light and matter. An example is a polariton. Such quasiparticles have names ending
in ‘...ons’, just as electrons, neutrons, protons, and photons. They take a life of their
own and behave as genuine particles. Such quasiparticles will be discussed in Chapters
46 and 47. So it is very important that you search whether the Hamiltonian can be
solved exactly before applying perturbation theory. If you can do it, you are in luck,
because you have discovered a new particle in nature! Typically the diagonalization
requires transformation to a suitable basis. If we fail to diagonalize the Hamiltonian, we
do perturbation theory.
25.2
Light-matter interaction as a perturbation
The method of applying perturbation theory is familiar by now: we identify the part of
the Hamiltonian for which we know exact solutions, and treat the rest as the perturbation. In Equation 25.4, we realize that we know the exact solutions to Ĥmatter and Ĥlight
independently. Let the eigenstate solutions for them be Ĥmatter |
the electron eigenvalue, and Ĥlight |
ph i
= Eph |
ph i.
ei
= Ee |
ei
with Ee
Without the term Ŵ in Equation
25.4 there is no coupling between them because p̂ commutes with â and ↠. The annihilation and creation operators â and ↠act exclusively on the photon Fock states, and
do nothing to the p̂ and the r of electron states. That means [Ĥmatter , Ĥlight ] = 0, i.e.,
they commute. Thus, the net quantum system can exist in a simultaneous eigenstate of
both Hamitonians. It is then prudent to consider the net unperturbed Hamiltonian to be
the sum Ĥ0 = Ĥmatter + Ĥlight , with the eigenstates the products |
e i| ph i
=|
e;
ph i
=
|e; n1 , n2 , ..., n , ...i. Note that we have used the Fock-space or occupation number representation of the photon field in this notation. Thus Ĥmatter |e; n1 , n2 , ..., n , ...i =
Chapter 25. Optical Transitions in Quantum Mechanics
184
P
~! (n + 12 )|e; n1 , n2 , ..., n , ...i,
P
giving us the net unperturbed Hamiltonian Ĥ0 |e; n1 , n2 , ..., n , ...i = [Ee +
~! (n +
Ee |e; n1 , n2 , ..., n , ...i, and Ĥlight |e; n1 , n2 , ..., n , ...i =
1
2 )]|e; n1 , n2 , ..., n
, ...i.
We focus our attention on the perturbation term Ŵ now. The vector potential field
operator  in this term in Equation 25.4 is a sum over all modes , just as in Ĥlight ,
and was derived explicitly in Chapter 21 Equation 21.37. It is written as a field operator
with photon creation and annihilation operators for each mode :
Â(r, t) =
The last term is
X
 =
e2 ·Â
2m
=
X
P
s
~
ê [â ei(k
2✏0 ⌦!
·r ! t)
+ ↠e
i(k ·r ! t)
].
~
~
[ 2✏0e⌦m!
(↠â + 12 ) + 4✏0e⌦m!
(â2 e2i✓ + (↠)2 e
2
2
(25.5)
2i✓
)], where
✓ = k ·r ! t. The first term here modifies the energy dispersion of the photon part of
the unperturbed Hamiltonian. The second term causes the photon number to change by
±2, because of the â2 and (↠)2 terms, as will become clear in the following discussion.
Since the electron momentum p̂ does not appear in this interaction, the electron state
cannot change because of this interaction term. This entire interaction is non-linear, and
is small for typical field strengths. So we will neglect this entire term in the following
discussion, but the reader is encouraged to explore its consequences.
The rest of the perturbation for mode
is Ŵ =
e
2m (p̂
· Â + Â · p̂). Because of our
choice of the Coulomb gauge r · Â = 0, for any scalar function f we have p̂ · (Â f ) =
 · (p̂f ) + f (p̂ ·  ). The last term is zero because p̂ ·  =
i~(r · Â ) = 0. Thus,
p̂ ·  =  · p̂, and [p̂,  ] = 0, i.e., p̂ and  commute, and the perturbation becomes
Ŵ =
e
2m (p̂
· Â + Â · p̂) =
e
m Â
in terms of the vector potential as
e
e
Ŵ = Â · p̂ =
m
m
s
· p̂. The perturbation then is written out explicitly
~ ⇣ ik
â e
2✏0 ⌦!
·r
†
i! t
e| {z
} +â e
ik ·r
⌘
t
e|+i!
{z } ê · p̂
(25.6)
emission
absorption
where we identify the time-dependence of the oscillating perturbation. Now from our
discussion of Fermi’s golden rule for transition rates in Chapter 24, section 24.3 Equation
24.23, we identify the e
i! t
term as responsible for the electron energy change Ef =
Ei + ~! , which is an absorption process. Similarly, the e+i!
emission processes, for which Ef = Ei
t
term is responsible for
~! . Fermi’s golden rule for the transition rates
is then captured in the equations
1
2⇡
⇡
⇥ |hf |Ŵ abs |ii|2 [Ef
⌧abs
~
(Ei + ~! )],
(25.7)
Chapter 25. Optical Transitions in Quantum Mechanics
185
and
1
⌧em
⇡
2⇡
⇥ |hf |Ŵ em |ii|2 [Ef
~
(Ei
~! )].
(25.8)
Let us first find the absorption matrix element. The perturbation term is
Ŵ
abs
e
=
m
s
~
eik
2✏0 ⌦!
·r
â (ê · p̂),
(25.9)
which we sandwich between the combined electron-photon unperturbed states to obtain
the matrix element hef ; m1 , m2 , ..., m , ...|Ŵ abs |ei ; n1 , n2 , ..., n , ...i, which is
hf |Ŵ
abs
e
|ii =
m
s
~
hef ; m1 , m2 , ..., m , ...|eik
2✏0 ⌦!
·r
â (ê · p̂)|ei ; n1 , n2 , ..., n , ...i,
(25.10)
where it is clear that the light polarization vector ê is a constant for a mode, the term
eik
·r p̂
acts only on the electronic part of the state, and the annihilation operator â
acts only on the photon part of the state. The matrix element thus factors into
hf |Ŵ
abs
e
|ii =
m
s
~
ê · hef |eik ·r p̂|ei i hm1 , m2 , ..., m , ...|â |n1 , n2 , ..., n , ...i,
{z
}
2✏0 ⌦!
|
{z
}|
p
pif
n
m ,n
1
(25.11)
where we identify the momentum matrix element as pif between the initial and final
electronic states. This is a vector which includes the photon momentum through the k
term. The photon matrix element has the annihilation operator â destroying one photon
only in mode leaving all other modes unaltered, leading to h..., m , ...|â |..., n , ...i =
p
p
n h..., m , ...|..., n
1, ...i = n m ,n 1 because of the orthogonality of the photon
modes. It is clear that if the electromagnetic field was in the vacuum state, the matrix
element is zero because â |0i = 0. This is just another way of saying that there cannot
be an absorption event if there is nothing to absorb. In general (including when n = 0)
then, the matrix element is given by
hf |Ŵ abs |ii =
e
m
s
p
~
(ê · pif ) n ·
2✏0 ⌦!
m ,n
1,
(25.12)
and it enters Fermi’s golden rule as the absolute square
|hf |Ŵ |iiabs |2 =
e2
~
(ê · pif )2 n ,
m2 2✏0 ⌦!
(25.13)
which is proportional to the photon number n in mode . This proportionality is of
utmost importance. For the momentum matrix element, we should retain the photon
wavevector k when necessary - for example in the Compton e↵ect of scattering of a
Chapter 25. Optical Transitions in Quantum Mechanics
186
photon o↵ an electron. However, for many cases of interest, the photon momentum
is much smaller than that of the electron, and may be neglected. For those cases,
eik
·r
= 1 + ik · r + ... ⇡ 1, which is called the dipole approximation. With this
approximation, we can write the momentum matrix element in a modified form. We have
hef |p̂|ei i = mhef | dr̂
dt |ei i =
im
~ hef |r̂Ĥ0
Ĥ0 r̂|ef i =
im
~ (Eef
Eif )hef |r̂|ef i = im!f i rif ,
where ~!f i = Eef Eei is the change in the electron energy eigenvalue in the absorption
R
? (r)r ? (r) is the dipole matrix element which is obtained using
event, and rif = d3 r ef
ei
the known electron eigenstates. Then, the square of the absorption matrix element is
written compactly as
e2 n ~!f i
(ê · rif )2 .
2✏0 ⌦ !
2
|hf |Ŵ abs |ii|2 =
(25.14)
Following the same procedure, we compute the matrix element for emission of photons
by electrons. The perturbation term is
Ŵ
em
e
=
m
s
~
e
2✏0 ⌦!
ik ·r †
â (ê · p̂),
(25.15)
which has ↠, the creation operator for photons. The emission matrix element is
e
hf |Ŵ em |ii =
m
s
~
ê · hef |e ik ·r p̂|ei i hm1 , m2 , ..., m , ...|↠|n1 , n2 , ..., n , ...i
2✏0 ⌦!
|
{z
}|
{z
}
p
pif
n +1
m ,n +1
(25.16)
where the matrix element of the creation operator now gives something di↵erent from
p
the annihilation operator: h..., m , ...|↠|..., n , ...i = n + 1h..., m , ...|..., n + 1, ...i =
p
n + 1 m ,n +1 . This is from the stepping property of the creation operator. The
matrix element for emission is then
e
hf |Ŵ em |ii =
m
s
p
~
(ê · pif ) n + 1 ·
2✏0 ⌦!
m ,n +1 ,
(25.17)
With the dipole approximation, the square of the emission matrix element becomes
e2 (n + 1) ~!f i
(ê · rif )2 .
2✏0 ⌦
!
2
|hf |Ŵ em |ii|2 =
(25.18)
We note that the square of the emission matrix element is proportional not to the photon
number n in mode , but to the photon number plus 1: n +1. On this crucial di↵erence
between the emission and absorption processes rests the explanation of spontaneous
Chapter 25. Optical Transitions in Quantum Mechanics
187
emission2 . What it really means is that an emission process is allowed even when there
are no photons n = 0 in the electromagnetic field - which is spontaneous emission. Let
use now evaluate the transition rates to see how.
The transition rate for photon absorption is obtained by using Equation 25.14 in Fermi’s
golden rule Equation 25.7 for absorption from a single photon mode :
1
2⇡
=
|hf |Ŵ abs |ii|2 [Ef
abs (! )
~
⌧i!f
(Ei + ~! )].
(25.19)
Since there are a many modes available in the electromagnetic spectrum to cause this
transition, the net rate should be the sum of the individual rates due to each mode :
1
abs
⌧i!f
=
X
X ⇡e2 n !f2i
1
=
(ê · rif )2 [Ef
abs (! )
✏0 ⌦ !
⌧i!f
(Ei + ~! )],
(25.20)
where we have substituted Equation 25.14. The sum over the photon modes is recognized
as an integration over the photon density of states, as was discussed in Chapter 21
section 21.2. The sum over the photon density of states is computed by the recipe
R
P
⌦! 2
(...) ! d(~! )DE (~! )(...), DE (~! ) = ~⇡2 c3 , which was obtained in Equation
21.16. This summation process thus yields the net absorption rate
1
abs
⌧i!f
=
Z
2
⇡e2 n !f i
d(~!)DE (~! )
(ê · rif )2 [Ef
✏0 ⌦ !
(Ei + ~! )],
(25.21)
which simplifies to
1
abs
⌧i!f
e2
=
⇡✏0 ~c3
Z
d! · (ê · rif )2 n !f2i ! [(!f
!i )
! )].
(25.22)
Note that the macroscopic volume ⌦ has cancelled out, as it should. The Dirac delta
function ensures energy conservation by picking out only those modes ! of the photon
field that satisfy !f i = !f
!i = ! , where Eef = ~!f and Eei = ~!i are the final and
2 cos2 ✓
initial electron energies in the transition. Note that the factor (ê · rif )2 = rif
where ✓ is the angle between the dipole vector rif and the mode polarization unit vector
ê changes as we sample the photon mode space. Similarly, the number of photons in
mode
also changes. Let us label the average contribution after the integration by
n (ê · rif )2 = n(!f i )(ê · rif )2 , which we will evaluate shortly. We then obtain the net
2
Sakurai, in his book ‘Advanced Quantum Mechanics’ has to say this about Equations 25.12, 25.14,
25.17 and 25.18 of absorption and emission matrix elements: ‘This simple rule is one of the most
important results of this book’. He could not emphasize it more!
Chapter 25. Optical Transitions in Quantum Mechanics
188
absorption rate to be
1
abs
⌧i!f
=
e2 !f3i
⇡✏0
~c3
(ê · rif )2 n(!f i ) =
Here we have defined a characteristic transition rate
1
⌧0
1
n(!f i ).
⌧0
e2 !f3 i
(ê
⇡✏0 ~c3
=
exactly the same procedure, we obtain the net emission rate to be
1
em
⌧i!f
25.3
=
1
1
[n(!f i ) + 1] = [ n(!f i ) +
⌧0
⌧0 | {z }
stimulated
1
|{z}
(25.23)
· rif )2 . By following
].
(25.24)
spontaneous
The fundamental optical processes
Equations 25.23 and 25.24 capture much of the important physics of photon absorption
and emission by matter, and consequently that of light-matter interaction. The rate
of photon absorption in Equation 25.23
1
abs
⌧i!f
=
1
⌧0 n(!f i )
is proportional to the number
of photons already present in the photon field, which is reasonable. This process is
responsible for photodetectors and solar cells; we will discuss this topic in greater detail
in Chapter 26 in the context of semiconductors.
But the rate of emission
1
em
⌧i!f
=
1
⌧0 [n(!f i )
+ 1] in Equation 25.24 is proportional to
n(!f i ) + 1. This implies that an electron in an excited state in an atom is perfectly capable of emitting a photon into vacuum - and populate the vacuum state |0, 0, ...i with
n(!f i ) = 0 with a photon and cause it to transition to the state |0, ..., 0, 1 , 0, ...i. What
this means is that an electron in the excited state is capable of emitting a photon even
when there is no electromagnetic radiation incident on it! An explanation of this process of spontaneous emission is fundamentally impossible from classical electrodynamics
because when there is no radiation field, A = 0, and there is no light-matter interaction.
It is also impossible to obtain from quantum mechanics if one quantizes just the electron
part of the Hamiltonian, and not the electromagnetic field - in a semi-classical picture3 .
That is because in the absence of electromagnetic radiation, again A · p̂ = 0, and an
electron in an eigenstate is a stationary state - it simply has no reason to change its
energy. The simple explanation of spontaneous emission obtained by quantization of
the electromagnetic field A ! Â is a triumph of quantum field theory. The process of
spontaneous emission is responsible for photon emission from light bulbs and semiconductor light-emitting diodes (LEDs); we will have more to say about these applications
in Chapter 28.
3
If the electron part of the Hamiltonian is also a harmonic oscillator, however, it is indeed possible
to obtain a spontaneous emission lifetime because of a crucial cancellation of ~! The physical meaning
of the lifetime is the radiation damping of a charged harmonic oscillator.
Chapter 25. Optical Transitions in Quantum Mechanics
Since the emission rate is
1
em
⌧i!f
=
1
⌧0 [n(!f i ) + 1],
189
the emission rate is increased if there are
already photons n(!f i ) in the modes into which the electron is allowed to emit. Thus, the
presence of photons in the field stimulates the electron to emit more, which is the reason
for calling the process stimulated emission. This means if somehow one can force the
number of photons in one single mode to be significantly higher than all competing equalenergy modes, then the process can self-propagate, leading to light which is composed of
photons that are all in one mode, and coherent. This process is called light amplification
by the stimulated emission of radiation, and the device where this is realized is called
the LASER. The laser, the precious child of field quantization and the driver of quantum
electronics, has revolutionized many areas of science, engineering, and applications. We
will have more to say about the laser in Chapters 27 and 28. Figure 25.1 summarizes
our results for the fundamental processes of light-matter interaction.
Figure 25.1: Illustration of the processes of absorption, spontaneous emission, and
stimulated emission from the perspective of a quantized electromagnetic field.
For now, let us discuss the absorption and emission rates quantitatively, and get a feel for
the timescales of optical transitions. The characteristic timescale is
1
⌧0
=
e2 !f3 i
(ê
⇡✏0 ~c3
· rif )2 .
To evaluate the angular factor (ê · rif )2 , let us orient the dipole vector rif = rif ẑ along
the z axis, and consider a situation when the allowed mode polarization vectors ê
vary isotropically. Since ê · rif = rif cos ✓, we obtain the average over solid angles4 ⌦
in spherical coordinates d⌦ = sin ✓d✓d to be
(ê · rif )2 =
R
d⌦(ê · rif )2
2
R
= rif
d⌦
R 2⇡
and the characteristic rate thus becomes
R⇡
2
=0 d
✓=0 d✓ sin ✓(cos ✓)
R 2⇡
R⇡
=0 d
✓=0 d✓ sin ✓
1 2
= rif
,
3
2
e2 !f3i rif
rif 2
1
900 nm 3
=
⇡(
) ⇥(
) ⇥ (109 sec),
⌧0
3⇡✏0 ~c3
1 nm
where we have used !f i =
2⇡c
(25.25)
(25.26)
, and scaled the wavelength and the dipole matrix element
to characteristic scales. For a wavelength of 900 nm and a dipole length of 1 nm, ⌧0 ⇡ 1
em = ⌧ . The
ns. This is the spontaneous emission lifetime, because when n(!f i ) = 0, ⌧i!f
0
4
Here ⌦ is the solid angle, not the volume of the radiation field.
Chapter 25. Optical Transitions in Quantum Mechanics
190
timescale we have found is very close to what is measured for the hydrogen atom: it
takes an excited electron roughly a nanosecond to transition to the ground state and
emit a photon spontaneously. However, note that the accurate value is sensitive to the
frequency (or wavelength) of the photon, and the dipole matrix element. The nature of
the 3D photon DOS makes the rate increase as the energy of transition increases. The
stronger the dipole matrix element, the faster the transition.
25.4
Einstein’s derivation of Planck’s Law
Before we end the discussion of this chapter, let us see how Einstein used the above
picture of light-matter interaction to re-derive Planck’s law of blackbody radiation. We
had lacked clarity of the nature of the ‘matter’ and its temperature in our prior discussion
in Chapter 21, subsection 21.4.4, and had alluded that field-quantization will resolve this
confusion.
Figure 25.2: Einstein’s derivation of Planck’s blackbody radiation law treated interaction of light and matter, and invoked the concepts of absorption, and spontaneous
and stimulated emission.
Consider now that there are various kinds of atoms present in the cubic box we had
used to index the modes of the electromagnetic field, and re-sketched in Figure 25.2.
The atoms are at temperature T . Electrons in the atoms can make transitions between
various energies by either absorbing photons from, or emitting photons into the electromagnetic field, which co-exists, and is in equilibrium with the atoms. Of the various
kinds of atoms, let there be N atoms of a particular species for which the electron eigenvalues are E2 and E1 , with E2
E1 = ~!. From Boltzmann statistics for electrons,
the N atoms are distributed into N = N1 + N2 , where
N1 =
~!
e kT
~!
e kT +1
N , and N2 =
1
~!
e kT +1
N2
N1
= e
E2 E1
kT
= e
~!
kT
. So
N . The fact that the photons are in equilibrium with
the atoms means that there is no net exchange of energy between them, which is possible
if
N1 ⇥ (Rate of energy absorption) = N2 ⇥ (Rate of energy emission).
(25.27)
Chapter 25. Optical Transitions in Quantum Mechanics
Now the net absorption rate of photons is given by Equation 25.23, i.e.,
191
1
⌧ abs
=
n(!)
⌧0 ,
and each photon absorption brings in an energy ~!. With a similar process for emission
from Equation 25.24 we get
~!
⇣ e kT
e
~!
kT
⌘ ⇣ n(!)~! ⌘ ⇣
⌘ ⇣ [n(!) + 1]~! ⌘
1
1
N ⇥
=
N ⇥
=) n(!) = ~!
~!
⌧
⌧
0
0
+1
e kT + 1
e kT 1
(25.28)
where the photon occupation number n(!) in mode ! is obtained as the now-familiar
Bose-Einstein distribution function. Now the values of E2 and E1 are not specific meaning there are other levels available for single atoms, and there are many kinds of
atoms. Thus, the electron energy di↵erences for atoms span a range of ~! values. In
that case, the energy content in the photon field in equilibrium with the atoms is
u! (!)d! =
⌦! 2
~!
DE (~!)
d(~!) ⇥ [n(!)~!] =
(~d!)( ~!
),
2
3
⌦
⌦~⇡ c
e kT 1
(25.29)
which immediately yields Planck’s blackbody spectrum
u! (!) =
!3~
1
.
~!
2
3
⇡ c e kT 1
(25.30)
This is the same result as in Chapter 21 Equation 21.47, but now the interaction of
light and matter is made explicit by Einstein’s derivation. Note that though we call
this Einstein’s derivation, he actually introduced phenomenological A and B coefficients
in his 1917 paper and invoked the concepts of spontaneous and stimulated emissions
based on pure reasoning. It took at least 10 more years after his paper for the basics
of quantum mechanics and perturbation theory to be in place, and then for Dirac to
quantize the electromagnetic field. So the quantization of the electromagnetic field gave
Einstein’s heuristic derivation a rigorous quantitative underpinning, and enabled us to
treat the subject with the incredible benefit of hindsight.
Chapter 26
Optical Transitions in Bulk
Semiconductors
26.1
Introduction
In this chapter, we explore fundamental optical transitions in bulk 3-dimensional semiconductors. We approach the topic by first investigating the optical absorption spectrum. The spectrum will direct us to a rich range of electron state transitions a↵ected
by the electron-photon interaction. Then, we explore the most important of these transitions: interband (valence ! conduction) transitions in more detail. We derive expres-
sions for the equilibrium interband absorption coefficient ↵0 (~!) for bulk semiconductors. With the understanding of the physics of optical absorption, in the next chapter
we extend the concept to non-equilibrium situations to explain optical emission, optical
gain, inversion, and lasing conditions. The key to understanding these concepts is a
clear quantum-mechanical picture of optical transitions, and the role of non-equilibrium
conditions. We begin with the fundamental quantum-mechanical optical transitions by
recalling the electron-photon Hamiltonian.
26.2
Electron-photon matrix elements for semiconductors
In Chapter 21, we justified that the Hamiltonian
H=
1
(p̂ + eA)2 + V (r)
2m0
198
(26.1)
Chapter 26. Optical Transitions in Bulk Semiconductors
199
captures the interaction of electrons with light. We used this Hamiltonian in Chapter
25 to investigate the interaction of light with atoms, and explained the optical spectra
of atoms. We found that the spectra of atoms are typically very sharp because of
the discrete energy eigenvalues of electrons. Here, we apply the same idea to bulk
semiconductors, in which the energy eigenvalues form bands separated by energy gaps.
We recall that the electromagnetic wave enters the Hamiltonian via the magnetic vector
potential A, which is related to the electric field via
r ⇥ E(r, t) =
@
B(r, t)
@t
!
|{z}
@
A(r, t),
@t
E(r, t) =
B(r,t)=r⇥A(r,t)
and we work in the Coulomb gauge
r · A = 0.
(26.2)
(26.3)
This enables the vector potential A to commute with the momentum operator p̂
[p̂, A] = 0 ! p̂ · A = A · p̂,
(26.4)
which leads to the electron-photon Hamiltonian
p̂2
e
e2 A2
H=[
+ V (r)] +
A · p̂ +
.
2m
m
2m
| 0 {z
} | 0 {z } | {z0}
W
Ĥ0
(26.5)
neglect
We have written out the Hamiltonian in terms of the electron Hamiltonian Ĥ0 , and the
‘perturbation’ term seen by the electron due to the electromagnetic wave. For an electron
in a semiconductor crystal, the potential energy term in the unperturbed Hamiltonian
is the periodic crystal potential V (r + a0 ) = V (r), where a0 is a lattice constant. We
neglect the perturbation term that goes as the square of the magnetic vector potential
for ‘weak’ intensities of light. This is justified when the condition |eA| << |p| ⇠ ~⇡/a0
is met; in other words, we neglect the term
e2 A2
2m0
w.r.t.
p̂2
2m0 .
The net Hamiltonian we
retain then has the electron experiencing a perturbation
Ŵ =
e
A · p̂
m0
(26.6)
Chapter 26. Optical Transitions in Bulk Semiconductors
200
due to its interaction with light. The magnetic vector potential for an EMag wave is of
the form1
A(r, t) = êA0 cos(kop · r !t)
A0 +ikop ·r i!t
A0
= ê e
e
+ ê e ikop ·r e+i!t ,
2
2
(26.7)
(26.8)
where ! is the angular frequency of the EMag wave, ê is the unit vector along the electric
(and vector potential) field, and kop is the propagation wave vector of magnitude 2⇡/ .
The electron-photon interaction Hamiltonian is then given by
e
A · p̂
m0
= Ŵ (r)e i!t + Ŵ + (r)e+i!t
Ŵ (r, t) =
(26.10)
eA0 eikop ·r
ê · p̂
2m0
(26.11)
eA0 e ikop ·r
ê · p̂
2m0
(26.12)
Ŵ (r) =
Ŵ + (r) =
(26.9)
The electron-photon matrix elements for bulk semiconductors are thus of the form
hkc |Ŵ |kv i and hkc |Ŵ + |kv i, where the unperturbed electron states |kc i and |kv i are solutions of the unperturbed Hamiltonian Ĥ0 =
p̂2
2m0 +V
(r). But this is precisely what we dis-
cussed in chapters 16 and 17 for semiconductors. The electron states in the valence and
conduction bands in the e↵ective mass approximation are
c (r)
ikc ·r
= hr|kc i = ( epV )uc (r)
for bulk semiconductors. The term in the round bracket is a slowly varying envelope
function, and uc (r) is the periodic part of the Bloch function. The e↵ective mass approximation transforms the unperturbed electronic Hamiltonian into the much simpler
form
p̂2
2m?
p̂2
2m0
c (r)
+ V (r) !
= (E
Ec )
p̂2
2m?c ,
and the corresponding e↵ective-mass Schrodinger equation is
c (r).
We will work in this e↵ective-mass theory. The advantage of
working in the e↵ective-mass theory is that the light-matter interaction matrix elements
for electrons confined in low-dimensional structures such as quantum wells, wires, or
dots follows in a simple way from the bulk results. We will need the matrix elements
shortly to explain the absorption spectra of bulk semiconductors, which we discuss next.
1
This approach of treating the electron-photon interaction is semi-classical, justified for classical
electromagnetic fields when the number of photons is much larger than unity. It is semi-classical because
electrons receive the full quantum treatment, but we neglect the quantization of the electromagnetic
field, treating it as a classical infinite source or sink of energy, albeit in quantum packets of ~!. The
electromagnetic field will be quantized in Chapters 46 and beyond in this book.
Chapter 26. Optical Transitions in Bulk Semiconductors
26.3
201
The absorption spectrum of bulk semiconductors
We learn early of the Beer-Lambert ‘law’, which states that if light of intensity I0
is incident on a material that absorbs, the intensity will decay inside the material as
I(z) = I0 e
↵z .
Here ↵ is the absorption coefficient, in units of inverse length. Typically
the unit used for ↵ is cm
1.
Let us consider the following experiment: take a piece of
bulk semiconductor, say GaAs or GaN, and using a tunable light source, measure ↵ as a
function of the photon energy ~!. Then we obtain the absorption spectrum ↵(~!). The
absorption spectrum of most semiconductors looks like what is shown in the schematic
Figure 26.1.
101
100
10-3
10-2
Interband
10-1
Exciton
VB to donor
k
acceptor to CB
102
VB
Optical Phonons
103
acceptor
VB to acceptor
104
CB
donor
donor to CB
105
E
free carrier absorption
Absorption Coefficient (cm-1)
106
100
101
Photon Energy (eV)
Figure 26.1: Schematic absorption spectrum ↵(~!) of bulk semiconductors. The
insets depict various state transitions upon absorption of photons.
The inset of Figure 26.1 indicates the electron bandstructure of the bulk semiconductor,
including states corresponding to donor and acceptor dopants. The transitions between
electron states caused by photon absorption are indicated. The floor of the absorption
spectrum is due to intraband transitions caused by the absorption of low energy (⇠ few
meV) photons by free carriers. Transitions between dopant and band states are shown, in
addition to the below-bandgap excitonic transition. Such optical measurements provide
a sensitive experimental determination of dopant and excitonic energies with respect
Chapter 26. Optical Transitions in Bulk Semiconductors
202
to the fundamental band edge energies. Photons can excite mechanical vibrations of
the bulk semiconductor crystal by creating optical phonons: the absorption peak for
this process is rather strong, and forms the basis or Raman spectroscopy. By far, the
strongest absorption occurs for interband transitions, which is the focus of this chapter.
The absorption spectrum is quantitatively defined as
↵(~!) =
Number of photons absorbed per unit volume per second
R(~!)
=
.
Number of photons incident per unit area per second
Nph (~!)
(26.13)
In the next section we derive an expression for the denominator Nph (~!), and in the
following section we deal with the numerator R(~!).
The number of photons in light
26.4
Consider a monochromatic EMag wave of frequency ! and corresponding wavevector
kop =
2⇡
n̂. For a plane wave, the magnetic vector potential is
A(r, t) = êA0 cos(kop · r
!t),
(26.14)
from where the electric field is obtained by using
@
A(r, t)
@t
ê!A0 sin(kop · r !t),
E(r, t) =
=
(26.15)
(26.16)
and the magnetic field intensity is
H(r, t) =
=
1
r ⇥ A(r, t)
µ
1
kop ⇥ êA0 sin(kop · r
µ
Here we have used r ⇥ (...) ⌘
!t).
(26.17)
(26.18)
kop ⇥ (...) for plane waves, as described in Chapter 20.
Then, the energy carried by the plane wave per unit area per unit time is given by the
Poynting vector
Chapter 26. Optical Transitions in Bulk Semiconductors
S(r, t) = E(r, t) ⇥ H(r, t)
!A20
= kop
sin2 (kop · r !t)
µ
203
(26.19)
(26.20)
Where we use the identity ê⇥kop ⇥ê = kop . Since the frequency of typical UV-visible-IR
light is very high, we time-average the Poynting vector over a period to obtain
hS(r, t)i =
!A20
kop ,
2µ
(26.21)
and its magnitude is
S = |hS(r, t)i| =
where µ = µ0 and nr =
!A20
nr c✏0 ! 2 A20
E2
kop =
= 0
2µ
2
2⌘
(26.22)
p
µr ✏r is the refractive index of the media, in which the speed
p
of light is c/nr . Also note that E0 = !A0 , and ⌘ = µ/✏ is the field impedance. This
relation gives us a way to find the magnitude of the vector potential A0 if we know the
power carried per unit area by the electromagnetic wave. Since energy in electromagnetic
waves is carried in quantum packets (photons) of individual energy ~!, the number of
photons that cross unit area per unit time is then given by
Nph (~!) =
S
nr c✏0 ! 2 A20
E02
=
=
.
~!
2~!
2⌘~!
(26.23)
The intensity of light is proportional to the square of the electric (or magnetic) field
amplitude, and thus the number of photons is a measure of the intensity of radiation.
Equation 26.23 provides the denominator of the expression for absorption coefficient
Equation 26.13. The numerator term is discussed in the next section.
26.5
Photon absorption rate in bulk semiconductors
To find the rate of photon absorption in the bulk semiconductor, we apply Fermi’s golden
rule derived in Chapter 24. We first note that the numerator of Equation 26.13 has units
of number of photons absorbed per unit volume per second. Consider Figure 26.2. An
electron in the valence band state |ai absorbs a photon of energy ~! and transitions into
state |ai in the conduction band. Each such transition results in the annihilation of a
Chapter 26. Optical Transitions in Bulk Semiconductors
204
Figure 26.2: The absorption process of a single photon by interband transition.
photon from the EMag field. The rate at which this happens is given by Fermi’s golden
rule as
1
2⇡
=
|hb|W (r)|ai|2 [Eb
⌧a!b
~
(Ea + ~!)],
(26.24)
where hb|W (r)|ai is the perturbation matrix element, and the Dirac-delta function is a
statement of energy conservation in the process. The reverse process of photon emission
is also allowed, which results in the creation of a photon in the EMag field at the rate
1
⌧b!a
=
2⇡
|ha|W (r)|bi|2 [Ea
~
(Eb
~!)],
(26.25)
which must be subtracted because an emission process makes a negative contribution to
the number of photons absorbed. The above results are for the single states |ai and |bi.
A semiconductor crystal has a large number of states in the respective bands, so let’s
sum the rates for all possible transitions, and divide it by the net volume V to obtain
the absorption rate per unit volume (in s
gs = 2 for each k
state2 .
1 ·cm 3 ).
Add in the electron spin degeneracy
For the absorption process to occur, the lower state |ai has
to be occupied (probability = fa ) and the higher state |bi has to be empty (probability
2
Photons carry an angular momentum of ±~ depending upon their polarization. Therefore, the
conservation of angular momentum couples specific spin states. Here we are considering light with
photons of mixed polarization. Anglular momentum conservation dictates which bands can be involved
in the absorption or emission process, thus providing a way to selectively excite say the light hole, heavy
hole, or split-o↵ bands because they di↵er in their net angular momentum.
Chapter 26. Optical Transitions in Bulk Semiconductors
= (1
205
fb )), where f ’s are the occupation functions. The net absorption rate per unit
volume is then given by
Rabs =
2 X X 2⇡
|Wba |2 [Eb
V
~
ka
(Ea + ~!)]fa (1
fb ),
(26.26)
~!)]fb (1
fa ).
(26.27)
kb
and the net emission rate per unit volume is
Rem =
2 X X 2⇡
|Wab |2 [Ea
V
~
ka
(Eb
kb
The summation runs over all valence band electron states ka and conduction band electron states kb , including those that do not meet the criteria Eb
Ea = ~!. The energy
conservation requirement is automatically taken care of by the Dirac-delta functions.
We note now that the Dirac-delta functions are the same for emission and absorption
process because [+x] = [+x], |Wab | = |Wba |, and fa (1
fb )
fb (1
fa ) = fa
fb .
Therefore, the net photon absorption rate per unit volume is the di↵erence
R(~!) = Rabs
Rem =
2 X X 2⇡
|Wab |2 [Eb
V
~
ka
kb
(Ea + ~!)] ⇥ (fa
fb )
(26.28)
To evaluate the sum over states, we must first obtain an expression for the matrix
element, which is given by the electron-photon perturbation term
Wab = hb|
e
A · p̂|ai.
m0
(26.29)
At this stage, we need to know the wavefunctions corresponding to the band states |ai
and bi. In the e↵ective mass approximation, the electron wavefunction = (envelope
function) ⇥ (Bloch function). The valence band state wavefunction is then
a (r)
= C(r)uv (r) =
eikv ·r
p
V
| {z }
Envelope C(r)
and the conduction band state wavefunction is
uv (r),
| {z }
(26.30)
Bloch
eikc ·r
0
p uc (r).
(r)
=
C
(r)u
(r)
=
c
b
V
(26.31)
Chapter 26. Optical Transitions in Bulk Semiconductors
206
Since the spatial part of the vector potential for the EMag wave is A = ê A20 eikop ·r , we
obtain the matrix element Wab = hb| me0 A · p̂|ai to be
Wab
=
=
=
eA0
ê ·
2m0
Z
[e
eA0
ê ·
2m0
Z
eA0
ê ·
2m0
Z
eA0
=
ê · (
2m0
Z
? ikop ·r
p̂ a d3 r)
be
eikc ·r
eikv ·r
[ p uc (r)]? (eikop ·r p̂)[ p uv (r)]d3 r
| {z }
V
V
Z
(26.33)
operator
[e
d
ikc ·r ?
uc (r)] (eikop ·r p)[e+ikv ·r uv (r)]
|
{z
operator
}
ei(
eA0
ê ·
2m
| 0
3r
V
d3 r
V
3
d r
kc +kop +kv )·r ?
[uc (r)uv (r)](~kv )
+
V
{z
}
ikc ·r ?
uc (r)](eikop ·r )[e+ikv ·r (~kv uv (r)
eA0
=
ê ·
2m0
|
(26.32)
Z
i~ruv (r))]
(26.34)
(26.35)
(26.36)
forbidden
ei(
kc +kop +kv )·r
[u?c (r)p̂uv (r)]
{z
allowed
d3 r
V
}
(26.37)
The first term is labeled forbidden because the integral is ⇡ ~kv hkc |kv i = 0 if we neglect
the photon momentum. This is because the states belong to di↵erent bands, and are
orthogonal. The ‘allowed’ transition matrix element is:
Z
eA0
d3 r
Wab =
ê ·
ei( kc +kop +kv )·r [u?c (r)p̂uv (r)]
2m0
V
V
Z
eA0
d3 r
?
op +kv )·r
=
ê ·
e|i( kc +k
[u
(r)p̂u
(r)]
v
c
{z
}|
{z
} |{z}
2m0
N⌦
V
slow
=
eA0
ê ·
2m0
Z
periodic
ei( kc +kop +kv )·r
V
|
N
{z
slow
V
d3 r
[u?c (r)p̂uv (r)]
{z
} ⌦
}|
(26.38)
(26.39)
(26.40)
periodic
To visualize the slow and periodic parts inside the integral, refer to Figure 26.3. The
periodic term functions uc (r)p̂uc (r) repeats in every unit cell in real space of volume ⌦
of the crystal. But the slowly varying function of the form eik·r hardly changes inside a
unit cell, it changes appreciably only over many many cells. Then, we treat the slowly
varying function as constant inside a the unit cell located at Ri , but the value to change
from cell to cell. Then, the integral decomposes to
Chapter 26. Optical Transitions in Bulk Semiconductors
207
vol:
unit cells
in real space
origin
Figure 26.3: Explanation of the decomposition of the optical matrix element. Because
the matrix element consists of a product of a plane wave part that varies slowly over
unit cells, and a part that is periodic in unit cells, the product decomposes into a sum
and a cell-periodic integral.
Wab
PN
i( kc +kop +kv )·Rn Z
eA0
d3 r
n=1 e
=
ê · [
] [u?c (r)p̂uv (r)]
.
2m0
N
⌦
|
{z
} |⌦
{z
}
(26.41)
pcv
kc ,kv +kop
The sum runs over all unit cells in real space. Since
PN
n=1 e
i( kc +kop +kv )·Rn
is the sum
of the complex exponential at every unit cell site Rn , and there are a lot of them, let
us visualize this sum. Refer to Figure 26.4 to see why the sum
zero for all cases except when
The complex numbers
ei✓n
PN
n=1
ei(
kc +kop +kv )·Rn
N
is
kc + kv + kop = 0, in which case it is evidently unity.
are all on the unit circle on the complex plane, and if there
are a lot of them, they distribute uniformly around the origin. Thus, their sum tends to
have zero real and imaginary parts; further, they are divided by a large number N . But
when ✓n = 0, all the points fall at ei0 = 1 + 0i, and thus the sum is unity.
The optical matrix element is thus given by the very important result
Wab =
eA0
[
2m0
kc ,kv +kop ](ê
· pcv )
(26.42)
Chapter 26. Optical Transitions in Bulk Semiconductors
208
Figure 26.4: The sum of complex exponentials
of the form ei✓n . If the sum is over
P i✓n
a large number
of phases, the sum
is zero, unless
ne
P i✓
P i✓✓nn = 0, in which case
n
e
=
N
.
This
statement
is
captured
in
= N ✓n ,0 .
n
ne
Note that the Kronecker-delta function ensures momentum conservation because ~kv +
~kop = ~kc . With this form of the optical matrix element, the net absorption rate from
equation 26.28 becomes
R(~!) = (
eA0 2 2 X X 2⇡
)
|ê·pcv |2
2m0 V
~
kc
2
kc ,kv +kop
[Ec (kc ) (Ev (kv )+~!)]⇥[fv (kv ) fc (kc )]
kv
(26.43)
We note that the square of the Kronecker-delta function is the same as the Kroneckerdelta
2
kc ,kv +kop
=
kc ,kv +kop .
We also note at this point that |kc |, |kv | >> kop . This
is because the band-edge states occur around reciprocal lattice vectors 2⇡/a0 , and the
lattice constants a0 <<
, the wavelength of light. This is the rationale behind the
commonly stated fact: direct optical transitions are vertical in E(k) diagrams. Using the
Kronecker delta function to reduce the summation over k states assuming kc = kv = k,
and kop ⇡ 0, and taking the term ê · pcv out of the sum because it does not depend on
k, we obtain the net absorption rate per unit volume to be given by the following form,
which actually holds also for lower-dimensional structures such as quantum wells, wires,
or dots:
R(~!) =
2⇡ eA0 2
2 X
(
) |ê · pcv |2
[Ec (k)
~ 2m0
V
k
(Ev (k) + ~!)] ⇥ [fv (k)
fc (k)] (26.44)
Chapter 26. Optical Transitions in Bulk Semiconductors
209
The Equilibrium Absorption Coefficient ↵0 (~!)
26.6
We are now ready to evaluate the absorption coefficient. Using the expression for R(~!)
with the photon flux Nph (~!) from Equation 26.23, the expression for the absorption
coefficient from Equation 26.13 becomes
↵(~!) = (
⇡e2
2 X
)|ê · pcv |2
[Ec (k)
2
V
nr c✏0 m0 !
k
| {z }
(Ev (k) + ~!)] ⇥ [fv (k)
fc (k)]. (26.45)
C0
Notice that the absorption coefficient thus formulated becomes independent of the intensity of the incident photon radiation I / A20 because both Nph (~!) / A20 and
R(~!) / A20 , and the A20 factor thus cancels in the ratio. This is a signature of a
linear process - i.e., the linear absorption coefficient of the semiconductor is a property
of the semiconductor alone, and does not dependent on the excitation intensity. With
the coefficient C0 =
⇡e2
nr c✏0 m20 !
we re-write the absorption coefficient again as the follow-
ing compact expression which will be used also for lower-dimensional structures such as
quantum wells, wires, or dots in chapter 27:
↵(~!) = C0 |ê · pcv |2
2 X
[Ec (k)
V
k
(Ev (k) + ~!)] ⇥ [fv (k)
fc (k)]
(26.46)
To evaluate the k sum, we need to identify the occupation functions fv (k) and fc (k). If
the semiconductor is in equilibrium, there is one Fermi level EF , and the occupation is
given by the Fermi-Dirac function f (E) = (1 + exp [(E
EF )/kB T ])
1
at temperature
T . When the semiconductor is pushed to non-equilibrium by either optical excitation or
electrical injection of excess carriers, the occupation functions are conveniently modeled
by retaining the Fermi-Dirac form. But the single Fermi-energy EF splits to two quasiFermi levels: one for electrons in the conduction band Fc , and the other for electrons in
the valence band Fv . The occupation functions are then given by
fv (k) =
fc (k) =
1
1 + exp ( Ev (k)
kT
Fv
)
Fc
)
1
1 + exp ( Ec (k)
kT
(26.47)
(26.48)
Chapter 26. Optical Transitions in Bulk Semiconductors
210
We will consider non-equilibrium conditions in the next chapter. Under thermal equilibrium, Fc = Fv = EF , and there is only one Fermi level3 . For an undoped semiconductor, EF locates close to the middle of the bandgap. Then, as T ! 0 K, fv (k) ! 1
and fc (k) ! 0. Actually, these conditions hold fine even at room temperature for wide-
bandgap semiconductors with little error. Converting the sum to an integral using the
usual prescription, we get the equilibrium absorption coefficient to be
2
↵0 (~!) = C0 |ê · pcv |
⇥
V
2
Z
d3 k
k
(2⇡)3
V
[Ec (k)
(Ev (k) + ~!)]
(26.49)
Note that the volume term V cancels.
3D (bulk) semiconductor
Figure 26.5: Definition of various energies, and the equilibrium absorption spectrum
of bulk (3D) semiconductors ↵0 (~!).
From Figure 26.5, the optical transition can occur only for k states that satisfy
Ec (k) = Eg +
Ev (k) =
Ec (k)
3
~2 k 2
2m?e
~2 k 2
2m?h
~2 k 2
2m?r
1
1
1
= ?+ ?
?
mr
me
mh
Ev (k) = Eg +
(26.50)
(26.51)
(26.52)
(26.53)
When photons are incident on the semiconductor, it is by definition not in equilibrium and Fc 6= Fv .
But we assume that the intensity of the EMag wave is low enough to ensure that Fc ⇡ Fv ⇡ EF .
Chapter 26. Optical Transitions in Bulk Semiconductors
211
Using spherical coordinates in the 3D k space, d3 k = k 2 sin ✓dkd✓d , we convert the
~2 k 2
2
2m?r , we break up k sin ✓dkd✓d
p
?
3
1 2mr 2
EdE, the second part being
2 ( ~2 )
variables from wavevector to energy. Assuming E =
?
?
1
r
r 2 p
into three parts: k 2 · dk = ( 2m
)E · 12 ( 2m
) dEE =
~2
~2
sin ✓d✓ and the third part d . When we integrate over all k space, the angular parts
R⇡
R 2⇡
evaluate to 0 sin ✓d✓ = 2 and 0 d = 2⇡.
The absorption coefficient then becomes
2
1 2m?r 3
↵0 (~!) = C0 |ê · pcv |
·
(2⇡)
·
(2)
·
(
)2
(2⇡)3
2 ~2
2
Z
1
|0
p
dE E ⇥ [E
{z
p
(~!
~! Eg
Eg )] (26.54)
}
which reduces to
↵0 (~!) = C0 |ê · pcv |2
2
2m?r 3 p
(
) 2 ~!
(2⇡)2 ~2
|
{z
Eg
}
⇢r (~! Eg )
(26.55)
where we have defined the joint optical density of states function for bulk 3D semiconductors as
⇢r (u) =
gs
2m?r 3 p
·
(
)2 · u
(2⇡)2
~2
(26.56)
Figure 26.5 shows the equilibrium absorption spectrum ↵0 (~!) of a bulk 3D semiconductor. Using typical values of e↵ective masses and material constants, it may be verified
that the absorption coefficient for GaN for example are of the order of ⇠ 105 cm
1,
as
indicated in Fig 26.1 at the beginning of this chapter. The absorption coefficient is zero
for photon energies below the bandgap of the semiconductor, as is intuitively expected.
Instead of leaving the expression for the absorption coefficient in terms of the unphysical
parameter C0 , we use the fundamental Rydberg energy R1 =
aB =
~
m0 c↵ ,
and the fine structure constant ↵ =
e2
4⇡✏0 ~c
e2
4⇡✏0 (2aB ) ,
the Bohr radius
to write the absorption coefficient
as
4⇡ 2 ↵
↵0 (~!) = (
) · (R1 a2B ) · (
nr
2|ê·pcv |2
m0
~!
) · ⇢r (~!
Eg )
(26.57)
Chapter 26. Optical Transitions in Bulk Semiconductors
212
where we have split o↵ the dimensionless term 2|ê · pcv |2 /m0 ~!. Note that as discussed
in chapter 14, the rough order of 2|ê · pcv |2 /m0 ⇡ 20 eV for most bulk semiconductors.
The coefficients decompose to reveal a proportionality to the fine-structure constant.
The term R1 a2B has units eV.cm2 , and the reduced density of states is in 1/eV.cm3 ,
which leads to the net units cm
1.
This general form of the equilibrium absorption
coefficient holds even for low-dimensional structures with the suitable DOS ⇢r (~!
0
Eg ),
0
where Eg accounts for ground state quantization shifts in the bandgap. Many interesting
e↵ects happen when the semiconductor is pushed out of equilibrium: it is the subject of
the next chapter.
Debdeep Jena: www.nd.edu/⇠djena