De Broglie's postulate - wave-like properties of particles
Wave-particle duality
•
Electromagnetic radiation can exhibit both particle and wave properties. It appears dicult to reconcile
these facts.
The full paradox of particle-wave duality is illustrated by the following two types of
experiments. The resolution is achieved by a probabilistic interpretation of wave mechanics.
Photon absorption experiment
•
Electromagnetic waves have polarization. Polarizer is a material which is transparent to light of only
one particular polarization. Consider two slabs of a polarizer material with a preferred axis. A light
beam shined on the rst slab gets ltered, so that only a linearly polarized beam along the preferred
axis direction comes out. This beam falls on the second slab, called analyzer. If the preferred axis of
the second slab is parallel to that of the rst slab, the polarized light will pass through without any
intensity loss. If the two preferred axes are at a right angle, no light will pass through the analyzer.
•
If the two axes are at an angle
I = I0 cos2 θ,
where
I0 ∝ E02
θ,
then the intensity of the light ltered through the analyzer will be
is the incident light intensity on the analyzer. This can be understood
by decomposing the linearly polarized electric eld vector in two components, one parallel and one
perpendicular to the analyzer axis. The parallel component has magnitude
The above relationship follows from
•
E0 cos θ and passes through.
I ∝ E2.
This behavior is experimentally veried (and can be easily demonstrated using 3D glasses). But, what
happens when the incident light intensity is lowered so much that only one photon passes through the
system at a time?
•
The properties of blackbody radiation and photoelectric eect could be explained only by assuming
that photons are absorbed and emitted in their entirety. There is no way to absorb or emit a part of
a photon. Therefore, the analyzer must either let a photon through, or completely absorb it.
•
I = I0 cos2 θ for an arbitrary angle θ when the light
2
time? Since 0 ≤ cos θ ≤ 1 we cannot allow the same
How can one reproduce the experimental nding
intensity is carried by a single photon at a
thing to happen to every photon that arrives at the analyzer. Some photons must pass through, and
some must be absorbed. The most fair (general) situation, given that photons are identical, is that
absorption be a random process: the probability that a photon will pass through the analyzer is
•
cos2 θ.
The microscopic mechanism responsible for such a probabilistic dynamics is not known. However, we
have no choice but to accept the experimentally observed reality and seek the best possible way to
mathematically describe it, after which we can make new physical predictions.
Double-slit experiment
•
Consider a beam of monochromatic light incident onto a plate with two parallel slits. Light can pass
through the slits onto a screen placed behind the plate, where its intensity is measured as a function
of position. The separation
•
d
between the slits is comparable with the wavelength of the light
λ.
Classically, light propagates as a wave. The two slits act as point-like sources of coherent light, so an
interference pattern will be observable on the screen. Let us set up a coordinate system so that the
y -axis, the plate is as y = 0, the screen is at y = l and the slits are at
x = ±d/2, y = 0. The electric eld at the screen is a superposition of elds which propagate
radially (in the xy -plane) from the slits:

 s


 s
2
2
d
2π
d
2π
 + cos 

l2 + x −
l2 + x +
E(x, l) = E0 cos 
λ
2
λ
2
incident light beam is parallel to
locations
23
We can simplify this by assuming
measured light intensity
•
and by using
√
1 + x = 1 + x/2 + O(x2 ):
2 !
2 !#
d
2πl
π
d
E(x, l) = E0 cos
x−
+ cos
+
x+
2
λ
λl
2
2πl 2π
d2
2πd
= 2E0 cos
+
x2 +
cos
x
λ
λl
4
λl
α+β
cos α−β
. The oscillatory dependence
also used cos α + cos β = 2 cos
2
2
"
where we
l x, d
I∝E
2πl
π
+
λ
λl
2
on
x
of the
on the screen is a manifestation of wave interference.
Now, reduce the incident light intensity until a single photon at a time passes through the slits. If a
photon were merely a point-like particle, it would either fall on the plate and get absorbed, or pass
through one of the slits and land on the screen. We might not be able to predict which slit a successful
photon will pass through, but we could assume that the slits are identical so that a photon has an
equal probability of
1/2
to pass through either slit. The most natural expectation is that the photon
would then land on the screen in close proximity to the slit it passed through. Therefore, if we counted
over a long period of time how often photons land at dierent positions on the screen, we should nd
a probability prole with two bright maximums at
x = ±d/2
just behind the slits, and the rest of the
plate should cast a shadow.
•
Clearly, this scenario contradicts the experimentally observed oscillatory interference pattern for large
light intensities when a large number of photons impede per unit time.
But, since photons do not
interact among themselves, it must not matter what the light intensity is. Each photon must behave
the same and independently from other photons contribute to the interference pattern. Therefore, when
a single photon passes through the slits, it must land at a random location on the screen but with the
probability which reects the interference pattern. This is the same logic as in the polarizer/analyzer
experiment.
•
Therefore, the naive interpretation of particle-like motion of photons is incorrect. We can reconcile the
existence of photons with interference only if we give up the notion that the photon passes through
only one slit at a time. Even a single photon must be regarded as a wave when it goes through the
slits.
•
But, we know that photons can behave like particles. So, let us place a photon detector at each slit.
We shall imagine that these detectors are designed to not block a slit for light passage.
However,
since a measurable physical interaction of light and matter must involve photon absorption, a detector
must absorb an incoming photon and then re-emit it after registering the event. Then, if a photon
were found to pass through the slit at
x = +d/2,
the upper detector would generate a signal, while
the lower detector would not. This is necessary because a photon cannot be partially absorbed, say
a half on each detector. Analogous would be true for a photon detected to pass through the slit at
x = −d/2.
Now notice that the random photon detection events trigger re-emission of photons by
either one detector, but not both at the same time. The re-emitted photons are collected at the screen
in a manner consistent with particle dynamics, and there can be no interference patterns.
•
The above experiment is the most general manner in which one could detect a position of a photon.
We see that by learning about a photon position we destroy its wave-like properties.
Alternatively,
in order to see interference we must not have information about photon's particle properties, such as
position.
Matter waves
•
Planck's and Einstein's postulates were extremely successful in explaining the properties of electromagnetic radiation which could not be understood within the classical physics framework.
These
postulates introduce a quantization of electromagnetic eld: electromagnetic eld is emitted, absorbed
and transmitted in lumps of energy, called photons.
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•
In classical physics a wave is characterized by its frequency
λν ,
ν,
wavelength
λ,
and phase velocity
c=
but it does not have the properties of instantaneous position and velocity. A (point) particle is
characterized by its position and velocity, but clearly not by frequency, etc. The actual experimentally
veried truth about electromagnetic radiation is that it has both wave-like and particle-like properties
in dierent circumstances. Its wave-like properties are best illustrated by interference and diraction,
while the photoelectric eect and Compton scattering illustrate its particle-like properties.
•
Classical physics regards matter as something that has particle-like properties, and radiation (elds)
as something that has wave-like properties. The true properties of radiation blurred this distinction.
Also, Einstein's theory of relativity blurs the distinction between particles and radiation by stating
that energy carried by the mass of particles can be converted into radiation (E
= mc2 ),
which found
a dramatic conrmation in the discovery of nuclear reactions.
De Broglie's postulate
•
Radiation can behave like matter. Shouldn't matter be able to behave like radiation?
•
De Broglie's generalization of Einstein's postulate to matter assigns a property of frequency and wavelength to matter. A particle with energy
wavelength
•
λ = h/p,
E
and momentum
p
should have frequency
ν = E/h
and
just like a photon. Matter should behave like a wave in certain situations.
This postulate was experimentally conrmed by observing diraction of electrons, slow neutrons, atoms
and even molecules. The rst such observation was electron diraction on crystals.
•
Our everyday experience does not give us any reason to think that matter has wave properties. But,
the same is true regarding light in geometrical optics. A collection of lenses, mirrors, etc. will generally
deect and scatter light as if a light beam were nothing but a stream of particle-like photons. In order
to see wave-like phenomena it is necessary to present obstacles to light whose dimensions (size) are
comparable with the wavelength. For example, diraction on a grating will be noticeable only when
the wavelength becomes comparable with the size of openings in the grating. Momentum carried by
matter particles or objects is typically much larger than that carried by radiation, so even sub-atomic
distances are much larger than the de Broglie wavelength and do not give rise to diraction.
Probabilistic interpretation of wave dynamics
•
De Broglie's matter waves are postulated to have exactly the same kind of particle-wave duality as
radiation.
The matter wave dynamics is mathematically captured by a
wavefunction ψ(r, t)
which
plays the same role as electric eld in electrodynamics. However, the wavefunction is a complex scalar
rather than a real vector like electromagnetic eld, because matter does not have polarization.
•
ψ is a matter of mathematical convenience. We do not have any physical
ψ , it is not a measurable quantity. We only use it to calculate superposition of matter
Choosing complex numbers for
interpretation for
waves.
•
We assume that the observed matter particles are quanta of de Broglie waves, just like photons are
quanta of electromagnetic waves. We have interpreted the intensity of light
I(r, t) ∝ |E(r, t)|2
as the
number of photons passing through a beam cross-section per unit time. We saw that in order to dene
a continuously varying intensity in terms of discrete photons we had to redene intensity as the number
of photons times the probability that a photon participates in the beam. The number of photons is a
discrete (integer) number, but probability is not. We must resort to the same trick for matter waves.
•
We assign the following physical meaning to the wavefunction:
a particle would be detected at location
I ∝ |E|2
r
at time
t.
|ψ(r, t)|2
is the probability density that
Note the analogy between this denition and
of electrodynamics in the limit of a single photon. The wavefunction is meant to describe a
single particle.
•
If we attempt to measure the position of a particle, we might nd a dierent result every time we
measure, just like in the case of light in the double-slit experiment.
probability of measurement outcomes by calculating
|ψ|2 ,
We can only determine the
but we cannot make a precise prediction
of where a particle is located (unless the wavefunction is zero everywhere except at a single point in
space).
•
The wavefunction must be normalized, implying that a particle must be somewhere at any given time:
ˆ
d3 r|ψ(r, t)|2 = 1
•
The fundamental dynamics of matter must describe wave propagation of de Broglie waves. We will
construct the equation of motion later. Here, we will focus on the kinematics of de Broglie waves.
Heisenberg uncertainty principle
•
A plane matter wave is given by the wavefunction
(|k|
= 2π/λ)
and
ω = 2πν
•
p = h/λ,
and its
k
where
is a wavevector
angular frequency. This wavefunction describes a particle with completely
|ψ(r, t)|2 = |ψ0 |2 = const.
energy is E = hν .
undetermined position, since
equal to
ψ(r, t) = ψ0 ei(kr−ωt) ,
A localized particle is described by the wavefunction
However, its momentum is well dened, being
ψ(r, t) = δ(r − r0 ),
where
δ(r)
is the Dirac delta-
function which vanishes when its argument is not equal to zero. If we Fourier transform this function
ˆ
we nd:
δ(r) =
d3 k ikr
e
(2π)3
Therefore, a fully localized wavefunction is an equal-amplitude superposition of all possible plane waves.
Since a wavevector is related to particle momentum, we see that a localized particle has a completely
undetermined momentum.
•
We see that wave kinematics together with the relationships
p = h/λ
and
E = hν
has the ingredients
necessary to describe the phenomenon seen in the double-slit experiment: it is not possible to simultaneously observe the wave-like and particle-like behavior. Observing interference prohibits knowing
through which slit a particle goes through, while measurements of the latter destroy the interference
pattern.
•
In classical physics one assumes that both position and momentum can be measured at the same time
with arbitrary accuracy. This is not possible in quantum mechanics. One can be measured accurately
only at the expense of not knowing the other one accurately.
Wavepackets
•
How does a wave which minimizes the uncertainty of both momentum and position look like? Such a
wave is called a wavepacket.
•
Let us for now ignore time dependence and consider a one-dimensional wavepacket given by the wavefunction
ψ(x).
of-mass point
This wavefunction is localized mostly within a nite distance
x0 .
Its Fourier transform
around the wavevector
•
ψ(k )
is also localized in
k -space
∆x
around the center-
within a nite distance
∆k
k0 .
The Fourier and inverse-Fourier transforms can be dened as:
ˆ
dk
√
ψ(k)eikx
2π
ψ(x) =
ˆ
ψ(k) =
dx
√
ψ(x)e−ikx
2π
These transformations look practically the same, there is a certain symmetry between real-space coordinates
x
and momentum coordinates
k.
26
•
Consider a function
x0 − ∆x ≤ x ≤ x0 + ∆x. For example:
(
1 , x0 − ∆x ≤ x ≤ x0 + ∆x
1
ψ(x) = √
2∆x 0 , otherwise
ψ(x)
localized within
Its Fourier transform is:
ψ(k)
x0ˆ+∆x
1
ie−ikx0
dx
√ e−ikx = √
e−ik∆x − eik∆x
2π
2∆x
2k π∆x
x0 −∆x
r
∆x −ikx0 sin(k∆x)
e
π
k∆x
√
=
=
The function
ψ(k)
sin(α)/α
is oscillatory under an envelope which decays as
1/α
α → ∞.
as
Therefore,
is localized in momentum space. The localization interval in momentum space can be estimated
k at which α becomes large enough, say α = 1 (this is an arbitrary choice). Hence we
∆k ∝ ∆x−1 . This is a very general conclusion, independent of the precise choice of ψ(x).
as the value of
nd that
It also works in the opposite direction: we could have assumed localization in momentum space and
deduced localization in real space from the inverse Fourier transform.
•
The uncertainty in real space is inverse-proportional to the uncertainty in momentum space. We could
write this as
∆x∆k = α.
What is the smallest possible value for
α?
The answer reveals a fundamental
limit for the accuracy of simultaneous measurements of position and momentum.
•
It is not hard to guess that the symmetry between Fourier and inverse-Fourier transforms implies
α occurs when ψ(x) and ψ(k) have the same dependence on their arguments. What
ψ(x) has the Fourier transform which looks the same as ψ(x)? Such a function is Gaussian
x0 = 0 for simplicity, without loss of generality, and temporarily give up proper normalization
that the smallest
function
(we set
in order to achieve a complete formal symmetry between the function and its Fourier transform):
ψ(x) = √
Its Fourier transform is:
ψ(k) = √
1
2πδx
x2
1
e− 2δx2
2πδx
ˆ∞
−∞
x2
dx
√ eikx e− 2δx2
2π
One calculates a Gaussian integral by completing the square in the exponent:
−
1
k 2 δx2
1 2
2 2
x
+
ikx
=
−
(x
−
ikδx
)
−
2δx2
2δx2
2
and by using the formula:
ˆ∞
dxe
−ax2
r
=
π
a
−∞
Fe nd:
ψ(k) = √
where we have written
k2
k2 δx2
1 √
1
1
2πδx2 √ e− 2 = √
e− 2δk2
2π
2πδx
2πδk
δx = 1/δk . The Fourier-transform has exactly the same form as the original
k instead of x. This assures us that the obtained δkδx = 1 is the best possible.
function, but in terms of
•
As a nal touch, we note the relationship between
δx
´∞
∆x2 = h(x − x0 )2 i =
−∞
and the root-mean-square
dx (x − x0 )2 |ψ(x)|2
´∞
−∞
27
dx |ψ(x)|2
∆x
dened as:
∆x2
is the variance of the probability distribution given by
require the proper normalization of
tion. For the Gaussian with
ˆ∞
x0 = 0
ψ(x)
- the denominator automatically xes the correct normaliza-
used above we have:
ˆ∞
1
dx |ψ(x)| =
2πδx
2
√
∆x = 2 π
ˆ∞
2
x
− δx
2
dx e
1
=
2π
−∞
−∞
2
|ψ(x)|2 , written in the form which does not
ˆ∞
02
1
dx0 e−x = √
2 π
−∞
ˆ∞
√
x2 − x22
δx2
dx
e δx = 2 π ×
2πδx
2π
−∞
0
02 −x02
dx x e
√
√
δx2
π
δx2
=2 π×
×
=
2π
2
2
−∞
(after integration-by-parts the integral in the second line is reduced to the integral of
e−x
2
).
customary in literature to express uncertainty relationships in terms of the root-mean-square
√
δx/ 2
and
√
∆k = δk/ 2.
Then,
δkδx = 1
turns into
It is
∆x =
∆x∆k = 1/2.
The uncertainty principle
•
We want a wavepacket to describe a particle with energy
classical trajectory
r0 (t) = vt
where we have introduced Dirac's constant
•
v . We must
~ = h/2π .
with velocity
∆p.
∆x∆k ≥ 1/2.
p which moves along a
E = p2 /m = ~ω and p = mv = ~k0 ,
and momentum
∆x and the uncertainty of wavevector ∆k are at best
∆k by ~ in order to obtain the uncertainty of momentum
We found that the uncertainty of position
constrained by
E
have
We now multiply
We discover that:
∆x∆p ≥
~
2
This is known as the Heisenberg uncertainty principle.
It is possible to measure the position of a
particle, or momentum of a particle with an arbitrary accuracy. However, both cannot be measured
with arbitrary accuracy at the same time.
•
An analogous relationship follows by considering time dependence of a wavefunction:
Since
E = ~ω
∆t∆ω ≥ 1/2.
we nd:
∆t∆E ≥
~
2
This relationship gives as a lower bound on the amount of time
of particle energy with accuracy
∆t
needed to complete a measurement
∆E .
Matter wave kinematics
ψ(x, t) = ψ0 ei(kx−ωt) is characterized by the phase velocity vp = λν = (2π/k)(ω/2π) =
ω/k . This is the velocity vp = dx/dt at which a point of xed phase θ = kx − ωt = const. moves as
the wave oscillates. For electromagnetic radiation vp = c is the speed of light.
•
A Plane wave
•
For relativistic matter waves we have:
hE
E
vp =
=
=
ph
p
p
s
2
p2 c2 + (mc2 )2
mc
=c 1+
p
p
Using the relativistic expression for the momentum
mv
p= q
2
1 − vc2
we nd:
vp =
28
c2
v
where
v
is the particle velocity. We see that the phase velocity is dierent from particle velocity, unless
the particle moves at the speed of light, like a photon. Even worse,
vp > c
which is forbidden by the
theory of relativity.
•
The above problem is related to the notion that the wavefunction in its own right does not have a
physical meaning. Only
•
|ψ|2 is physical, but it does not betray the phase velocity (|ψ|2 = |ψ0 |2 = const.)
How do we obtain the physical velocity
ˆ
v
of a particle from its wavefunction? Consider a wavepacket
dk dω
ψ(k, ω)ei(kx−ωt) ≡
2π 2π
ψ(x, t) =
ˆ
dk
ψ(k)ei(kx−ω(k)t)
2π
Since momentum and energy are related by an equation of motion, there is a denite relationship
ω = ω(k).
between frequency and wavevector,
regarding
|ψ|2
The most likely position of the particle is calculated by
as the probability density:
ˆ
dx|ψ(x, t)|2 x
x(t) =
We now Fourier transform both the
ˆ
ˆ
x(t) =
The only dependence on
x
ψ
and
ψ∗
in the this integral:
0
0
dk dk 0 ∗
ψ (k)e−i(kx−ω(k)t) ψ(k 0 )ei(k x−ω(k )t) x
2π 2π
dx
in this integral comes from the phase factors. We can integrate out
using Dirac delta-functions:
x
by
ˆ∞
dxeikx = 2πδ(k)
−∞
ˆ∞
dxe
ikx
ˆ∞
d
x = −i
dk
−∞
dxeikx = −2πiδ 0 (k)
−∞
Therefore,
ˆ
x(t)
ˆ
0
0
dk dk 0 ∗
ψ (k)ψ(k 0 )e−i(ω(k )−ω(k))t dxei(k −k)x x
2π 2π
ˆ
0
dk dk 0 ∗
ψ (k)ψ(k 0 )e−i(ω(k )−ω(k))t (−2πi)δ 0 (k 0 − k)
2π 2π
=
=
The Dirac delta-function has the following property:
ˆ
ˆ
0
dxf (x)δ (x) = −
where
f (x)
is an arbitrary function. This follows from integration by parts and the denition of the
delta-function,
(k
0
dxf 0 (x)δ(x) = −f 0 (0)
´
dxδ(x) = 1,
and
δ(x) = 0 (∀x 6= 0).
Substituting in the integral for
v
we obtain
= k ):
x(t)
ˆ
0
dk dk 0 d ∗
ψ (k)ψ(k)e−i(ω(k )−ω(k))t 2πi
0
2π 2π dk
k0 =k
ˆ
dk ∗
dω
≈
ψ (k)ψ(k) t
2π
dk
=
where we assumed that the dominant contributions to this integral come from amplitudes
exhibit little dependence on
k.
Lastly, we assume that
dω/dk
ψ(k)
which
is practically constant in the range of
k
which gives the main contribution to the integral. The normalization condition on the wavefunctions
leaves us with:
x(t) =
29
dω
t
dk
and the particle velocity is:
v=
dx
dω
=
dt
dk
In wave dynamics this is called group velocity. For a relativistic particle we have:
v=
dE
dp 2 2
c2
d(E/~)
pc2
=
=
=
p c + (mc2 )2 =
d(p/~)
dp
dp
E
vp
which is in accord with the expression for the phase velocity obtained above. It is satisfying to see
that the measurable predictions of wave dynamics are consistent with the theory of relativity, since
one always nds
v ≤ c.
30