QUANTUM PHYSICS
DR. N. VENKATANATHAN
QUANTUM PHYSICS
CLASSICAL PHYSICS
Classical physics is the branch of physics based on the principles developed before the rise of
relativity and quantum mechanics. It deals with the set of laws describing motion of
macroscopic bodies along with their forces. It shows extremely accurate results if we look
about the

large objects

speed which is not approaching the velocity of light “c”
When the object approaches velocity of light “c” classical mechanics enhanced by the special
theory of relativity (i.e.) general relativity unifies classical mechanics and special relativity.
Quantum Physics

Deals with microscopic bodies

Speed approaches or equals to the velocity of light
LIMITATIONS OF CLASSICAL PHYSICS
 Constancy of speed of light - "... light is always propagated in empty space with a
definite velocity [speed] c which is independent of the state of motion of the
emitting body."
 Black body radiation - Absorbs and emits all type of wavelength
 Photoelectric effect - The photoelectric effect is the observation that metals emit
electrons when light incident them. Electrons emitted in this manner can be called
photoelectrons. According to classical electromagnetic theory, this effect can be
attributed to the transfer of energy from the light to an electron in the metal.
From this perspective, an alteration in either the amplitude or wavelength of
light would induce changes in the rate of emission of electrons from the metal.
Furthermore, according to this theory, a sufficiently dim light would be expected
to show a lag time between the initial shining of its light and the subsequent
emission of an electron. However, the experimental results did not correlate with
either of the two predictions made by this theory
 Compton Scattering - inelastic scattering - because the wavelength of the
scattered light is different from the incident radiation. The effect is important
because it demonstrates that light cannot be explained purely as a wave
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QUANTUM PHYSICS
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phenomenon. Thomson scattering, the classical theory of an electromagnetic wave
scattered by charged particles, cannot explain low intensity shifts in wavelength. Quantum of radiations carrying energy as well as momentum scatters off an
electron.
Dual nature of electron - According to this theory, small particles like electrons
when in motion possess wave properties.
Spectral lines emitted by the hydrogen spectrum - the emission spectrum of
atomic hydrogen are divided into a number of spectral series.
Zeeman Effect - splitting a spectral line into several components in the presence of
a static magnetic field.
Anomalous stability of atoms and molecules - In case of atoms, electrons are
revolving in the extra nuclear space. According to the electro-magnetic theory of
radiations a revolving charged particle radiates energy in the form of
electromagnetic radiations. If a revolving electron emits a radiation, then it
continuously loses its energy and it will collapse with the nucleus. This shows the
instability of atom. But experimental studies show that atoms are stable.
Wave and particle duality - Classical physics can deal with wave or particle.
Various experiment like interference, photo electric effect, electron diffraction
shows that waves sometimes act as if they were stream of particles and stream
particles sometimes act as if they were waves. But which couldn’t explained by
classical physics
WIEN’S LAW
Wavelength of the most intense radiation is inversely proportional to the absolute
temperature (0 K) of an emitting body.
𝜆𝑚𝑎𝑥 =
1
𝑇
𝜆𝑚𝑎𝑥 . 𝑇 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Rayleigh – Jeans Law
Energy radiated by a black body/second/unit area cross section between the frequency
ν and ν + dν,
𝐸𝜆 =
8𝜋𝐾𝑇
𝜆4
Failure of Wien’s & Rayleigh Jeans Law
Both laws are based on classical mechanics. According to Classical mechanics, the
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oscillating particles can have any value of frequency, hence any amount of vibration energy.
(i.e.) The exchange of energy between radiation and oscillators should be a perfectly
continuous process and energy of the oscillator must vary continuously. This leads to
Ultraviolet catastrophe.
 Ultraviolet catastrophe - According to classical physics energy density of an
electric field in vacuum is infinite due to the divergence of energy by the shorter
wavelength modes. But if we see experimentally no divergence of energy and the
total energy is finite
Black body radiation
Black body - an opaque and non-reflective body – contributed mainly to the
breakdown of classical physics - Black-body radiation has a characteristic, continuous
frequency spectrum.
Thermal radiation emitted by black body has following properties,
 The thermal radiation is independent of the material
 The thermal radiation strongly depends only on the body's temperature
 Temperature increases – radiated energy and frequency of most intense radiation
increases.
Spectral radiance – energy emitted per unit time per unit area for the frequency between
ν and ν + dν - 𝑅𝑇 (𝜈)𝑑𝜈
Since black body emits radiation with all possible frequency, the total radiance is,
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𝛼
𝑅𝑇 = ∫ 𝑅𝑇 (𝜈)𝑑𝜈
0
Black body neither reflects nor transmits any radiation, therefore appear black. On
heating a black body emits all possible wavelengths.
Distribution of energy in black body Spectrum
Black body radiation – identical to light radiation – difference is wavelength – wave
length of visible light is smaller than thermal radiation. Wavelength is function of body
temperature.
Conclusion from energy spectrum
1. Energy – not uniformly distributed – it is discrete.
2. For a given temperature – intensity of radiation increases with increase in
wavelength – maximum for particular value of wavelength – further increase in
wavelength, intensity of radiation decreases.
3. The wavelengths corresponding to the maximum energy represented by the peak
of the curve shifts toward shorter wavelength as temperature increases.
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Planck’s theory
Matter is composed of large number of oscillating particles which vibrate with
different frequency – natural frequency of the particle – its overtones.
Postulates
 Black body contains number of harmonic oscillators at molecular dimension –
vibrate with all possible frequency – natural frequency & overtones.
 Frequency of radiation of oscillator emitted = frequency of vibration of
oscillator.
 Oscillator emits discrete energy, it is in the order of E = nhν; where n = 1, 2, 3,
…
 Oscillator absorbs or emits radiation in the order of hν. (i.e.) 1hν, 2hν, 3hν, …
The energy changes – takes place – discontinuously & discrete manner – always an
integral multiple of a small, indivisible unit (or) packet of energy – called quanta.
The quantum of energy is not fixed – directly proportional to frequency. E = hν.
Theory
1. The matter is composed of a large number of oscillating particles – vibrate with
different frequency.
2. Energy of oscillating particle is quantized. E = nhν
3. Oscillator emits energy, when it moves from higher quantized state to lower quantized
state and absorbs energy when it moves from lower quantized state to higher
quantized state.
4. It neither absorbs nor emits energy as long as it remains in the same state.
5. Emission (or) absorption – energy equals to hν – called pockets of energy.
Properties of Photon
1. Photon has mass and momentum
𝐸 = 𝑚𝐶 2 ; 𝑚 =
𝐸
ℎ𝜈
ℎ𝐶
ℎ
= 2=
=
2
2
𝐶
𝐶
𝜆𝐶
𝐶𝜆
𝑝 = 𝑚𝐶 =
ℎ
ℎ
𝑥𝐶 =
𝐶𝜆
𝜆
2. Energy of a photon; E = hν
This quantum value is not same for all kinds of radiation, which means ν differs for
each case of radiation.
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3. Non electrical nature of photon – photons are electrically neutral – not deflected by
electric and magnetic fields, hence they do not ionize.
4. Speed of photon = Speed of light
5. Similar to electron, photon also existed – like electron has ‘e’ & ‘m’ – photon has ‘h’
& ‘ν’
MATTER WAVES
Light radiation possesses dual nature. Interference, diffraction & polarisation
explained by – wave nature of light. Photo electric effect, Compton Effect explained by –
Particle nature of light.
DUAL NATURE OF LIGHT RADIATION AS WAVE AND PARTICLE
In 1924, Louis de Broglie suggested, like light radiation MATTER has dual property.
Matter like protons & electrons – made up of discrete particles – should exhibit wave nature.
DE BROGLIE CONCEPT OF MATTER WAVES (OR) POSTULATES
1. Nature loves symmetry
Nature has two manifestations – matter & radiation. Since radiation possesses
dual nature, matter might also possess dual nature.
2. The close parallelism between mechanics & optics
Principle of least action in Mechanics – moving particle always chooses – path
for which action is less.
Fermat’s principle in optics – light always chooses a path for which the transit
is minimal.
3. Bohr’s theory of atomic structure
According to Bohr, the stable states of electrons in the atom are governed by
“integer rules”.
Only phenomena involving integers in physics – modes of vibration of
stretched strings – imply wave motion.
These similarities suggested to de Broglie that an electron (or) any other matter
particle must exhibit wave property in addition to particle property.
The de Broglie wavelength
Particle – undergo periodic change – give rise to matter waves (ψ)
𝜓 = 𝜓0 sin 2𝜋𝜈0 𝑡
𝜓0 = 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
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2𝜋𝜈0 = 𝜔
When t = t0,
𝜓 = 𝜓0 sin 2𝜋𝜈0 𝑡0
(1)
𝑣𝑥
𝑡− 2
𝐶
𝑡0 =
(2)
2
√1− 𝑣 2
𝐶
𝑣𝑥
𝜓 = 𝜓0 sin
2𝜋𝜈0 (𝑡− 2 )
𝐶
(3)
2
√1− 𝑣 2
𝐶
Standard equation of wav motion,
2𝜋
𝑥
𝑦 = 𝐴 sin {( 𝑇 ) (𝑡 − 𝑢)}
(4)
A- Amplitude; T – time period & u – velocity of wave in x – direction
Comparing eq 3 & 4,
2𝜋
=
𝑇
1
𝑇
2𝜋𝜈0
2
√1 − 𝑣 2
𝐶
= 𝜈=
𝜈0
(5)
2
√1− 𝑣 2
𝐶
𝑡−
𝑥
𝑣𝑥
=𝑡− 2
𝑢
𝐶
𝑥
𝑣𝑥
= 2
𝑢
𝐶
𝑢=
𝐶2
𝑣
(6)
According to Einstein’s equation,
𝐸 = 𝑚𝐶 2 ;
𝐸 = ℎ𝜈;
(7)
(8)
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ℎ𝜈 = 𝑚𝐶 2
For t = t0, ℎ𝜈0 = 𝑚0 𝐶 2
𝜈0 =
𝑚0 𝐶 2
ℎ
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Substitute in equation (5),
𝑚0 𝐶 2
𝜈=
ℎ√1 −
𝜈=
𝑚𝐶 2
ℎ
𝑣2
𝐶2
Since 𝑚 =
𝑚0
2
√1− 𝑣 2
𝐶
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
The wavelength of the matter wave (λ) = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =
𝑢
𝜈
Substitute the value of “u” and “ν”
=
𝐶2
𝑣
𝑚𝐶2
=
ℎ
𝑚𝑣
ℎ
𝜆=
ℎ
𝑚𝑣
Alternate method
𝐸 = 𝑚𝐶 2 ;
𝐸 = ℎ𝜈;
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ℎ𝜈 = 𝑚𝐶 2
ℎ
𝐶
= 𝑚𝐶 2
𝜆
ℎ
ℎ
𝜆 = 𝑚𝐶 𝜆 = 𝑝
De Broglie wave length in term of K.E,
𝐾. 𝐸 =
1
𝑚𝑣 2 =
2
𝑃2
𝑝 = 𝑚𝑣; 𝑝2 = 𝑚2 𝑣 2 ; 𝐾. 𝐸 =
2𝑚
𝑚2 𝑣 2
2𝑚
1
= 2 𝑚𝑣 2
𝑝 = √2𝑚𝐸
Substitute in λ,
𝜆=
ℎ
√2𝑚𝐸
Calculation of de Broglie wavelength of material particles like electrons:
ℎ
From the equation, 𝜆 = 𝑚𝑣
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Electrons are accelerated to various velocities, so various wave lengths – higher the electron
velocity – smaller the de Broglie wave length.
If velocity (v) is given to electron by accelerating it through a P.D. of “V”, then
Work done = Ve
This work can be converted into K.E.
1
Therefore 2 𝑚𝑣 2 = 𝑉𝑒
𝑣2 =
2𝑉𝑒
2𝑉𝑒
;𝑣 = √
𝑚
𝑚
𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚𝑣 = √2𝑚𝑉𝑒
𝜆=
ℎ
ℎ
=
𝑚𝑣
√2𝑚𝑉𝑒
If m = m0 (by ignoring relativistic consideration)
𝜆=
ℎ
√2𝑚0 𝑉𝑒
PHASE VELOCITY (OR) WAVE VELOCITY
Each particle of matter (like electron & proton etc.,) may be regarded as consisting of
a group of waves (or) a wave pocket.
Each component wave propagates with a definite velocity called wave velocity (or)
phase velocity.
GROUP VELOCITY
When a disturbance consists of number of component waves, each travelling slightly
different velocity, then the resultant velocity is called “Group Velocity”.
EXPERIMENTAL STUDY OF MATTERWAVES
Consider a free electron which starting from rest, falls through a P.D. of “V” volts.
𝜆=
=
ℎ
√2𝑒𝑉𝑚𝑒
6.625 𝑥 10−34
√2 𝑥 1.6 𝑥 10−19 𝑥 9.11 𝑥 10−31 𝑥 𝑉
DR. N. VENKATANATHAN
=
12.27
√𝑉
𝑥 10−10 𝑚 =
12.27
√𝑉
𝐴. 𝑈.
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QUANTUM PHYSICS
The wavelength of the de Broglie waves for P.D. of 100 V in vacuum is 1.227 A.U. Since the
wave length comparable to the wavelength of X - rays, de Broglie waves should be diffracted
by the crystals like X – rays.
G. P. Thomson’s Experiment
Aim: To prove electrons possess wave nature.
Construction
A beam of cathode rays are accelerated by means of induction coil in a discharge tube
(AC). Cathode rays directed towards a positive electrode (A) and made to pass through a fine
hole in the anode, so that it incident on thin gold foil (F). The thickness of the gold foil is in
the order of 10-8 m. The electrons emerging out from the gold foil will incident on the
fluorescent screen (S). After the visual examination is done photographic plate (P) can be
inserted to make permanent record. Very high vacuum should be maintained at the portion
“FP”, while small amount of air is allowed to leak into section “AC” of discharge tube. It can
be done with the help of needle valve. This is to produce beam of required voltage, which in
turn makes the discharge tube sufficiently soft.
C
A
To Vacuum Pump
F
S
P
Experimental Procedure
Beam of electrons of known velocity is made to fall on photographic plate after
traversing thin gold foil. When photographic plate is developed, symmetrical pattern of
concentric rings can be seen.
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The pattern is similar to the pattern produced by the X rays, when it is passed through
powdered crystal.
To prove electrons got diffracted.
In order to confirm that the cathode rays are involved in the formation diffraction
pattern, magnetic field is applied after the gold foil. Entire pattern in the screen get shifted
confirms electrons get diffracted and secondary X rays are not produced.
Proves wave nature of electrons
Removing the gold foil “F”, the pattern get disappeared, clearly shows the presence of
the foil is essential.
If the electrons are in corpuscles nature, it might be scattered after the film “F”. Thus
production diffraction pattern demonstrates electrons are behaving like a wave also.
Verification of de Broglie wave
If high voltage in the order of 50000 V is applied to accelerate electrons, then very
high speed electrons will behave as a wave.
𝜆=
12.27
√𝑉
The relativistic mass correction should be done for very large values of “V”.
HEISENBERG’S UNCERTAINITY PRIN CIPLE
It is impossible to determine precisely and simultaneously the values of both the
numbers of a pair of physical variables, which describe the motion of an atomic system. For
example, the position and momentum of an electron cannot be determined accurately.
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∆𝑥. ∆𝑝 =
ℎ
2𝜋
; ∆𝐸. ∆𝑡 =
ℎ
2𝜋
Such pair of variables is called as canonically conjugate variables*. It is not the
statement about inaccuracy of measurement. It is arises from wave property - integral part of
quantum mechanical description.
*Pair of variables mathematically defined in such a way that they become Fourier
transforms# duals of one another.
#Mathematical operation – expresses a mathematical function of time as a function of
frequency. Function of time often called as the time domain and function of frequency often
called as the frequency domain.
THE WAVE FUNCTION
Wave on a string can be described by the displacement y(x, t). In case of sound wave
in air, pressure varies in space and time P(x, t). For EM waves, E and B are varying in space
and time.
To characterise de Broglie waves associated with a material particle, which require a
quantity that varies in space and time. That quantity is called as “Wave Function”,
designated by “ψ”, which is function of co-ordinates (x, y, z) and time “t”.
The displacement can be positive (or) negative. (i.e.) ψ(x, y, z, t) can be positive 0r
negative. But according to uncertainty principle, finding quantum particle is a probability
function, hence probability cannot be negative. (i.e.) wave function cannot be negative.
Ψ (x, y, z, t) is not a direct measure of presence of particle, since it is not observable.
Particle having a well-defined momentum will have infinite uncertainty in co-ordinate
(position).
Assume a free particle, which is moving along x – axis, having well defined
momentum. Then,
𝜓(𝑥, 𝑡) = 𝐴𝑒 [𝑖(𝑘𝑥−𝜔𝑡)]
𝑘=
2𝜋
; 𝜔 = 2𝜋𝜈
𝜆
For all three coordinates,
𝜓(𝑟, 𝑡) = 𝐴𝑒 [𝑖[𝑘𝑟−𝜔𝑡]]
Such a particle is a non –localised particle*. For a localized particle, super pose the
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waves and the wave function is represented by wave packet.
*cannot represent a particle whose wave function is non – zero in an limited region of space.
TIME DEPENDENT SCHRӦDINGER EQUATION
In classical physics the particle and waves are described by the equations of motion,
similarly in quantum mechanics, particle described by wave function 𝜓(𝑟, 𝑡) and the equation
of motion is given by Schrödinger.
𝜆=
ℎ
ℎ
; 𝐸 = ℎ𝜈 = ђ𝜔 𝑆𝑖𝑛𝑐𝑒 ђ =
; 𝜔 = 2𝜋𝜈
𝑝
2𝜋
Free particle wave equation in 1D
Free particle which is moving along x – axis having definite momentum can described
by an infinite plane wave,
𝜓(𝑥, 𝑡) = 𝐴𝑒 [𝑖(𝑘𝑥−𝜔𝑡)] --------- (1)
𝑘=
2𝜋
2𝜋𝑝
𝑝
𝐸
=
= 𝑎𝑛𝑑 𝜔 =
𝜆
ℎ
ђ
ђ
Substitute these values in equation 1,
𝜓 (𝑥, 𝑡) = 𝐴𝑒
𝑖
ђ
[ (𝑝𝑥−𝐸𝑡)]
------------ (2)
Differentiate with respect to x,
𝑖
𝜕𝜓
𝑖𝑝
[ (𝑝𝑥−𝐸𝑡)]
= ( ) 𝐴𝑒 ђ
𝜕𝑥
ђ
𝜕𝜓
𝑖𝑝
=
𝜓
𝜕𝑥
ђ
Multiplying by –iђ on both sides,
−𝑖ђ
𝜕𝜓
= 𝑝𝜓
𝜕𝑥
On differentiating,
𝑖
𝜕 2𝜓
𝑖𝑝 2
[ (𝑝𝑥−𝐸𝑡)]
ђ
=
(
)
𝐴
𝑒
𝜕𝑥 2
ђ
= −
DR. N. VENKATANATHAN
𝑝2
𝜓
ђ2
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QUANTUM PHYSICS
Multiplying on both sides by –
ђ2 𝜕2 𝜓
− 2𝑚
𝜕𝑥 2
=
𝑝2
2𝑚
ђ2
2𝑚
𝜓 -------------- (3)
Differentiating equation (2) with respect to ‘t’
𝑖
𝜕𝜓
𝑖𝐸
[ (𝑝𝑥−𝐸𝑡)]
= −[ 𝐴 𝑒 ђ
]
𝜕𝑡
ђ
=−
𝑖𝐸
𝜓
ђ
Multiplying iђ on both sides,
𝜕𝜓
𝑖ђ 𝜕𝑡 = 𝐸𝜓 --------------------- (4)
R.H.S of the equation (3) & (4) are equal, since for a classical particle,
𝑃2
𝐸=
2𝑚
Hence equating L.H.S,
ђ2 𝜕2 𝜓
𝜕𝜓
𝑖ђ 𝜕𝑡 = − 2𝑚
𝜕𝑥 2
----------------- (5)
The above equation is called as one dimensional time dependent Schrödinger equation.
For three dimensional cases the equation becomes,
𝜕𝜓
ђ2 𝜕 2 𝜓 𝜕 2 𝜓 𝜕 2 𝜓
ђ2 2
𝑖ђ
= −
(
+
+
)= −
∇ 𝜓
𝜕𝑡
2𝑚 𝜕𝑥 2
𝜕𝑦 2
𝜕𝑧 2
2𝑚
∇2 𝜓 =
𝜕2
𝜕2
𝜕2
+
+
𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2
The General wave Equation
The energy ‘E’ is called as Hamiltonian (total energy of the system) of a particle in a
potential V(r, t)
𝐸=
𝑝2
2𝑚
+ 𝑉(𝑟, 𝑡)----------- (7)
Substituting E, p and V(r,t),
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𝑖ђ
𝜕𝜓
𝜕𝑡
= − [−
ђ2
2𝑚
∇2 + 𝑉(𝑟, 𝑡)] 𝜓(𝑟, 𝑡) ----------------- (8)
Time – dependent Schrödinger equation for a particle moving in a potential V (r, t).
TIME – INDEPENDENT SCHRÖDINGER EQUATION
Potential ‘V’ is a function of co-ordinates only. We can separate the variables and can
obtain two equations.
One equation depending on variable ‘t’ only and the other equation on variable ‘r’,
𝜓(𝑟, 𝑡) = 𝜓(𝑟)𝜙(𝑡)----------- (9)
Ψ(r) – function of space coordinate & ϕ (t) function of time
Substitute equation (9) in (8),
𝜓(𝑟). 𝑖ђ
𝑑𝜙(𝑡)
𝑑𝑡
ђ2
= 𝜙(𝑡) [− 2𝑚 ∇2 + 𝑉(𝑟)] 𝜓(𝑟)
Dividing both sides by ψϕ,
1 𝑑𝜙
𝑖ђ 𝜙 𝑑𝑡 =
1
ђ2
[− 2𝑚 ∇2 + 𝑉(𝑟)] 𝜓(𝑟)
𝜓
⇓
Function of t only
⇓
Function of r only
This is possible only if both sides are equal to a constant “E”.
𝑖ђ
𝑑𝜙(𝑡)
= 𝐸𝜙(𝑡)
𝑑𝑡
ђ2 2
−[
∇ + 𝑉(𝑟)] 𝜓(𝑟) = 𝐸𝜓(𝑟)
2𝑚
This called time independent Schrödinger equation.
HAMILTONIAN
−[
ђ2 2
∇ + 𝑉(𝑟)] 𝜓(𝑟) = 𝐸𝜓(𝑟)
2𝑚
ђ2
− [2𝑚 ∇2 + 𝑉(𝑟)] --------- Hamiltonian (H)
𝐻𝜓(𝑟) = 𝐸𝜓(𝑟)
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
The function cannot be cancelled out of this equation, since “H” is not a simple scalar
multiplier, while “E” is the value of energy.
(Operator H) acting on function ψ = (total energy) multiplying with function ψ.
PROBABILITY INTERPRETATION OF THE WAVE FUNCTION
Wave function ψ(r, t) is not observable, but ψ must in some way be the index of
presence of the particle at (r, t).
Statistical interpretation is now universally accepted.
𝜓 ∗ 𝜓 = |𝜓|2
Probability density P(r, t),
𝑃(𝑟, 𝑡) = |𝜓(𝑟, 𝑡)|2
Probability of finding the system in volume element dτ, |𝜓|2 𝑑𝜏
This quantity when integrated from –α to +α, then the total probability is equal to unity.
𝛼
2
∫|𝜓| 𝑑𝜏 = 1
−𝛼
If the probability of finding a particle in some region increases with time, the
probability of finding it outside this region decreases by the same amount.
Physical significance of the wave function ψ
1. The wave function ψ is a complex quantity, cannot be measured – not
observable.
2. The wave function relates the particle nature and wave nature of a matter
statistically.
3. The square of the wave function is determined by multiplying the wave
function by its complex conjugate. |𝜓|2 = 𝜓 𝑥 𝜓 ∗ = real quantity.
4. |𝜓|2 is the probability of finding the particle in the state, and it is a measure of
position probability density (P). 𝑃 = |𝜓|2 = 𝜓 𝑥 𝜓 ∗ .
5. The probability of finding a particle in a volume 𝑑𝜏 = 𝑑𝑥𝑑𝑦𝑑𝑧 is 𝑃 =
∭|𝜓|2 𝑑𝑥𝑑𝑦𝑑𝑧
6. Since “P” is the probability of finding the particle, the value lies between 0
and 1.
7. For the presence of the particle, P = 1 and for the absence, P = 0.
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
HAMILTONIAN OPERATOR
The Hamiltonian of a particle is its total energy. Hamiltonian of a particle of mass
“m” moving in a potential V(r),
𝐾. 𝐸. (𝑇) =
𝑝2
2𝑚
---------------- (1)
The operator for p is -iђ∇. Replacing ‘p’ by its operator in above equation,
𝑇𝑜𝑝 =
(−𝑖ђ∇). (−𝑖ђ∇)
ђ2 2
= −
∇
2𝑚
2𝑚
𝑃. 𝐸. (𝑉𝑜𝑝 ) = 𝑉(𝑟)
𝐻𝑜𝑝
ђ2 2
= 𝑇𝑜𝑝 + 𝑉𝑜𝑝 = −
∇ + 𝑉(𝑟)
2𝑚
𝐻= −
ђ2 2
∇ + 𝑉(𝑟)
2𝑚
EIGEN VALUES AND EIGEN FUNCTIONS OF AN OPERATOR
Operator ‘A’ operating on a function 𝑓𝑖 (𝑥) simply multiplies 𝑓𝑖 (𝑥) by a constant 𝑎𝑖 ,
𝐴 𝑓𝑖 (𝑥) = 𝑎𝑖 𝑓𝑖 (𝑥)
𝑓𝑖 (𝑥) is an Eigen function of the operator ‘A’ and Eigen value 𝑎𝑖
𝑑 5𝑥
𝑒 = 5𝑒 5𝑥
𝑑𝑥
𝑑
𝑒 5𝑥 − Eigen function of 𝑑𝑥
5 – Eigen value
PARTICLE IN A 1D INFINITE SQUARE WELL
Consider the solution of the time independent Schrödinger equation, for a particle
(e.g. electron) moving in a 1D square well of infinite depth. The particle is bouncing back
and forth between the walls of the box. The box has insurmountable potential barriers as x =
0 and x = a. i.e. the box is supposed to have walls of infinite height at x = 0 and x = a. The
particle has a mass ‘m’ and its position ‘x’ at any instant is given by 0<x<a.
α
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
‘a’
V
0
a
x
The potential energy V of the particle is infinite on both sides of the box. The
potential energy V of the particle can be assumed to be zero between x = 0 and x = a.
The potential V(x) is given by,
0
𝑉(𝑥) = {
𝛼
0<𝑥<𝑎
𝑥 ≤ 0 𝑎𝑛𝑑 𝑥 ≥ 𝑎
The Schrödinger equation for 1D case,
−
ђ2 𝑑 2 𝜓(𝑥)
= (𝐸 − 𝑉)𝜓(𝑥)
2𝑚 𝑑𝑥 2
In the regions x<0 and x>a (outside the box), the Schrödinger equation
ђ2 𝑑 2 𝜓
−
= (𝐸 − 𝛼)𝜓(𝑥) = −𝛼𝜓(𝑥)
2𝑚 𝑑𝑥 2
𝜓(𝑥) =
1 ђ2 𝑑 2 𝜓
=0
𝛼 2𝑚 𝑑𝑥 2
The wave function ψ(x) = 0 for the particle being outside the box.
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
Energy Eigen values and Eigen Functions
Consider the region 0 ≤ x ≤ a, (i.e.) for particle inside the box, then the time independent
equation,
ђ2 𝑑 2 𝜓
−
= 𝐸𝜓 ∵ 𝑉 = 0, ℎ𝑒𝑛𝑐𝑒 𝐸𝜓 𝑜𝑛𝑙𝑦
2𝑚 𝑑𝑥 2
Rearranging the equation,
𝑑2 𝜓
𝑑𝑥 2
= −𝑘 2 𝜓 ∵ 𝑘 2 =
2𝑚𝐸
ђ2
--------- (1)
The solution for this equation is,
𝜓 = 𝐴 sin 𝑘𝑥 + 𝐵 𝑐𝑜𝑛 𝑘𝑥 ------------ (2)
To determine the constants we have to use boundary conditions
𝜓 = 0 𝑎𝑡 𝑥 = 0 𝑎𝑛𝑑 𝑎𝑡 𝑥 = 𝑎
𝜓 = 0 𝑎𝑡 𝑥 = 0 𝑙𝑒𝑎𝑑𝑠 𝑡𝑜 𝐵 = 0 𝑠𝑖𝑛𝑐𝑒 cos 𝑘𝑥 = cos 0 = 1
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝜓 = 𝐴 sin 𝑘𝑥
The condition ψ = 0 at x = a gives 𝐴𝑠𝑖𝑛 𝑘𝑎 = 0
But A cannot be zero, since it is constant, which would make the function zero
everywhere.
∴ sin 𝑘𝑎 = 0 (𝑜𝑟) 𝑘𝑎 = 𝑛𝜋 𝑤ℎ𝑒𝑟𝑒 𝑛 = 1, 2, 3, … --------------- (3)
The value n = 0 is left out since it leads to ψ = 0.
From the equation 3 & 1,
𝑘=
𝑛𝜋
2𝑚𝐸𝑛
& 𝑘2 =
𝑎
ђ2
2𝑚𝐸𝑛 𝑛2 𝜋 2
= 2
ђ2
𝑎
𝐸𝑛 =
𝑛2 𝜋 2 ђ2
2𝑚𝑎2
; 𝑛 = 1, 2, 3, … ------------------ (4)
These are the energy Eigen values for the particle in the 1D box. With the above values of k,
the wave function ψ becomes,
𝜓 = 𝐴 sin
DR. N. VENKATANATHAN
𝑛𝜋𝑥
𝑛 = 1, 2, 3, …
𝑎
19
QUANTUM PHYSICS
For each value of ‘n’, there is energy level and corresponding wave function. Each
value of En is Eigen value and corresponding 𝜓𝑛 is Eigen function. Inside the box, the
particle can only have discrete energy and corresponding energy levels, it cannot be zero.
Use of normalization condition gives,
𝑎
|𝜓|2 = |𝐴|2 ∫ sin2
0
|𝐴|2 .
𝑛𝜋𝑥
𝑑𝑥 = 1
𝑎
𝑎
=1
2
𝐴= √
2
𝑎
2
Therefore the normalised wave functions of the particle 𝜓𝑛 (𝑥) = √𝑎 sin
𝑛𝜋𝑥
𝑎
; 𝑛 = 1, 2, 3, …
The normalised wave functions 𝜓1 , 𝜓2 𝑎𝑛𝑑 𝜓3 𝑎𝑟𝑒 𝑝𝑙𝑜𝑡𝑡𝑒𝑑.
With the help of simple particle-in-a-box problem, some important principles are revealed:
1. The energy comes out quantized.
2. The highest probability of finding the particle in the box is where the antinodes of the
sine function appear.
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
3. The concept of zero-point energy, which states that the energy of the system would
approach E1(x) if it were cooled to absolute zero. This implies that the particle would
still be able to move throughout the box (a contradiction classically).
Zero-point energy is the lowest possible energy that a quantum mechanical physical
system, it is the energy of its ground state. All quantum mechanical systems undergo
fluctuations even in their ground state and have associated zero-point energy, a
consequence of their wave-like nature. Due to the uncertainty principle every physical
system to have a zero-point energy greater than the minimum of its classical potential
well, even at absolute zero. For example, liquid helium does not freeze under atmospheric
pressure at any temperature because of its zero-point energy.
QUANTUM TUNNELLING
Quantum tunnelling refers to the quantum mechanical phenomenon where a particle
tunnels through a barrier that it classically could not overcome. This plays an essential role in
several physical phenomena, such as the nuclear fusion that occurs in main sequence stars
like the Sun. It has important applications to modern devices such as the tunnel diode,
quantum computing, and the scanning tunnelling microscope. Tunnelling is often explained
using the Heisenberg uncertainty principle and the wave–particle duality of matter.
𝑉0
0≤𝑥≥𝑎
0
𝑥 < 0 𝑎𝑛𝑑 𝑥 > 𝑎
Consider a particle of mass m and energy E < Vo incident on the barrier from left.
Classically, all the particles will be reflected at x = 0, if E< Vo and all will be transmitted into
the region x > a, if E > Vo.
𝑉(𝑥) = {
DR. N. VENKATANATHAN
21
QUANTUM PHYSICS
However quantum mechanically there will be finite probability for the particles to be
in “region 3” even if E < Vo.
This is called as barrier penetration (or) quantum tunnelling. This is possible due to
the wave nature of the material particles.
Particles tunnels through the region 0<x>a, never be observed as its kinetic energy is
negative in that region. (Like roller coaster)
The solutions of Schrödinger equation for the particle give the wave function for the 3
regions.
Region 1: Wave function corresponds to two waves. One is travelling from left to right (i.e.)
incident, the other from right to left (reflected by the barrier).
Region 2: Two solutions are possible, one is an exponentially increasing function and the
other one is an exponentially decreasing function.
Region 3: The wave function is finite, which is travelling only from left to right.
Thus quantum mechanically there is a probability for the particle to be in the “region
3”. For that the particle should have the solution, which is exponentially increasing function.
𝑑2 𝜓
Region 1: 𝑑𝑥 21 +
Region 2:
Region 3:
𝑑2 𝜓2
𝑑𝑥 2
𝑑2 𝜓3
𝑑𝑥 2
+
+
2𝑚𝐸
ђ2
𝜓1 = 0 (𝑆𝑖𝑛𝑐𝑒 𝑉 = 0)
2𝑚(𝐸−𝑉)
ђ2
2𝑚𝐸
ђ2
𝜓2 = 0 (𝑆𝑖𝑛𝑐𝑒 𝑉 = 𝑉)
𝜓3 = 0 (𝑆𝑖𝑛𝑐𝑒 𝑉 = 0)
2𝑚𝐸
2𝑚(𝐸 − 𝑉)
= 𝛼2 &
= 𝛽2
2
ђ
ђ2
Therefore,
Region 1:
𝑑2 𝜓1
𝑑𝑥 2
+ 𝛼 2 𝜓1 = 0
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
Region 2:
Region 3:
𝑑2 𝜓2
𝑑𝑥 2
𝑑2 𝜓3
𝑑𝑥 2
− 𝛽 2 𝜓2 = 0
+ 𝛼 2 𝜓3 = 0
The solutions of these equations are,
𝑅𝑒𝑔𝑖𝑜𝑛 1: 𝜓1 = 𝐴𝑒 𝑖𝛼𝑥 + 𝐵𝑒 −𝑖𝛼𝑥
𝑅𝑒𝑔𝑖𝑜𝑛 2: 𝜓2 = 𝐹𝑒 −𝛽𝑥 + 𝐵𝑒 𝛽𝑥
𝑅𝑒𝑔𝑖𝑜𝑛 2: 𝜓3 = 𝐶𝑒 𝑖𝛼𝑥 + 𝐷𝑒 −𝑖𝛼𝑥
Where A, B, C, D, F & G are amplitude of the corresponding components of each wave.
A – amplitude of wave – incident on the barrier from the left.
B – Amplitude of the reflected wave in region 1.
F – Amplitude of the wave, penetrating the barrier in region 2.
G – Amplitude of the reflected wave in region 2.
C – Amplitude of the transmitted wave in region 3.
D – Amplitude of the reflected wave (non-existent) in region 3.
Probability density associated with a wave function is proportional to the square of
the amplitude of that function.
|𝐶|2
The barrier transmission coefficient, 𝑇 = |𝐴|2
The reflection coefficient for the barrier surface at x = 0, 𝑅 =
|𝐵|2
|𝐴|2
If a particle with energy E is incident on a thin energy barrier of height greater
than E, there is finite probability of the particle penetrating the barrier, called the
“tunneling effect”.
To explain quantum tunneling following phenomena in physics can be used.
1. The emission of α – particles from radioactive nuclei.
A potential well - The alpha decay of polonium-212 is the one that releases the most
important quantity of energy, 8.95 MeV. It is impossible for an alpha particle to go
from inside the nucleus in A to outside in B. It finds itself trapped at the bottom of a
"well" as shown by the curve that represents the potential energy of interaction
between the particle and the rest of the nucleus. To get from A to B, the particle must
DR. N. VENKATANATHAN
23
QUANTUM PHYSICS
pass through a restricted area where its kinetic energy would be negative. The
permitted areas are the well where nuclear attraction dominates, and outside the well
where the repulsion due to the nuclear charge prevails.
The quantum tunneling or “tunnel effect” describes the fact that a particle behaves as
both a particle and a wave in the infinitesimally small world where quantum mechanics
replaces classical mechanics.
The wave associated with an alpha particle trapped inside a nucleus has been
superimposed to the previous figure. We see that the wave extends slightly outside the
nucleus, where the oscillation amplitude has been amplified to make them visible. The square
of the amplitude of the oscillations is, in quantum mechanics, the probability of observing the
particle at a given position. So there is a probability of observing the alpha particle outside
the nucleus, that is to say decay.
2. Periodic inversion of ammonia molecule – atomic clock constructed on the basis.
The two energetically equivalent states of ammonia, NH3, can exchange more readily
than is classically predicted - this is because the molecule can tunnel through the potential
DR. N. VENKATANATHAN
24
QUANTUM PHYSICS
barrier at lower energies than are required to pass through the transition state.
3. Electronic devices – tunneling diode – Josephson junction
4. Tunneling in Smell Receptors - Until quite recently, it was believed that chemical
receptors in the nose (400 different kinds in humans) detected the presence of various
chemicals by a lock-and-key process, which identified the molecule's physical shape.
There are some issues with this theory, however. For example, ethanol and
ethanethiol, which have very similar shapes, smell completely different (ethanol is the
alcohol we drink - ethanethiol smells of rotten eggs). This suggests that some other
identifying mechanism is at work.
A theory which has been growing in popularity over the last decade or so is
that smell receptors rely in part on quantum tunneling to identify chemicals. The
receptors pump a small current across the odorant molecule, causing it to vibrate in a
characteristic way. In order for the current to flow, however, the electrons must tunnel
through the non-conducting gap between the cells of the receptor and the molecule.
DR. N. VENKATANATHAN
25
QUANTUM PHYSICS
Smell receptors can detect the differences between similarly-shaped molecules by tunneling a
small current across them, causing a characteristic vibration.
5. Electron tunneling – Scanning tunneling Microscope
Scanning Tunneling Microscopes - A Scanning Tunneling Microscope (STM) works by
scanning a very sharp conducting probe across the surface of a material. An electrical
current is passed down the tip of the probe, and tunnels across the gap into the material. As
the gap gets wider or narrower, the tunneling current gets smaller or larger, respectively.
Using this data, we can build an incredibly detailed picture of the surface, even to the point
of resolving humps in the surface due to individual atoms. This technique has allowed leaps
forward in our understanding of the physics and chemistry of surfaces.
Schematic of an STM - the tunneling current varies with the distance between the tip and the
atoms on the surface, allowing defects and even individual atoms to be mapped.
Flash drives - Data on flash drives is stored in a network of memory cells made up of
"floating-gate" transistors. These consist of two metal gates, a control gate and a floating
gate. The floating gate is trapped in an insulating layer of metal oxide. A floating-gate
transistor in its normal state registers a "1" in binary code. When an electron is attached to the
floating gate, it becomes trapped in the oxide layer, affecting a change in the voltage of the
control gate - a transistor in this state registers a "0" in binary. When data is erased from flash
DR. N. VENKATANATHAN
26
QUANTUM PHYSICS
memory, a strong positive charge applied to the control gate causes the trapped electron to
tunnel through the insulating layer, returning the memory cell to a "1" state.
Schematic diagram of a floating-gate transistor - Trapping an electron in the floating gate causes
a change in the voltage across the control gate. Each transistor stores a bit of information.
TIME DEPENDENT PERTURBATION THEORY
In quantum mechanics, perturbation theory is a set of approximation schemes directly
related to mathematical perturbation for describing a complicated quantum system in terms of
a simpler one. The idea is to start with a simple system for which a mathematical solution is
known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to
the system. If the disturbance is not too large, the various physical quantities associated with
the perturbed system (e.g. its energy levels and eigenstates) can be expressed as "corrections"
to those of the simple system. These corrections, being small compared to the size of the
quantities themselves, can be calculated using approximate methods such as asymptotic
series. The complicated system can therefore be studied based on knowledge of the simpler
one.
For example, by adding a perturbative electric potential to the quantum mechanical
model of the hydrogen atom, we can calculate the tiny shifts in the spectral lines of hydrogen
caused by the presence of an electric field (the Stark effect). This is only approximate
because the sum of a Coulomb potential with a linear potential is unstable although the
tunneling time (decay rate) is very long.
PROBLEMS
1. Calculate the energy of a photon of wavelength 2 Ao in electron volts. What is the
momentum of this photon?
E=
ℎ𝑐
𝜆
P=
=
6.62 × 10−34 × 3 × 108
2 × 10−10 × 1.602 × 10−19
6.62 ×10−34
2 ×10−10
= 6.198×103 eV (or) 6.198×103×1.602×10-19 = 9.93×10-16 J
= 3.31 × 10-24 kg.m.s-1
2. Determine the energy values of an electron confined in a box of width 1 Ao.
DR. N. VENKATANATHAN
27
QUANTUM PHYSICS
𝑛2 ×(6.62 ×10−34 )
𝑛2 ℎ 2
2
E= 8𝑚𝑎2 = 8 ×9.1 ×10−31 ×(10−10 )2 = 6 × 10-18 n2 J
=
6 ×10−18 𝑛2
1.6 ×10−19
eV = 37.5 n2 eV
3. Find the lowest energy level and the momentum of an electron in one dimensional
potential well of width 1 Ao.
𝑛2 ℎ 2
𝑝2
1
En = 8𝑚𝑎2 = 2 𝑚𝑣 2 = 2𝑚
12 ×(6.62 ×10−34 )
𝑛2 ℎ 2
2
E1 = 8𝑚𝑎2 = 8 ×9.1 ×10−31 ×(10−10 )2 = 6 × 10-18 J
=
6 ×10−18 𝑛2
eV = 37.5 n2 eV
1.6 ×10−19
𝑝2
𝑛2 ℎ 2
En = 2𝑚 = 8𝑚𝑎2
ℎ
p = 2𝑎 =
(6.62 ×10−34 )
2 ×10−10
= 3.31 × 10-24 kg.m.s-1.
4. A particle is confined to one dimensional infinite potential well of width 0.2 × 10-9 m.
It is found that when the energy of the particle is 230 eV its Eigen functions have 5
antinodes. Find the mass of the particle.
Number of antinodes = energy level. Here 5 antinodes means, fifth level (i.e.) E5
E5 = 230 eV = 230 × 1.6 × 10-19 J; a = 0.2 × 10-9 m.
𝑛2 ℎ 2
52 ℎ 2
ℎ2
En =8𝑚𝑎2 ; E5 = 8𝑚𝑎2 = 52 × 8𝑚𝑎2 = 25 E1
𝐸
E1 = 525 =
230×1.602×10−19
25
ℎ2
= 14.7 × 10-19 J.
(6.62 ×10−34 )
ℎ2
E1 =8𝑚𝑎2 ; m =8E
1
2
= 8×14.7 ×10−19 ×(0.2 ×10−9 )2 = 9.3 × 10-31 kg.
𝑎2
5. A particle is moving in one dimensional potential box of infinite height of width
50Ao. Calculate the probability of finding the particle within an interval of 10Ao at the
centre of the box when it is in its least energy.
Given: n = 1; a = 50 × 10-10 m; x = 10 × 10-10 m.
Probability of finding the particle (P) = |𝜓(𝑥)2 | ∆𝑥
2
We know ψ(x) = √𝑎 sin
𝑛𝜋𝑥
𝑎
2
𝜋𝑥
= √𝑎 sin 𝑎 (Since n = 1).
At the centre of the box, x = 𝑎⁄2
|𝜓(𝑥)2 | =
2
𝑎
2
[𝑠𝑖𝑛
𝜋(𝑎⁄2) 2
𝑎
𝑎
𝜋
2
] = 2 sin2 2 = 𝑎
2
20
P = 𝑎 ∆𝑥 =50×10−10 × 10 × 10−10= 50 = 0.4
The probability of finding the particle is 0.4.
6. If a dust particle of 1 µg requires 100 s to cross a distance of 1 mm which is the
separation between two rigid walls of the potential, determine the quantum numbers
described by it.
Mass of the particle (m) = 1 µg = 10-9 kg; t = 100 s; a = 1 mm = 10-3 m.
1
𝑛2 ℎ 2
E = 2 𝑚𝑣 2 and E = 8𝑚𝑎2
The particle moves 10-3 m in 100s.
DR. N. VENKATANATHAN
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QUANTUM PHYSICS
Therefore the velocity of the particle in 1 second =
10−3
100
= 10−5
1
E = 2 × 10−9 × (10−5 )2 = 5 × 10−20 𝐽
n2 =
8𝑚𝑎2
ℎ2
𝐸=
8×10−9 ×(10−3 )
(6.62 ×10−34 )2
2
× 5 × 10−20 = 9.11×1032
Therefore quantum state (n) = 3×1016.
7. An electron is bound by a potential box of infinite height having width of 0.25 Å.
Calculate the lowest three permissible energies that the electron can have.
[6 eV, 24 eV, 54 eV]
8. Find the lowest energy of a neutron confined to a box 10-14 m across. [2.1 MeV]
9. Determine the wave length associated with an electron having K. E. equal to 1 MeV.
[0.00726Å].
10. A proton and α – particle have the same K. E. a) How do their speeds compare? b)
How do their de Broglie wavelengths compare? c) How do their momenta compare?
Given - 𝑚𝛼 = 4𝑚𝑝 . [a) 𝑣𝑝 = 2𝑣𝛼 ; 𝑏) 𝑃𝑝 =
1
𝑃 ; 𝑐)
2 𝛼
𝜆𝑝 = 2𝜆𝛼 ]
11. Calculate the wavelength associated with electrons whose speed is 0.01 the speed of
light. [2.4 x 10-10 m]
12. Calculate the de Broglie wavelength of an alpha particle accelerated by a P. D. of
25000 volts. [6.390 x 10-14 m]
13. The equivalent wavelength of a moving electron is 0.24 x 10-10 m. What voltage is
applied between two grids will bring it to rest? [2616 V].
14. An electron is confined to move between two rigid walls separated by 10-9 m. Find
the de Broglie wavelengths representing the first three allowed energy states of the
electron and the corresponding energies. {(20 Å, 10 Å, 6.7 Å); (0.38 eV, 1.52 eV,
3.42 eV)}
DR. N. VENKATANATHAN
29