UNIT 1

1. QUANTUM MECHANICS
SRIJITH S
ASSISTANT PROFESSOR
DEPARTMENT OF PHYSICS
SREE NARAYANA COLLEGE, KOLLAM.

2. CLASSICAL MECHANICS (Newton's mechanics) and MAXWELL'S EQUATIONS
(electromagnetics theory) can explain macroscopic phenomena such as motion of
billiard balls, cars, planets..etc.
QUANTUM MECHANICS is used to explain microscopic phenomena such as structure
of atom, nuclear phenomenon, photon-atom scattering and flow of the electrons in
a semiconductor.
QUANTUM MECHANICS is a collection of postulates based on a huge number of
experimental observations.

3. Quantum mechanics arose out of the need to provide explanations for a range of
physical phenomena that could not be accounted for by ‘classical’ physics:
 Black body spectrum
 photoelectric effect
 Spectra of the atoms..
 Specific heat of solids . . .

4. Black body Spectrum
 Known since centuries that when a material is heated, it radiates heat and its color depends
on its temperature
Example: heating elements of a stove:
– Dark red: 550ºC
– Bright red: 700ºC
– Then: orange, yellow and finally white (really hot !)
 The ability of a body to radiate is closely related to its ability to absorb radiation . A material
is constantly exchanging heat with its surrounding (to remain at a constant temperature). It
absorbs and emits radiations.
 The blackbody spectrum is the spectrum of the radiation emitted by the object when it is in
thermal equilibrium with its surroundings.
 At room temperature most of the radiation is in the infrared part of the spectrum and hence
is invisible.

5. Black body
Blackbody is a cavity, such as a metal box with a
small hole drilled into it.
 Incoming radiations entering the hole keep
bouncing around inside the box with a negligible
change of escaping again through the hole =>
Absorbed.
 The hole is the perfect absorber, e.g. the
blackbody Radiation emission does not depend
on the material the box is made of => Universal
in nature.
A black body is any object that absorbs all
radiation, whatever frequency, that falls on it and
emits all frequencies, at thermal equilibrium.
Ferry Black body

6. Rayleigh-Jeans law
 Considering the radiation inside a cavity of absolute temperature T whose walls
are perfect reflectors to be a series of standing em waves. The condition for
standing waves in such a cavity is that the path length from wall to wall,
whatever the direction must be a whole number of half wavelengths, so that
node occurs at each reflecting surface.
 An 1D harmonic oscillator has two degrees of freedom, corresponding to KE and
PE. Since each standing wave in a cavity originates in an oscillating electric
charge in the cavity wall, two degrees of freedom are associated with the wave
and it should have an average energy per standing wave is KT.

7. Rayleigh-Jeans law

8. Rayleigh-Jeans law
 Classical physics can be used to derive an equation which describes the intensity of
blackbody radiation as a function of frequency for a fixed temperature — the result is
known as the Rayleigh-Jeans law. Although the Rayleigh-Jeans law works for low
frequencies, it diverges as f2; this divergence for high frequencies is called the ultraviolet
catastrophe
 Radiated density means total energy per unit volume in the cavity in the frequency
interval from ὑ to ὑ+dὑ

9. Planck’s Radiation law
 Before Einstein, Planck postulated from study of radiation from hot bodies that the
radiating atoms can only radiate energy in discrete amounts or that atoms exist only in
discrete states, called Quantum states. Oscillator energies=nhὑ
 He used Boltzmann statistical formula to find average energy per standing wave.
 The Royal Academy of Sciences has decided to award the Nobel Prize for Physics, for the
year 1918, to Dr. Max Planck, professor at Berlin University, for his work on the
establishment and development of the theory of elementary quanta.
E(ὑ)dὑ =
8πhὑ3
1
c2
eh
ᶹ/kT
− 1

10. Wien’s Displacement Law (Awarded Nobel Prize in 1911)
 Differentiating Planck’s function and setting the derivative equal to zero yields the
wavelength of peak emission for a blackbody at temperature T;
λmax T= 2.898x10−3 m K
where λmax is expressed in meters and T in degrees kelvin.
 On the basis of this equation, it is possible to estimate the temperature of a
radiation source from a knowledge of its emission spectrum.
 For example: The wavelength of maximum solar emission is observed to be
approximately 0.475µm.
Hence T = 2900/ λmax = 2900 /0.475 = 6100K

12. The Stefan Boltzmann Law
 The blackbody flux density obtained by integrating the Planck function E(λ) over
all wavelengths, is given by
E = σ𝑇4 where σ is a constant equal to 5.67×10−8W𝑚−2 𝐾−4.
 If a surface emits radiation with a known flux density, this equation can be
solved for its equivalent blackbody temperature, that is, the temperature a
blackbody would need to have in order to emit the same flux density of
radiation.

13. Einstein Explanation
 Einstein preferred to believe that the formula hὑ was a property of the
electromagnetic field.(Planck thought it was a property of the atoms in the
blackbody).
 Einstein proposed that light of frequency ὑ is absorbed or emitted in packets (i.e
quanta) of energy E, where E = hὑ.
 Sometime later he extended this idea to say that light also had momentum, and
that it was in fact made up of particles now called photons.
 Einstein used this idea to explain the photo-electric effect (in 1905).
 A first example of wave-particle duality: light, a form of wave motion, having
particle-like properties.

14. Photoelectric Effect

15. Photoelectric Effect
Why was red light incapable of knocking electrons out of certain materials, no
matter how bright
yet blue light could readily do so even at modest intensities
Einstein explained in terms of photons, and won Nobel Prize in 1921
 The photoelectric effect was first observed in 1887 by Heinrich Hertz during
experiments with a spark gap generator

16. Photoelectric Effect
 When the power supply is set to a low voltage it traps the least energetic
electrons, reducing the current through the micro ammeter.
 Increasing the voltage drives increasingly more energetic electrons back until
finally none of them are able to leave the metal surface and the micro ammeter
reads zero. The potential at which this occurs is called the stopping potential.
 It is a measure of the maximum kinetic energy of the electrons emitted as a
result of the photoelectric effect.
 At frequencies below a certain threshold frequency, ὑ0, which is a characteristic
of each particular metal, no electrons are emitted.
 There must be a minimum energy Φ for an electron to escape from a particular
metal surface with zero velocity. This energy is called the work function of the
metal;
i.e Φ =hὑ0

17. Photoelectric Effect
 The higher the frequency of light, the
more energy the photoelectrons have.
Blue light results in faster electrons than
red light.
 As the frequency increases, the critical
retarding potential increases; i.e
𝐾𝐸 𝑚𝑎𝑥 = eV
 All photons of frequency, ὑ have the
same energy , so changing the intensity
of a monochromatic light beam will
change the number of photoelectrons
but not their energies
 Photoelectron current is proportional to
light intensity for all retarding voltages

18. Photoelectric Effect
Electromagnetic theory
 Energy in an em wave is supposed to
be spread across the wavefronts
 The more the intense the light, the
greater the energies of the electrons
Einstein Explanation
 Energy in light is not spread out in wave
fronts but is concentrated in small packets
or photons
 A bright light yield more photoelectrons
than a dim one of the same frequency ,
but energies remains the same.
 Energy was not only given to em waves in
separate quanta but was also carried by
the waves in separate quanta. i.e E= nhὑ
 A first example of wave-particle duality:
light, a form of wave motion, having
particle-like properties.

19. Laws of Photoelectric effect and Einstein Photoelectric
equation
 The strength of the photoelectric current is directly proportional to the intensity of
the incident light, provided the frequency is greater than the threshold frequency
 The velocity and hence the energy of the emitted photoelectrons is independent
of the intensity of light and depends only on the frequency of the incident light
and nature of the metal.
 Photoelectric emission is an instantaneous process.
h=Φ + 1/2𝑚𝑣 𝑚𝑎𝑥
2
h=hὑ0 +1/2𝑚𝑣 𝑚𝑎𝑥
2

20. Spectra of atoms
 Rutherford proposed that an atom consisted of a small positively charged nucleus
with electrons in orbit about this nucleus.
 According to classical EM theory, the orbiting electrons are accelerating and
therefore ought to emit EM radiation.
 The electron in a hydrogen atom should spiral into the nucleus in about 10−12
sec
 model cannot satisfactorily explain the stability of atom and the observed sharp
spectral lines of atoms.

21. Spectra of Atoms- Bohr Postulates (Awarded Nobel
Prize in 1922)
Condition for orbit stability: An electron can circle
a nucleus only if its orbit contains an integral
number of de broglie wavelengths. i.e; nλ= 2πr
Bohr proposed existence of stable orbits (stationary
states) of radius r such that angular momentum =
mvr = nħ, n= 1,2,3…
The atom emits EM radiation by making a
transition between stationary states, emitting a
photon of energy hὑ where
hὑ = 𝐸 𝑚 − 𝐸 𝑛=
𝑍2 𝑚𝑒4
8ℎ2Ɛ0
2 (
1
𝑛2 −
1
𝑚2)

22. Bohr model….
 The Bohr model of the atom started the progress toward a modern theory of the atom
with its postulate that angular momentum is quantized, giving only specific allowed
energies. Then the development of the quantum theory and the Schrodinger
equation refined the picture of the energy levels of atomic electrons.
 The centripetal force holding the electron in an orbit r from the nucleus is provided by
the coulomb interaction;
m
𝑣2
𝑟
=
1
4𝜋𝜀0
𝑒2
𝑟2
 Electron velocity, V=
𝑒
4𝜋𝜀0 𝑚𝑟
, Orbital radii in Bohr atom, 𝑟𝑛 =
𝑛2∈0ℎ2
𝜋𝑚𝑒2 n=1,2,3…
 Total Energy=-
𝑒2
8𝜋𝜀0 𝑟
; (negative sign indicates that atomic electron is bound to the
nucleus).

23. Bohr Model….
This is often expressed in terms of the inverse wavelength or "wave number" as follows:

24. Bohr model- Hydrogen spectrum

25. Einstein Explanation….
 Einstein later showed that the the photon had to carry both energy and
momentum — hence it behaves very much like a particle.
 Einstein also showed that the time at which a transition will occur and the
direction of emission of the photon was totally unpredictable.
 The first hint of ‘uncaused’ randomness in atomic physics.

26. Demerits of Bohr model
 Bohr’s model worked only for hydrogen-like atoms, It failed miserably for
helium
 The Bohr model treats the electron as if it were a miniature planet, with definite
radius and momentum. This is in direct violation of the uncertainty
principle which dictates that position and momentum cannot be simultaneously
determined.
 The Bohr model gives us a basic conceptual model of electrons orbits and
energies. The precise details of spectra and charge distribution must be left to
quantum mechanical calculations, as with the Schrodinger equation..

27. Bohr correspondence principle…
 When Quantum physics is applied to macroscopic systems, it must reduce to the
classical physics. Therefore, the non classical phenomena, such as uncertainty
and duality, must become undetectable. Niels Bohr codified this requirement
into his Correspondence principle.
 In other words, it says that for large orbits and for large energies, quantum
calculations must agree with classical calculations.
 In limit that n → ∞, quantum mechanics must agree with classical physics

29. Compton effect (awarded the Nobel Prize in 1927)
 Arthur H. Compton observed the scattering of x-rays from electrons in a carbon
target and found scattered x-rays with a longer wavelength than those incident upon
the target. The shift of the wavelength increased with scattering angle according to
the Compton formula.
 At a time (early 1920's) when the particle (photon) nature of light suggested by
the photoelectric effect was still being debated, the Compton experiment gave clear
and independent evidence of particle-like behavior.
 Compton explained and modeled the data by assuming a particle (photon) nature
for light and applying conservation of energy and conservation of momentum to the
collision between the photon and the electron. The scattered photon has lower
energy and therefore a longer wavelength according to the Planck relationship.

30. Compton effect
 λ’-λ =Δλ=
ℎ
𝑚𝑐
(1-cosφ)
 Compton wavelength, λ 𝑐 =
ℎ
𝑚𝑐
, for an electron λ 𝑐=2.426 pm

31. Compton effect
 Conservation of momentum;
In the direction of incident photon;
ℎ𝜗
𝑐
+0=
ℎ𝜗′
𝑐
cos φ+pcos𝜃
Perpendicular to the direction; 0=
ℎ𝜗′
𝑐
sin φ - psin𝜃
 Conservation of energy;
E=KE + m𝑐2 E= 𝑚2 𝑐4 + 𝑝2 𝑐2 KE= ℎ𝜗 − ℎ𝜗′

33. Compton effect
 Unlike the prediction of classical wave theory, the wavelength of the scattered
radiation does not depend on the intensity of radiation but depends on the
scattering angle and the wavelength of the incident beam.
 Compton effect can be explained by considering radiation to be of particulate
nature.
 Compton effect is best exhibited with short wavelength radiation like x-rays.
 A free electron cannot absorb a photon because it is not possible to simultaneously
satisfy energy momentum conservation.

34. Wave nature of particle…
 In 1924 Prince Louis de Broglie(awarded the Noble Prize in 1929) asked a philosophical
question, if waves can sometimes behave like particles, can particles sometimes exhibit
wave-like behaviour

35. De broglie wave…

36. Electron as wave….
 GP Thomson (1892–1975)
Electrons undergo diffraction. They behave as waves with wavelength h/p
GP Thomson (son of JJ) won Nobel Prize in 1937 for demonstrating that the
electron is a wave.
 JJ Thomson won the Nobel Prize for Physics in 1906 for demonstrating that the
electron is a particle.
 And they were both right!

37. Davisson and Germer Experiment
In 1926 Davisson and Germer (G P Thompson
independently carried out the same experiment)
accidently showed that electrons can be diffracted
(Bragg diffraction)
 Fired beam of electrons at nickel crystal and
observed the scattered electrons.
 They observed a diffraction pattern identical to that
observed when waves (X-rays) were fired at the
crystal.
 The wavelengths of the waves producing the
diffraction pattern was identical to that predicted by
the de Broglie relation.
 Fitting de Broglie waves around a circle gives Bohr’s
quantization condition.

38. Wave Particle duality…
Waves as particles:
Max Plank work on black-body radiation, in which he assumed that the
molecules of the cavity walls, described using a simple oscillator model, can
only exchange energy in quantized units.
1905 Einstein proposed that the energy in an electromagnetic field is not
spread out over a spherical wave front, but instead is localized in individual
clumbs - quanta. Each quantum of frequency n travels through space with
speed of light, carrying a discrete amount of energy and momentum
=photon => used to explain the photoelectric effect, later to be confirmed by
the x-ray experiments of Compton.

39. Wave Particle duality…
Particles as waves
 Double-slit experiment, in which instead of using a light source, one
uses the electron gun.
 Davisson Germer experiment.
 light as an electromagnetic wave.

40. THANK YOU….