1. M. Sc Physics, 2nd Semester
Quantum Mechanics-I (PH-515)
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Contents of Course:
Linear Vector Space and Matrix Mechanics
Stationary State Approximate Methods
Angular Momentum

3. Books Recommended:
Quantum Mechanics, concept and applications by
Nouredine Zetili
Introduction to Quantum Mechanics by D.J. Griffiths
Quantum Mechanics, Vol 1, Cohen Tannoudji
The Principle of Quantum Mechanics by P A M Dirac
Feynmann Lectures, Vol 3, Quantum Physics
Quantum Mechanics by Eugen Merzbacher

4. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.1
About the Birth of Quantum Mechanics

5. In classical physics we can define the state of physical system
(macroscopic system) with arbitrary accuracy and can study
its future behaviour completely.
For e.g. Consider particle of mass m subject to force F(x,t)
and moving along x-axis.
Once we know the position x(t), we can find velocity
(v = dx/dt), momentum (p=mv) or any other dynamical
variable.
We can use the Newton law of motion to study the system.

6. Everything was fine with the classical physics upto the end
of 19th centaury.
However, with the advent of advance technology, as one
goes to the study of physical systems at microscopic level
the classical physics runs into difficulties and contradictions.
This led to the birth of quantum mechanics.
Quantum mechanics is the description of the behaviour of
matter and light in all its details and about the description
of phenomena at atomic scale.

7. We list here few experimental observations which
led to the birth of quantum mechanics.
Black body radiations and ultraviolet catastrophe
Photoelectric effect
Compton effect
Discrete spectral line
Atomic stability
Specific heat of solids

8. Black Body Radiations:
A blackbody is a perfect absorber as well as a perfect
emitter of radiation. When solids are heated they emit the
radiations with continuous distribution of frequency whereas
when gases are heated then we get a discrete distribution
of spectrum.
Figure on next slide shows the experimentally observed
spectral energy density of blackbody radiation at different
temperatures as a function of frequency.

10. Using the classical physics Wilhelm Wein (in 1889) and
Rayleigh (in 1900) tried to explain the observed spectrum
of radiations.
Wein failed to explain the spectrum at low frequency
whereas the Rayleigh-Jeans Law was failed at high
end frequencies (ultraviolet catastrophe).
Max-Planck introduced the concept of quantum of energy
in Experimental facts about the black body
Radiations are explained by Planck using the concept that
the energy exchange between radiation and its
surroundings takes place in discrete or quantized amounts.
See figure on next slide.

11. 14th December 1900,
Birthday of Quantum Mechanics

12. Photoelectric effect: This effect was experimentally
observed in 1887 by Hertz and theoretically explained
by Einstein in 1905 using the idea of quantization of light.

13. Compton Effect: Scattering of X-rays by electron and
explained by considering the X-rays as particle (photon)
and then collision of X-rays photon with electron.

14. Wave-particle duality is not restricted to radiations
Wave Aspect of Particles:
Wave-particle duality is not restricted to radiations
but must be universal. All material particles must
also possess wave-particle duality.
Each material particle of momentum p behaves as a
group of waves whose wavelength and wave vector
are given by

15. Davisson Germer Experiment:
This experiment prove the wave nature of the material particle.

16. In optics, when wavelength is very small compared with size
of object, we deal with geometrical optics or rays optics.
Wave nature of light is not detectable in this case.
However, when we cannot neglect the wavelength, one
deal with the physical optics.
Similarly, when the de Broglie wavelength of object are
comparable to size of objects, then these are detectable
and cannot neglected.
However, when the size of objects exceed considerably as
compared to the wavelength, then the wave behaviour is
not detectable.

17. Particle Vs Waves:
Double slit experiment: The experiment is used to
illustrate the particle and wave properties from classical and quantum physics point of view.
Experimental set-up consist of:
Source S (Source can be of material particles or waves)
A wall with two slits S1 and S2
A back screen equipped with counters that record
whatever arrive on it from slits

18. In Fig 1 Source S fire bullets (macroscopic particle). I1, I2 and
I are the intensities of bullets observed on the when S1 or S2 or
Both slits are open.
Fig 1

19. Fig 2 In Fig 2 S is Source of waves. Note that when both the slits
are open then the total intensity is not just sum of I1 and I2. A
oscillating term has to be added. This is because of interference.
Fig 2

20. Classically wave exhibit interference pattern but
particles do not. When two non-interacting stream of
particles combine in the same region of space their
intensities add. When wave combine their amplitude
add but their intensities do not.

21. Fig 3 In Fig 3 S is Source of electron. Note that when both the slits
are open then the total intensity is not just sum of I1 and I2. A
oscillating term has to be added. This is because of interference
which is a property of waves. It means the material particles
electron behave as wave. .
Fig 3

22. In Fig 4 S is Source of electron
In Fig 4 S is Source of electron. Here a light source is placed behind the wall containing S1 and S2. Now note that
interference pattern is destroyed and the total intensity is
I = I1+I2
Fig 4

23. Note that in fig (4), when we use the light source to watch
the electrons, the interference pattern destroys. In quantum
Mechanics it means measurements interfere with the states
Of microscopic objects.
We slow down the light source, so that some of the electrons
are not seen but we heard on counter. The profile
corresponding to unseen electron (but we heard on counter )
shows interference pattern.
For the electrons which show the interference pattern, it
is not possible to know that through which slit they pass.
This means the microscopic world is indterministic.

24. In classical physics, we can follow the path of macroscopic
Particle. But in case of microscopic particles this is not
possible. This led Heisenberg to postulate uncertainty
principle:
It is impossible to design an apparatus which allows us to
determine the slit that the electron went through without
disturbing the electron enough to destroy the
interference pattern.
Conclusion: The electron behave both as wave
as well as particle. When we keep watching them with the
help of light source, they behave as particle. But when
we see only the pattern of their distribution on screen
after many measurement, they behave as wave.

25. Atomic Structure:
Using the idea of Rutherford’s atomic model, Planck’s
quantum concept and Einstein’s Photon, Bohr in 1913
introduced model of hydrogen atom. Atoms can be found
in discrete states of energy and emission or absorption of
radiations by atom takes place only in discrete amount of hv.

26. Above outlined new ideas in physics led to the Schrodinger
and Heisenberg to formulate a consistent theory known
as Quantum mechanics.
In 1925 Heisenberg formulated Matrix Mechanics version of
Quantum mechanics. In this, the energy, position,
momentum, etc are expressed in terms of matrices and
Eigen-value equations are solved to describe the dynamics
of microphysical system.
Second formulation, known as Wave Mechanics was
Introduced by Schrodinger in generalizing
de Broglie postulates.
In this the dynamics of microphysical systems are described
by the Wave equations known as Schrodinger’s equation.

27. Dirac suggested a more general formalism of quantum
Mechanics in which one deal with the abstract objects
Such as kets, bras and operators .
The representation of Dirac formalism in continuous basis
i.e. in position or momentum representation give back
Schrödinger’s Wave mechanics.
By representing the Dirac formalism in Discrete basis
we get Heisenberg’s matrix mechanics.