Faculty: Dr. Padmaja Patnaik

Course Name : Advanced Quantum Mechanics

Code(Credit) : CUTM1412 (2-0-2)

Course Objectives

• Learn methods to solve Schrodinger’s equation by WKB method, Variational method and perturbation method.
• Learn Practical application of these methods to real time problems
• Learn to apply these methods to solve several problems.

Learning Outcomes

On completion of this course students will be able to

• Solve Schrodinger's equation for different systems using WKB method, Variational method and perturbation method.
• Develop Python code to solve Schrodinger's equation and find energy eigen values

Course Syllabus

Module-I

Time independent Perturbation Theory: Energy shifts and perturbed eigen states, nondegenrate and degenerate perturbation theory, spin orbit coupling
Assignment 1: (Any one)

• Develop  solution  for shifting and splitting of spectral lines of atoms - Stark effect
• Develop  solution  for shifting and splitting of spectral lines of atoms - Zeeman effect

 Module-II

Pictures of quantum mechanics: The Schrodinger picture, Heisenberg picture, the interaction picture.
Variational methods: General formalism, ground state of one-dimensional harmonic oscillator, first excited state of one-dimensional harmonic oscillator,
Assignment 2: (Any one)

• Solve the problem for tunneling of a particle through a potential barrier
• Find out the energy of second excited states of  harmonic oscillator

Module-III
WKB Approximation: General formalism, validity of WKB approximation method, bound states for potential wells with no rigid walls.
Assignment 3: (Any one)

• Gamow's theory of alpha decay – Finding solution with WKB method
• Find out the energy of particle in bound states for potential wells with one rigid wall

Module-IV

Time dependent perturbation theory: Introduction, transition probability, transition probability for constant perturbation, transition probability for harmonic perturbation, adiabatic approximations, sudden approximations.
Assignment 4:

• Calculate the transition probability rate for an excited electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor by using Fermi golden rule.

Module-V

Applications of time dependent perturbation theory: Interaction of atoms with radiation, classical treatment of incident radiation, transition rates for absorption and emission of radiation,
Assignment 5: (Any one)

• Light absorption and emission - mathematical formulation using electric dipole radiation
• The quantum mechanical selection rules for electric dipole transitions
• Find out expression for transition rates within the dipole approximation

Module-VI

Assignment 6:
Group Project (Any one to be done)

1. One-electron phenomena in strong Laser fields – derivation and Python programming- application of adiabatic approximation
2. Two-electron phenomena in strong Laser fields - application of adiabatic approximation
3. Fermi’s golden rule applied to find tunneling current of a scanning tunneling microscope
4. Derivation and Python programming for bound states for potential wells with two rigid walls
5. Ground state of Hydrogen atom - solve using Python programming.
6. Study of the neutron quantum states in the gravity field - Python programming
7. Find an expression for electric-dipole two-photon absorption selection rules and use it  to summarize the rules for two photons of unequal frequency.
8. Simulation using Python programming for tunneling through a potential barrier
9. Quantum harmonic oscillator using Python
10. Gamow's theory of alpha decay – solution and simulation sing python
11. Applications of quantum tunneling - Python programming.

Textbook:

1. Advanced Quantum Mechanics by Satyaprakash, S Chand Publications

Reference Books:

1. Quantum Mechanics: Concepts and Applications by Nouredine Zettili
2. Introduction to Quantum Mechanics, D J Griffith, Pearson, 2014.
3. Modern Quantum Mechanics, J.J. Sakurai, Pearson, 2013.

Session Plan

Session 1

Time dependent perturbation theory, energy shifts and perturbed eigen states
https://www.youtube.com/watch?v=B4fpfhCC_cM
https://www.slideshare.net/razorgreen/time-independent-perturbation-theory

Session 2

Perturbation theory, nondegenrate and degenerate perturbation theory
https://www.youtube.com/watch?v=aNS38D1Hxu0
https://www.youtube.com/watch?v=5cK5dT86S_U
https://slideplayer.com/slide/1473236/

Session 3

Spin orbit coupling
https://www.youtube.com/watch?v=j7VMZk1sISU
https://www.youtube.com/watch?v=k6EuwfbuPvE
https://slideplayer.com/slide/5682703/

Session 4

Normal and anomalous Zeeman Effect
https://www.youtube.com/watch?v=qmbuHAo2uzM
https://slideplayer.com/slide/5078795/

Session 5

Assignment 1: (2 hours)
(Any one)

• Develop  theoretical solution for shifting and splitting of spectral lines of atoms - The Stark effect
• Develop  theoretical solution for shifting and splitting of spectral lines of atoms - Zeeman effect

https://www.youtube.com/watch?v=BXg_jWZivo0&t=3s
https://www.youtube.com/watch?v=7oqAjuJ3YL4
https://slideplayer.com/slide/5078795/

Session 6

The Schrodinger picture,  Heisenberg picture, interaction picture. Variational methods: Introduction and general formalism
https://www.powershow.com/view1/1ef4e6-ZDc1Z/Schrdinger_Heisenberg_Interaction_Pictures_powerpoint_ppt_presentation
https://www.youtube.com/watch?v=l7n8gQHHFyg
https://www.slideserve.com/tender/review-three-pictures-of-quantum-mechanics

Session 7

Ground state of Hydrogen atom, Write the hamiltonian, solve the Schrodinger's equation by variational method, find the energy value
https://www.youtube.com/watch?v=mUbxvtTj6Dg
https://www.slideshare.net/AnilkumarShoibam/hydrogen-atom-15172172

Session 8

Ground state of one-dimensional harmonic oscillator, write the hamiltonian, Solve the Schrodinger's equation by using variational method, find energy eigen values
https://www.youtube.com/watch?v=Bcjywu0u1SA
https://www.youtube.com/watch?v=YQFVq08tutM
https://www.slideshare.net/ahmed7aider/harmonic-oscillator

Session 9

First excited state of one-dimensional harmonic oscillator,write the hamiltonian, solve the Schrodinger's equation by using variational method, find energy eigen values
https://www.youtube.com/watch?v=hqG2e7JFEYY
https://slideplayer.com/slide/7872366/

Session 10

Assignment 2: (2 hours)
(Any one)
1. Solve theoretically for tunneling of a particle through a potential barrier
2. Solve theoretically for harmonic oscillator – second excited state
https://www.youtube.com/watch?v=RF7dDt3tVmI
https://www.slideshare.net/ahmed7aider/harmonic-oscillator

Session 11

WKB Approximation: General formalism, condition to apply WKB approximation
https://www.youtube.com/watch?v=HQbJI9xzi5Q
https://www.slideshare.net/user0503/the-wkb-approximation-56368805

Session 12

Steps to solve a problem by WKB method, validity of WKB approximation method
https://www.youtube.com/watch?v=X7zb5xFdNY8
https://www.slideshare.net/user0503/the-wkb-approximation-56368805

Session 13

Bound states for potential wells with no rigid walls using WKB approximation, find out the energy expression
https://www.youtube.com/watch?v=-2CRQNInhVA
http://misc-lecture.tripod.com/SEC3b.pdf

Session 14

Bound states for potential wells with one rigid wall using WKB method, classical turning point, Find the energy eigen values
https://www.youtube.com/watch?v=-2CRQNInhVA
http://misc-lecture.tripod.com/SEC3b.pdf

Session 15

Assignment 3: (2 hours)
(Any one)

• Gamow's theory of alpha decay – theoretical solution
• Theoretical solution for bound states for potential wells with one rigid wall
• Time taken for a can of soft drink at room temperature to topple spontaneously- applications of quantum tunneling

Session 16

Time dependent perturbation theory: Introduction, transition probability, transition probability for constant perturbation
https://www.youtube.com/watch?v=Prgjila8ep0
https://www.slideshare.net/razorgreen/time-dependent-perturbation-theory

Session 17

Transition probability for harmonic perturbation, Fermi’s golden rule
https://www.youtube.com/watch?v=lzBEgkO5Ejw
https://www.youtube.com/watch?v=nCUG9tlsH4o

Session 18

Electric dipole radiation and selection rules
https://www.youtube.com/watch?v=-1TwrwNF0C8

Session 19

Adiabatic approximations, sudden approximations
https://www.youtube.com/watch?v=M4i3Lq0SwQA
https://www.youtube.com/watch?v=LyYAW153zZM
https://slideplayer.com/slide/8136412/

Session 20

Assignment 4: (2 hours)
Calculating the transition probability rate for an excited electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor by using Fermi golden rule
https://www.youtube.com/watch?v=uu3X0h6V6Nw

Session 21

Applications of time dependent perturbation theory: Interaction of atoms with radiation, classical treatment of incident radiation
https://www.youtube.com/watch?v=VUkcnnnY3cA
https://www.youtube.com/watch?v=21PWi1nPjBg
https://www.slideshare.net/vandana_rt/interaction-of-radiation-with-matter-dr-vandana

Session 22

Classical treatment of incident radiation
http://web.phys.ntnu.no/~stovneng/TFY4215_2019/lecturenotes/lecturenotes16.pdf

Session 23

Classical treatment of incident radiation continued
http://web.phys.ntnu.no/~stovneng/TFY4215_2019/lecturenotes/lecturenotes16.pdf

Session 24

Transition rates for absorption and emission of radiation
http://home.uchicago.edu/~tokmakoff/TDQMS/Notes/4.4.-4.5.%20Abs%20Stim%20Emission-Spontaneous%20Emission.pdf

Session 25

Assignment 5: (2 hours)
(Any one)

• Light absorption and emission - mathematical formulation using electric dipole radiation
• Derive the quantum mechanical selection rules for electric dipole transitions
• Find out expression for transition rates within the dipole approximation

Session 26

Assignment 6: (8 sessions of 2 hrs each )
Group project. (Any one to be done)

1. One-electron phenomena in strong Laser fields – derivation and Python programming- application of adiabatic approximation
2. Two-electron phenomena in strong Laser fields - application of adiabatic approximation
3. Fermi’s golden rule applied to find tunneling current of a scanning tunneling microscope
4. Derivation and Python programming for bound states for potential wells with two rigid walls
5. Ground state of Hydrogen atom - solve using Python programming.
6. Study of the neutron quantum states in the gravity field - Python programming
7. Find an expression for electric-dipole two-photon absorption selection rules and use it  to summarize the rules for two photons of unequal frequency.
8. Simulation using Python programming for tunneling through a potential barrier
9. Quantum harmonic oscillator using Python
10. Gamow's theory of alpha decay – solution and simulation sing python
11. Time taken for a can of soft drink at room temperature to topple spontaneously- applications of quantum tunneling - Python programming.

Course Materials

• Arduino_ftdi_chip-1