  Teacher Category

### Course Attendees

Still no participant

Still no reviews

# 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 1

Time dependent perturbation theory, energy shifts and perturbed eigen states

https://www.slideshare.net/razorgreen/time-independent-perturbation-theory

## 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://slideplayer.com/slide/5078795/

## Session 7

Ground state of Hydrogen atom, Write the hamiltonian, solve the Schrodinger's equation by variational method, find the energy value

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.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://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.slideshare.net/ahmed7aider/harmonic-oscillator

## Session 11

WKB Approximation: General formalism, condition to apply WKB approximation

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.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

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

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.slideshare.net/razorgreen/time-dependent-perturbation-theory

## Session 17

Transition probability for harmonic perturbation, Fermi’s golden rule

## Session 18

Electric dipole radiation and selection rules

## 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

## Session 21

Applications of time dependent perturbation theory: Interaction of atoms with radiation, classical treatment of incident radiation

## Session 23

Classical treatment of incident radiation continued

http://web.phys.ntnu.no/~stovneng/TFY4215_2019/lecturenotes/lecturenotes16.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.

### Our Main Teachers 