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- 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.

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

**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
- Find out the ground state energy of a hydrogen atom

**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:** (Any one)

- 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)

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

**Textbook:**

- Advanced Quantum Mechanics by Satyaprakash, S Chand Publications

**Reference Books: **

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

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

Perturbation theory, nondegenrate and degenerate perturbation theory

https://www.youtube.com/watch?v=aNS38D1Hxu0

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

The Schrodinger picture, Heisenberg picture, interaction picture. Variational methods: Introduction and general formalism

https://www.youtube.com/watch?v=l7n8gQHHFyg

https://www.slideserve.com/tender/review-three-pictures-of-quantum-mechanics

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

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

First excited state of one-dimensional harmonic oscillator,write the hamiltonian, solve the Schrodinger's equation by using variational method, find energy eigen values

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

3. Find out the ground state energy of hydrogen atom

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

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

Bound states for potential wells with no rigid walls using WKB approximation, find out the energy expression

Bound states for potential wells with one rigid wall using WKB method, classical turning point, Find the energy eigen values

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

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

Transition probability for harmonic perturbation, Fermi’s golden rule

Electric dipole radiation and selection rules

Adiabatic approximations, sudden approximations

https://www.youtube.com/watch?v=M4i3Lq0SwQA

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

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

Classical treatment of incident radiation

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

Classical treatment of incident radiation continued

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

Transition rates for absorption and emission of radiation

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
- Mathematical formulation of transition rates within the dipole approximation

Assignment 6: (8 sessions of 2 hrs each )

Group project. (Any one to be done)

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

Dr. Padmaja Patnaik has done her PhD in Physics from IITB, Mumbai under the guidance of Dr Gautam Mukhopadhya and Dr Prabhakar P Singh of IITB, Mumbai. She focuses on the application of theory behind many scientific research and applications in the field of Physics and Material Science to solve modern day problems and foster […]