Advanced Quantum Mechanics
Teacher
Dr. Padmaja Patnaik
Category
Core Courses
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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)
- 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.
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
Session 3
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
Session 6
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
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
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
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
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
Session 14
Bound states for potential wells with one rigid wall using WKB method, classical turning point, Find the energy eigen values
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
Session 18
Electric dipole radiation and selection rules
Session 19
Adiabatic approximations, sudden approximations
https://www.youtube.com/watch?v=M4i3Lq0SwQA
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
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
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)
- 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.
Our Main Teachers
Dr. Padmaja Patnaik
Associate Professor in School of Applied Sciences, Department of Physics
VIEW PROFILEDr. 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 […]