Quantum Mechanics and Applications

Teacher

Dr. Subrata Sarangi

Category

Core Courses

Course Attendees

Still no participant

Course Reviews

Still no reviews

Quantum Mechanics and Applications

Code(Credit) : CUTM1493(3-2-1)

Course Objectives

This course aims to

  • Train the B.Sc. (Physics Honours) students with an understanding of the basic concepts of Quantum Mechanics and its mathematical frame work  to describe the systems of microscopic objects (like atoms, molecules) and their properties
  • Illustrate, in detail, the procedure of solving the 1D and 3D Schrodinger equations for microscopic systems interacting with time independent Central Potentials.
  • Offer hands-on simulation experience of obtaining numerical solutions of Schrodinger Equation for various use-cases along with visualization of the solutions through Python
  • Illustrate some present day and some futuristic applications of Quantum Mechanics like Tunnel Diodes, Scanning Tunneling Microscopy, Bose-Einstein Condensation, Quantum Computers etc.

Course Outcomes

COs
Course outcomes
Mapping COs with POs (High-3, Medium-2, Low-1)
CO1
Solve, both analytically and numerically, time-independent and time-dependent Schrodinger Equations
PO1-3, PO2-3
CO2
Generate appropriate visualizations of the solutions of Schrodinger Equations and interpret them
PO2-3, PO5-2
CO3
Get a qualitative knowledge of how Quantum Mechanics operates in macroscopic devices like Tunnel Diodes, Scanning Tunneling Microscopes or large scale objects like the Sun as well as the evolving technologies like Quantum Computers and Bose-Einstein Condensates.
PO2-2, PO3-1,PO5-2, PO9-1

Course Syllabus

Module I: Basic Concepts-I

Introduction to Quantum Mechanics (the What and Why of QMech), The relevance of Quantum Mechanics in our day-to-day life, Postulates of Quantum Mechanics; The Time dependent Schrodinger equation and dynamical evolution of a quantum state; Properties of Wave Function. Interpretation of Wave Function Probability and probability current densities in three dimensions; Conditions for Physical Acceptability of Wave Functions. Normalization. Linearity and Superposition Principles. Eigenvalues and Eigenfunctions. Position, momentum and Energy operators; commutator of position and momentum operators; Expectation values of position and momentum. Wave Function of a Free Particle.

Practice-1 (2 Hours): Given candidate wave functions, check for validity of each candidate using computer visualization and compute the Position Probability Density in each valid case

Practice-2 (2 hours): Given candidate wave functions, check for valid wave functions using computer visualization and compute current probability density

Module II: Basic Concepts-II

Time independent Schrodinger equation-Hamiltonian, stationary states and energy eigenvalues; expansion of an arbitrary wave function as a linear combination of energy eigenfunctions; General solution of the time dependent Schrodinger equation in terms of linear combinations of stationary states; Application to spread of Gaussian wave-packet for a free particle in one dimension; wave packets, Fourier transforms and momentum space wave function; Position-momentum uncertainty principle.

Practice-3 (2 hours): Free Particle wave function, Gaussian wave packet and its time evolution

Module III: 1D Applications

General discussion of bound states in an arbitrary potential- continuity of wave function, boundary condition and emergence of discrete energy levels; application to one-dimensional problem-1.Infinite square well potential; 2. The case of Finite Square-Well Potential and Quantum Tunneling, The working principle of Tunnel Diodes, Scanning Tunnelling Microscope, Quantum mechanics of simple harmonic oscillator-energy levels and energy eigenfunctions using Frobenius method; Hermite polynomials; ground state, zero point energy and uncertainty principle for 1D Simple Harmonic Oscillator.

Practice-4 (2 hours): Solution of 1D time independent Schrodinger Equation: Infinite Square Well Potential, Emergence of quantized eigenfunctions, normalization of wave function

Practice-5 (2 hours): Solution of 1D Schrodinger Equation with Finite Square-Well Potential: Eigenvalues and Eigenfunctions

Practice-6 (2 hours): Solution of 1D Schrodinger Equation with Finite Square-Well Potential: Tunneling probability of wave function

Practice-7 (2 hours): Solution of 1D Schrodinger Equation with Harmonic Oscillator potential

Practice-8 (2 hours): Solution of 1D Schrodinger Equation with Harmonic Oscillator potential, computation of eigenvalues and eigenfunctions

Module-IV: 3D Applications-I (The H-Atom and Hydrogen-like cases)

Quantum theory of Hydrogen and Hydrogen-like atoms: time independent Schrodinger equation in spherical polar coordinates; separation of variables for second order partial differential equation; angular momentum operators and quantum numbers; Radial wave functions from Frobenius method; shapes of the probability densities for ground and first excited states; Orbital angular momentum quantum numbers l and m; s, p, d,.. shells.

Practice-9 (2 hours): Solution of radial Schrodinger equation with Coulomb Potential (Hydrogen atom)

Practice-10 (2 hours): Solution of radial Schrodinger equation with Coulomb Potential (Hydrogen atom), Computation of eigenvalues and eigenfunctions in Ground State and First Excited state

Module-V: 3D Applications-II (A Single Electron Atom in External EM  Fields)

Atoms in Electric & Magnetic Fields: Electron angular momentum. Space quantization. Electron Spin and Spin Angular Momentum. Larmor’s Theorem. Spin Magnetic Moment. Stern-Gerlach Experiment. Zeeman Effect: Electron Magnetic Moment and Magnetic Energy, Gyromagnetic Ratio and Bohr Magneton. Atoms in External Magnetic Fields (Qualitative discussions only): Normal and Anomalous Zeeman Effect. Paschen-Back and Stark Effect (Qualitative Discussion only).

Practice-11 (2 hours) Numerical solution of radial Schrodinger equation for Screened Coulomb Potential (He- and other atoms)

Module-VI: 3D Applications-III (Many Electron Atoms: Alkali Atoms)

Many electron atoms: Pauli’s Exclusion Principle. Symmetric & Anti-symmetric Wave Functions. Periodic table. Fine structure. Spin orbit coupling. Spectral Notations for Atomic States. Total angular momentum. Vector Model. Spin-orbit coupling in atoms-L-S and J-J couplings. Hund’s Rule. Term symbols. Spectra of Hydrogen and Alkali Atoms (Na etc.).

Practice-12 (2 hours) Numerical solution of radial Schrodinger equation with Morse Potential for H2 molecule

Module-VII:

Technology Applications: 

Tunnel Diode; Scanning Tunnel Microscope; Magnetic Resonance Imaging (MRI); Quantum Computations with Qubits; Bose-Einstein Condensates: A case of "Macroatoms" near T=0K.

Text Books:

  1. Quantum Mechanics: Concepts and Applications by N Zettili, John Wiley & Sons Ltd.

Reference Books:

  1. A Text book of Quantum Mechanics, P.M.Mathews and K.Venkatesan, 2nd Ed., 2010, McGraw Hill

Session Plan

Session 1

Introduction to Quantum Mechanics: the What and Why Questions, The relevance of Quantum Mechanics in our day-to-day life

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

Reference:  https://www.youtube.com/watch?v=TcmGYe39XG0

Session 2

Postulates of Quantum Mechanics

Time dependent Schrodinger equation and dynamical evolution of a quantum state; Properties of Wave Function. Interpretation of Wave Function Probability and probability current densities in three dimensions

https://www.youtube.com/watch?v=kUm4q0UIpio

https://www.youtube.com/watch?v=OFwskHrtYQ4

Reference: https://www.slideshare.net/khalilkhanjatoi/postulates-of-quantum-mechanics?qid=2cd7dd59-2711-469f-887e-3a7509f1c5e7

Session 3

Assignment-1:

Checking for validity of candidate wave functions, Normalization of wave functions, Calculation of probabilty density and probability current density for given wave functions

https://www.youtube.com/watch?v=wqfPqD66hAY

https://www.youtube.com/watch?v=bXgBk0RgheQ

Session 4

Practice-1 (2 hours):

Given candidate wave functions, develop Python codes to check for valid wave functions

Session 5

Practice-2 (2 hours):

Given wave functions, develop Python codes to calculate the position probability density and the  current probability density

https://www.youtube.com/watch?v=Fvi9A_tEmXQ&t=219s

Session 6

Conditions for Physical Acceptability of Wave Functions. Normalization. Linearity and Superposition Principles. Eigenvalues and Eigenfunctions.

https://www.youtube.com/watch?v=R-5hjmV-bdY&t=287s

https://www.youtube.com/watch?v=d4skxu7MpFI

https://www.youtube.com/watch?v=hkmoZ8e5Qn0

Session 7

Assignment-2:

The mathematical foundations of Quantum Mechanics: Linear Vector Spaces, the Hilbert Space, The Quantum States, The Dirac Notations, Operators in Quantum Mechanics

https://www.youtube.com/watch?v=fVfp82FpSO8

https://www.youtube.com/watch?v=7zx3MT9FgT0

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

https://www.youtube.com/watch?v=iR7B_Eq_oTo

Session 8

Position, momentum and Energy operators; commutator of position and momentum operators;

Expectation values of position and momentum. Wave Function of a Free Particle

https://www.youtube.com/watch?v=XQKV-hpsurs

https://www.youtube.com/watch?v=Egu4i8umpoM&t=16s

https://www.youtube.com/watch?v=KvS7Z0rEutE

https://www.youtube.com/watch?v=-r0pfHPvhg8&list=PLdgVBOaXkb9Bv466YnyxslT4gIlSZdtjw&index=7

Session 9

Assignment-3 (Flipped Class):

Momentum Commutation relation, Expectation values of position, momentum and other operators

https://www.youtube.com/watch?v=-r0pfHPvhg8

https://www.youtube.com/watch?v=-h2lU2FXy48&pbjreload=101

Session 10

Assignment-4 (Flipped Class):

Expectation values of position, momentum and other operators

Session 11

Time independent Schrodinger equation-Hamiltonian, stationary states and energy eigenvalues; expansion of an arbitrary wavefunction as a linear combination of energy eigenfunctions

General solution of the time dependent Schrodinger equation in terms of linear combinations of stationary states

https://www.youtube.com/watch?v=ipXNYnO7yRk

Reference:

https://www.google.com/urlsa=t&source=web&rct=j&url=https://jakevdp.github.io/blog/2012/09/05/quantumpython/&ved=2ahUKEwj9wu62vZbpAhUUyjgGHdFfAcEQFjAAegQIAhAB&usg=AOvVaw1MeWBssXJ5VuJIMQ7_dzWL

Session 12

Assignment-5 (Flipped Class):

Solution of time independent Schrodinger Equation, Stationary states, Linear Superposition of stationary states

https://www.youtube.com/watch?v=qEgv9gL8ozo

Session 13

Application to spread of Gaussian wave-packet for a free particle in one dimension

Wave packets, Fourier transforms in position and  momentum  space wave function;

https://www.youtube.com/watch?v=dzI5PddY6eE

https://www.youtube.com/watch?v=DvFb-D1zJTA

https://www.youtube.com/watch?v=vWGP5dogNm8

Session 14

Practice-3 (2 hours):

Free Particle wave function, Gaussian wave packet and its time evolution

https://www.youtube.com/watch?v=ipXNYnO7yRk

https://www.youtube.com/watch?v=50Tla309i7o&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=33

Session 15

Position-momentum uncertainty principle, Heisenberg's Uncertainty Relation

https://www.youtube.com/watch?v=MBnnXbOM5S4

https://www.youtube.com/watch?v=YIpc4RNhuK4

Session 16

Assignment-6 (Flipped Class):

Position and Momentum space representations, Time evolution of Gaussian Wave Packets, Position-Momentum Uncertainty Relation, The Generalized Uncertainty Relations for Operators

https://www.youtube.com/watch?v=wbEqsVvKggs&t=292s

https://www.youtube.com/watch?v=6BNCU36BcvM

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

Session 17

Solution of Schrodinger Equation for bound states with an arbitrary potential, Continuity of wave function, boundary condition and emergence of discrete energy levels; application to one-dimensional problem-1.Infinite square well potential

https://www.youtube.com/watch?v=gMHkf-107Sw

https://www.youtube.com/watch?v=wfr11_JydbY

Session 18

Practice-4 (2 hours):

Solution of 1D time independent Schrodinger Equation: Infinite Square Well Potential, Emergence of quantized eigenfunctions, normalisation of wave function

Session 19

Session 20

Assignment-7 (Flipped Class):

Solution of 1D Schrodinger Equation (evaluation of eigenvalues and eigenvectors) with

(i) Infinite Square Well Potential

(ii) Finite Square Well potential

(iii) Tunneling probability

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

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

https://www.youtube.com/watch?v=jd4es6Bo600

Session 21

Practice-5:

Solution of 1D Schrodinger Equation with Finite Square-Well Potential: Eigenvalues and Eigenfunctions

Session 22

Practice-6:

Solution of 1D Schrodinger Equation with Finite Square-Well Potential: Tunneling probability of wavefunction

Session 23

Quantum mechanics of simple harmonic oscillator-energy levels and energy eigenfunctions using Frobenius method; Hermite polynomials

Reference:

.https://www.youtube.com/watch?v=sxzFpOsvfgU&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=64

.https://www.youtube.com/watch?v=eNf8nH1yEYc&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=65

Session 24

Assignment-8(Flipped Class):

Solution of 1D Schrodinger Equation with Harmonic Oscillator potential, eigenvalues and eigenfunctions

Reference:

https://www.youtube.com/watch?v=RxWfrE3o-9k&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=66

https://www.youtube.com/watch?v=Y6Ma-zn4Olk&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=67

Session 25

Practice-7:

Solution of 1D Schrodinger Equation with Harmonic Oscillator potential

Session 26

Practice-8:

Solution of 1D Schrodinger Equation with Harmonic Oscillator potential, computation of eigenvalues and eigenfunctions

Session 27

Energy Eigenstates and Eigenvalues of 1D Quantum Harmonic Oscillator,  Hermite Polynomials

https://www.youtube.com/watch?v=Xj9PdeY64rA

Session 28

1D Quantum Harmonic Oscillator Schrodinger Equation : Algebraic Solution-I

https://www.youtube.com/watch?v=8CCFPgd_P1w

Session 31

Quantum theory of the Hydrogen atom: time independent Schrodinger equation in spherical polar coordinates; separation of variables for second order partial differential equation

https://www.youtube.com/watch?v=phIc-an6B2A

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

https://www.youtube.com/watch?v=P1kuW2Ziv54

Reference:

https://www.youtube.com/watch?v=GWMeYKUvj7Y&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=104

Session 32

The Theta Equation and the Phi Equation; Solution of the Phi Equation and the Projection Quantum Number m; Solution of the Theta equation and Angular Momentum Quantum Number l

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

https://www.youtube.com/watch?v=2yvgsyLjhIU

Session 33

Assignment-9 (Flipped Class):

Algebraic Solution of Schrodinger Equation with Harmonic Oscillator potential, The Step-up, Step-down opetators; Schrodinger Equation of H-atom

Solution of the Phi- and Theta equations

https://www.youtube.com/watch?v=hF7j4pePJSU

https://www.youtube.com/watch?v=EIvkoHsUbbE&t=8s

Session 34

Angular momentum operators and quantum numbers; Angular momentum algebra

https://www.youtube.com/watch?v=NwbvTa2xV9k

https://www.youtube.com/watch?v=xoCHe0mtxu0

Session 35

Radial wavefunctions from Frobenius method, Radial wave function; Probability densities for ground and first excited states (Shape of the curves included); Orbital angular momentum quantum numbers l and m; s, p, d,.. shells

https://www.youtube.com/watch?v=KfbvrGt3MlI

https://www.youtube.com/watch?v=3VXLIF2DpHI&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=106

https://www.youtube.com/watch?v=c5yzy1S3gPg&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=107

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

Session 36

Assignment-10 (Flipped Class):

Angular Momentum Algebra, Solution of radial Schrodinger equation with Coulomb and Screened Coulomb potential, Problems involving the Principal Quantum Number (n), Angular Momentum Quantum Number (l) and Projection Quantum Number (m)

Reference:

https://www.youtube.com/watch?v=xoCHe0mtxu0&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=96

https://www.youtube.com/watch?v=Mh8vUEStCQ8&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=97

https://www.youtube.com/watch?v=lWTUcojZ_gQ&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=98

Session 37

Practice-9 (2 hours):

Solution of radial Schrodinger equation with Coulomb Potential (Hydrogen atom)

Session 38

Practice-10 (2 hours):

Solution of radial Schrodinger equation with Coulomb Potential (Hydrogen atom), Computation of eigenvalues and eigenfunctions in Ground State and First Excited state

Session 39

Atoms in Electric and Magnetic Fields: Electron angular momentum. Space quantization

Electron Spin and Spin Angular Momentum. Larmor’s Theorem

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

https://www.youtube.com/watch?v=qZ9g3LlXRug&t=21s

https://www.youtube.com/watch?v=xlkocM0Crko

Session 40

Spin Magnetic Moment. Stern-Gerlach Experiment.

https://www.youtube.com/watch?v=PH1FbkLVJU4

Reference:

https://www.youtube.com/watch?v=AX9769eQV24

Session 41

Zeeman Effect: Electron Magnetic Moment and Magnetic Energy, Gyromagnetic Ratio and Bohr Magneton

https://www.youtube.com/watch?v=OzkcB1lkgGU

https://www.youtube.com/watch?v=WlZf4aOkNMQ

Session 42

Atoms in External Magnetic Fields:- Normal and Anomalous Zeeman Effect. Effect of large magnetic field: Paschen Back effect, Zeeman Effect and Sun Spots

https://www.youtube.com/watch?v=JaEQ4ZtGLmA

Session 44

Many electron atoms: Pauli’s Exclusion Principle. Symmetric & Antisymmetric Wave Functions.

https://www.youtube.com/watch?v=Z08pNaGQPv8

Session 45

Assignment-11 (Flipped Class):

The Stern-Gerlach Experiment, The Spin Angular Momentum, Spin Quantum Numbers, Spin Quantum States (Eigenstates, Linear superposition of spin states..), Zeeman, Paschen Back and Stark Effects

Session 46

Practice-11 (2 hours):

Numerical solution of radial Schrodinger equation for Screened Coulomb Potential (He- and other atoms)

Session 47

Practice-12 (2 hours):

Numerical solution of radial Schrodinger equation with Morse Potential for H2 molecule

Session 48

Periodic table. Fine structure of Hydrogen Atom. Spin orbit coupling. Spectral Notations for Atomic States. Total angular momentum

https://www.youtube.com/watch?v=mas9avjieP0&t=32s

Session 49

Total angular momentum. the Vector Atom Model. Spin-orbit coupling in atoms-L-S and J-J couplings. Hund’s Rule. Term symbols, Spectra of Hydrogen and Alkali atoms (Na etc.)

https://www.youtube.com/watch?v=C6afrc1QS6Y

Reference:

https://www.youtube.com/watch?v=ZTNip78TUvA

Session 50

Assignment-12(Flipped Class):

Many electron atoms, Antisymmetric eigenfunctions and eigenstates, L-S Coupling and J-J coupling, Vector Model of atoms, Eigenstates of Alkali atoms.

Session 52

The Magnetic Resonance Imaging

https://www.youtube.com/watch?v=rJ9gV4yFMi8

Session 54

Bose-Einstein Condensate: A case of "Macroatoms" near T=0K

Reference:

https://www.youtube.com/watch?v=NoO7XKVmZC8

Case Studies

Case Studies

Session Plan

Session 1

I am text block. Click edit button to change this text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut elit tellus, luctus nec ullamcorper mattis, pulvinar dapibus leo.

Session 2

I am text block. Click edit button to change this text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut elit tellus, luctus nec ullamcorper mattis, pulvinar dapibus leo.

Case Studies

Case Studies

Our Main Teachers

Dr. Subrata Sarangi has a Masters’ Degree in Physics from IIT, Kanpur and a Ph.D in Nuclear Structure Theory from Physical Research Laboratory, Ahmedabad. He has 25 years’ experience in teaching at UG, PG and PhD levels. He has published over 20 peer reviewed research articles in areas of Atomic Nuclei, Nuclear Matter, Materials Sciences […]