 # Quantum Mechanics and Applications Teacher

##### Dr. Subrata Sarangi Category

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# 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.

## Learning Outcomes

After successfully completing this course, the student should be able to

• Solve, both analytically and numerically, time-independent and simple time-dependent Schrodinger Equations for systems of particles interacting via time-independent Central Potentials along with appropriate Boundary and Initial Conditions
• Generate appropriate visualizations of the solutions of Schrodinger Equations and interpret them
• 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.

## 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 1

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

## 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

## 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

## 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

## Session 6

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

## 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

## 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

## Session 9

Assignment-3 (Flipped Class):

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

## 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:

## Session 12

Assignment-5 (Flipped Class):

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

## 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;

## Session 14

Practice-3 (2 hours):

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

## Session 15

Position-momentum uncertainty principle, Heisenberg's Uncertainty Relation

## 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

## 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

## 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 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

## 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:

## Session 24

Assignment-8(Flipped Class):

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

Reference:

## 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

## Session 28

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

## 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

Reference:

## 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

## 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

## Session 34

Angular momentum operators and quantum numbers; Angular momentum algebra

## 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

## 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:

## 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

## Session 40

Spin Magnetic Moment. Stern-Gerlach Experiment.

Reference:

## Session 41

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

## 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

## 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

## 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.)

Reference:

## 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

## Session 54

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

Reference:

## Session 1

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## Session 2

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## Case Studies

### Our Main Teachers ### Dr. Subrata Sarangi

##### Professor
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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 […]