Quantum Mechanics and Applications
Teacher
Dr. Subrata Sarangi
Category
Core Courses
Course Attendees
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Course Reviews
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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.
Course Outcomes
COs
|
Course outcomes
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Mapping COs with POs (High-3, Medium-2, Low-1)
|
CO1
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Solve, both analytically and numerically, time-independent and time-dependent Schrodinger Equations |
PO1-3, PO2-3
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CO2
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Generate appropriate visualizations of the solutions of Schrodinger Equations and interpret them |
PO2-3, PO5-2
|
CO3
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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:
- Quantum Mechanics: Concepts and Applications by N Zettili, John Wiley & Sons Ltd.
Reference Books:
- 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
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.
https://www.youtube.com/watch?v=R-5hjmV-bdY&t=287s
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
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
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 |
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; |
https://www.youtube.com/watch?v=dzI5PddY6eE
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
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
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 19
The case of Finite Square-Well Potential and Quantum Tunnelling
https://www.youtube.com/watch?v=WPZLRtyvEqo
https://www.youtube.com/watch?v=RF7dDt3tVmI
Reference:
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
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
Session 28
1D Quantum Harmonic Oscillator Schrodinger Equation : Algebraic Solution-I
Session 29
1D Quantum Harmonic Oscillator Schrodinger Equation : Algebraic Solution-II
https://www.youtube.com/watch?v=vnyxYtj0mfE&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=69
Session 30
Ground state, zero point energy and uncertainty principle. for 1D Simple Harmonic Oscillator
https://www.youtube.com/watch?v=Y6Ma-zn4Olk&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=67
https://www.youtube.com/watch?v=vnyxYtj0mfE&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=69
Reference for Further Reading:
https://www.youtube.com/watch?v=kefsxztSX74&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=70
https://www.youtube.com/watch?v=xmjvqbYvY9o&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=71
https://www.youtube.com/watch?v=BRFekCz4XQY&list=PLUl4u3cNGP60cspQn3N9dYRPiyVWDd80G&index=72
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 |
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
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
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
Session 40
Spin Magnetic Moment. Stern-Gerlach Experiment.
https://www.youtube.com/watch?v=PH1FbkLVJU4
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 43
Atoms in external electric field: the Stark Effect
Session 44
| Many electron atoms: Pauli’s Exclusion Principle. Symmetric & Antisymmetric Wave Functions. |
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.)
https://www.youtube.com/watch?v=C6afrc1QS6Y
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 51
The Tunnel Diodes, The Scanning Tunneling Microscope
https://www.youtube.com/watch?v=hNzLQdFW-FI
Session 53
Session 54
Bose-Einstein Condensate: A case of "Macroatoms" near T=0K
Reference:
Case Studies
Case Studies
Session Plan
Session 1
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Session 2
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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 […]