Quantum Mechanics-Basic concepts of quantum mechanics

Basic concepts of quantum mechanics

Quantum Mechanics

Basic concepts of quantum mechanics

A brief summary

Basic concepts of quantum mechanics and mathematical preliminaries, Eigenvalues, expectation values, Measurement and the uncertainty principle, Time evolution of wavepackets, 1-dimensional potential well problems, Simple harmonic oscillator Central force problems, Orbital angular momentum and spin, Hydrogen atom.



What you will learn

Motivations for studying quantum mechanics., Basic principles of quantum mechanics,Probabilities and probability amplitudes. Linear vector spaces , bra and ket vectors. Completeness, orthonormality, basis vectors. Orthogonal, Hermitian and Unitary operators, change of basis. Eigenvalues and expectation values, position and momentum representation. Measurement and the generalized uncertainty principle. Schrodinger equation, plane wave solution. Probability density and probability current Wavepackets and their time evolution. Ehrenfest relations. 1-dimensional potential well problems, particle in a box. Tunnelling through a potential barrier. The linear harmonic oscillator; Operator approach. The linear harmonic oscillator and the Hermite polynomials. Coherent states and their properties. Application to optics. Other interesting superpositions of basis states such as squeezed light. Motion in 3-dimensions; The central potential problem. Orbital angular momnetum and spherical harmonics.  Hydrogen atom ; its energy eigenvalues and eigenfunctions. Additional symmetries of the hydrogen atom. The deuteron ; Estimation of the size of the deuteron . The isotropic oscillator, energy degeneracy. Invariance principles and conservation laws. Spin and the Pauli matrices. Addition of angular momentum.  The spin-orbit coupling and its consequences. Charged particle in a uniform magnetic field; Energy eigenvalues and eigenfunctions. The Schrodinger, and Heisenberg pictures, Heisenberg equations of motion. The interaction picture. The density operator; pure and mixed states, with examples. An introduction to perturbation theory; its relevance, and physical examples. Time-independent perturbation theory : non-degenerate case. Time-independent perturbation theory:degenerate case. Time- dependent perturbation theory; atom- field interactions and the dipole approximation. Examples of time-dependent calculations . Summary of non-classical effects surveyed in the course.

Course Fee: Free

Certification By: Not Certified

Section 1

Quantum Mechanics -- An Introduction, Linear Vector Spaces - I, Linear Vector Spaces - II: The two-level atom, Linear Vector Spaces - III: The three-level atom, Postulates of Quantum Mechanics - I, Postulates of Quantum Mechanics - II, The Uncertainty Principle, The Linear Harmonic Oscillator, Introducing Quantum Optics, An Interesting Quantum Superposition: The Coherent State

Section 2

The Displacement and Squeezing Operators, Exercises in Finite Dimensional Linear Vector Spaces, Exercises on Angular Momentum Operators and their algebra, Exercises on Quantum Expectation Values, Composite Systems, The Quantum Beam Splitter, Addition of Angular Momenta - I, Addition of Angular Momenta - II, Addition of Angular Momenta - III, Infinite Dimensional Linear Vector Spaces

Section 3

Square-Integrable Functions, Ingredients of Wave Mechanics, The Schrodinger equation, Wave Mechanics of the Simple Harmonic Oscillator, One-Dimensional Square Well Potential: The Bound State Problem, The Square Well and the Square Potential Barriernit, The Particle in a one-dimensional Box, A Charged Particle in a Uniform Magnetic Field, The Wavefunction: Its Single-valuedness and its Phase

Section 4

The Central Potential, The Spherical Harmonics, Central Potential: The Radial Equation, Illustrative Exercises -I, Illustrative Exercises -II, Ehrenfest's Theorem, Perturbation Theory - I, Perturbation Theory - II, Perturbation Theory - III, Perturbation Theory - IV, Time-dependent Hamiltonians, The Jaynes-Cummings model

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