Quantum Mechanics-I

Waves and Particles: Introduction to the fundamental ideas of quantum mechanics:

Electromagnetic  waves  and  photon,  material  particles  and matter waves, quantum description of a particle, wave packets, particle in a time-independent scalar  potential,  order  of  magnitude of  the  wavelength associated with material particles, constraints imposed by uncertainty relations, one-dimensional Gaussian wave packet: Spreading of the wave packet, stationary states of a particle in one-dimensional square potential, behavior of a wave packet at a potential step

The Mathematical Tools of Quantum Mechanics:

One-particle wave function  space,  state  space,  Dirac  notation,  representations in  the  state space,  observable,  representations, review  of  some  useful  properties  of linear operators, unitary operators, study of the  and  representations, some general properties of two observable, Q and P, whose commutator is equal to iħ, the two-dimensional infinite well

The Postulates of Quantum Mechanics:

Statement of the postulates and their  physical  interpretation, the  physical  implications of  the  Schrodinger equation, the superposition principle, particle in an infinite potential  well, study  of  the  probability current  in  some  special  case,  root-mean-square deviations of two conjugate observables, the density and evolution operators, Schrodinger and Heisenberg pictures, Gauge invariance, bound states of a particle in a potential well of arbitrary shape, unbound states of a particle in the presence of a potential well or barrier of arbitrary shape, quantum properties of a particle in a one-dimensional periodic structure

Application of The Postulates to Simple Cases: Spin ½ And Two-Level Quantum   Systems: 

Spin   ½   particles,   quantization   of   the   angular momentum, illustration of the postulates in the case of a spin ½, general study of two level systems, Pauli matrices, diagonalization of a 2×2 hermitian matrix, System of two spin ½ particles, Spin ½ density matrix, Spin ½ particle in a static magnetic field and a rotating field, Magnetic resonance

The One-Dimensional Harmonic Oscillator:

Importance of the harmonic oscillator in physics, eigenvalues and eigenstates of the Hamiltonian, mean value and root-mean-square deviations of X and P in state ǀ , Some examples of harmonic oscillators, study of the stationary states in the   representation, Hermite polynomials, solving the Eigenvalues of the harmonic oscillators by the polynomial method, study of the stationary states in the  representation, isotropic three-dimensional harmonic oscillator, charged harmonic oscillator placed in a uniform electric field, coherent  states, Normal vibrational modes of coupled harmonic oscillators, vibrational modes of an infinite linear chain of coupled harmonic oscillators, phonons, one- dimensional harmonic oscillator in thermodynamics equilibrium at a temperature T

relations,  application  to  orbital  angular  momentum,  spherical  harmonics,
rotation  operators, rotation  of  diatomic molecules, angular  momentum of stationary states of a two-dimensional harmonic oscillator, charged particle in a magnetic field and Landau levels
 

Particle in a Central Potential:

The Hydrogen atom, Stationary states of a particle in a central potential, motion of the center of mass and relative motion for a system of two interacting particles, Hydrogen atom, Hydrogen- like systems, A solvable example of a central potential: the isotropic three- dimensional harmonic oscillator, probability currents associated with the stationary states of the hydrogen atom, The hydrogen atom placed in a uniform magnetic field, para-magnetism and diamagnetism, Zeeman effect, study of some atomic orbitals, vibrational-rotational levels of diatomic molecules.