Quantum Mechanics-I Problem Set

Concise collection of assignment questions on following topics in Quantum Mechanics-I. FAILURE OF CLASSICAL PHYSICS - blackbody radiation, atomic spectroscopy, photoelectric effect, Compton effect, de Broglie’s hypothesis, Davisson-Germer experiment, double slit experiment. WAVE STATE - Born interpretation and probability density, linear superposition, uncertainty principle, time-dependent and time-independent Schrodinger equation, free particle Gaussian, wavepackets and Fourier transform. 1-D PROBLEMS - stationary states, boundary conditions, step, barrier, well, and delta potential. SIMPLE HARMONIC OSCILLATOR - non-dimensionalisation, asymptotic analysis, Hermite polynomials, quantisation, generating function, Rodrigues’ formula, integral form, orthonormalisation. MATHEMATICAL PRELIMINARIES - wavefunction space, linear operators, closed orthonormal basis, Hilbert and dual spaces, Dirac bra-ket notation, projection, hermitian, adjoint, and unitary operators, matrix representation, eigenfunctions and eigenvalues of observables, complete set of commuting observables, position and momentum space representation. POSTULATES - probabilities and expectation values, probability current and continuity equation, Ehrenfest theorem, constants of motion, Bohr frequencies. DYNAMICS - Hamiltonian and time evolution unitary operator, Schrodinger, Heisenberg, and interaction pictures, momentum and space translation unitary operator, commutativity and invariance/conservation, parity operator, anti-linear time reversal operator. SIMPLE HARMONIC OSCILLATOR - scaling transformation, raising and lowering operators, eigenvalues and eigenfunctions of modified Hamiltonian, matrix representation, mean and rms values of position and momentum, stationary state time evolution, correspondence principle. SPIN 1/2 - spin vector eigenvalues and eigenfunctions, uniform magnetic field and Larmor precession, two-level system, perturbation and couple d Rabi oscillations. ANGULAR MOMENTUM - commutation relations, common eigenstates of L2 and Lz, angular momentum component matrices, orbital angular momentum operator in spherical coordinates, orthonormalised spherical harmonics, rigid rotator. 3-D PROBLEMS - Cartesian coordinates, free particle, rectangular box potential, and anisotropic harmonic oscillator, spherical coordinates and central potential, Bessel and Neumann radial wavefunctions of free particle and spherical well, two body problem, associated Laguerre radial polynomials, quantum numbers of hydrogen atom.