The Schrodinger Equation: some historical experiments, wave-particle duality, Schrodinger equation, interpretation of wave function, the probability current, normalization condition, Ehrenfest’s theorem; Time Independent Schrodinger Equation: separation of variable, time-independent Schrodinger equation, stationary states, infinite potential well, free particle, phase velocity, and group velocity, bound state and scattering state; Potential Barrier and Tunneling: delta function potential, potential step, potential barrier, quantum tunneling; Quantum Harmonic Oscillator: review of the classical harmonic oscillator, algebraic approach to diagonalize harmonic oscillator, creation, and annihilation operator, commutation relation, differential operator approach to solve harmonic oscillator, Hermite polynomials as eigenfunctions; Mathematical Formalism of Quantum
Mechanics: Hilbert space, linear algebra in bra-ket notation, Hermitian operators, properties, determinate state, eigenvalue problem, generalized statistical interpretation; The Uncertainty Principle: proof of generalized uncertainty principle, Heisenberg’s uncertainty principle, energy-time uncertainty, minimum uncertainty state, coherent state; Schrodinger Equation in Three Dimensions: separation of variables for central potentials, radial equation, Hydrogen atom, angular equation, spherical harmonics; Angular Momentum: commutation relations, quadratic Casimir, quantization of angular momentum, spherical harmonics as eigenfunctions of angular momentum operator; Spin: intrinsic angular momentum, commutation relation, SU(2) algebra, quadratic Casimir, quantization of spin angular momentum, bosons and fermions; Electron in Magnetic Field: Hamiltonian, Larmour precision, Stern-Gerlach experiment explained, the addition of angular momenta, singlet, and the triplet, Clebsch-Gordon coefficient; Time Independent Perturbation Theory: non-degenerate perturbation theory, degenerate perturbation theory; Applications of Time Independent Perturbation Theory: fine structure of Hydrogen, the Zeeman effect, hyperfine splitting of Hydrogen.