The state space of a quantum system is defined; kets and bras are introduced, and their inner and outer products are defined; adjoint, hermitian , and unitary operators are introduced; representations of states and operators on discrete and/or continuous bases are discussed; the properties of commuting hermitian operators are examined; tensor products are defined.

Applications of the postulates are discussed in a number of problems; the interaction representation is defined.

Derivation of the Born approximation and criteria for its validity; applications of the Born approximation to scattering in Coulomb and Yukawa potentials; derivation of the optical theorem; perturbative expansion of the scattering wave function and scattering amplitude; scattering in a hard-sphere potential at low and high energy; scattering in potential well and resonances; partial wave expansion of the integral equation for scattering in a central potential; scattering in a spin-dependent potential; phase shifts in Born approximation; effective range theory; phase shifts at high energy and the eikonal approximation for the scattering amplitude.

The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.

Examples of addition of two angular momenta; derivation of the addition formula for two spherical harmonics; implications of conservation of angular momentum and parity for two-body decays; addition of three angular momenta.

The lowering, raising, and number operators are defined and their properties are studied; the eigenfunctions of the harmonic oscillator Hamiltonian are derived; coherent or quasi-classical states are obtained and their properties examined. Several applications are discussed, including, among others, the isotropic harmonic oscillator in N dimensions and a model for a one-dimensional crystal.

Scattering in one dimension is discussed in terms of wave packets; reflection, transmission, and tunneling probabilities are defined; the WKB method to calculate these probabilities is introduced; the S-matrix for scattering in one dimension is defined; the phase-shift method for one-dimensional parity-invariant potentials is introduced; applications to various combinations of finite and infinite barriers with delta-function potentials are examined.

Spinor wave functions; classical Lagrangian and Hamiltonian of a charged particle in an electromagnetic field, and its quantum Hamiltonian; gauge invariance; spin magnetic moment in a uniform magnetic field; magnetic resonance; the Stern--Gerlach experiment; neutron interferometry and rotations of spinor wave functions; treatment of a particle in a uniform magnetic field with and without the inclusion of spin degrees of freedom; Ahronov--Bohm effect for a charged spinless particle confined in a cylindrical shell.

Several experimental facts cannot be explained by classical physics (Newtonian mechanics and Maxwell’s equations): the observed black-body radiation spectrum, the stability of atoms and associated spectral lines, the heat capacities of solids, and several others. The problems posed in this chapter are meant to illustrate and analyze the failure of classical physics in explaining these phenomena and how this failure points to the need for a radically new treatment.

Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.

General derivation of the eigenvalues and eigenstates of the square and z-component of the angular momentum; relationship between the angular momentum and the harmonic oscillator in two dimensions; transformation of states and operators under rotations; algebraic derivation of the hydrogen atom spectrum.

Bound states in one-dimensional finite and infinite wells and delta-function potentials, and combinations of these, are obtained; the WKB method for bound states is introduced; the consequences of a parity-invariant potential for the eigenfunctions are derived.

Exchange and permutation operators; symmetrizer and antisymmetrizer; two bosons or two fermions in a central potential; scattering amplitudes of two identical particles in a central potential; the Fermi gas; theory of white dwarf stars; the Thomas-Fermi approximation for many-electron atoms.

Wave-particle duality and the Davisson-Germer experiment are briefly discussed; wave packets are defined and their features, including phase velocity, group velocity, and spreading, are examined; the stationary phase method is presented; free-particle wave functions are introduced, and the equivalence between coordinate- and momentum-space representations of these wave functions is emphasized.

Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.

Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.

Construction ofunitary operators inducing space and time translations, and rotations; the anti-unitary operator inducing time reversal; consequences of invariance under a symmetry transformation; periodic potentials and Bloch waves; the Kronig-Penney model; the ammonia molecule and broken parity symmetry; consequences of time reversal invariance on the scattering amplitude of spinless particles; Kramers degeneracy.