After a qualitative introduction to quantum mechanical ideas, using optical analogies to familiarize the student with the new cocepts introduced, we will present, in a systematic fashion, the mathematical tools and postulates of quantum mechanics. The physical consequences of such postulates will be illustrated by means of simple model systems, such as two-level systems and the harmonic oscillator.

Course program:

1) Introduction to the fundamental ideas of quantum mechanics: light quanta, the Planck-Eistein relations, wave-particle duality, double-slit experiment, the de Broglie relations, wave packets, Schrödinger equation, phase velocity and group velocity, uncertainty relation for waves, Heisenberg uncertainty principle.

2) Solution of Schrödinger equation for one-dimensional problems: free evolution of a Gaussian wave packet, time-independent Schrödinger equation, stationary states, transmission and reflection coefficients, evanescent waves, delay time, bound states, scattering matrix. These concepts will be illustrated by means of model systems such as the potential step, the square potential barrier, the square well potential, the δ-function potential, the double δ-function potential, the harmonic oscillator.

3) The mathematical tools of quantum mechanics: one-particle wave function space, linear operators, discrete orthonormal bases and continuous orthonormal “bases”, closure relation, Dirac notation, kets and generalized kets, representations in the Hilbert space, change of representations in Dirac notation, observables, complete sets of commuting observables, the coordinate and momentum representations, Schrödinger equation in the coordinate and momentum representations, tensor product of Hilbert spaces, quantum non separability , tensor product of operators.

4) The postulates of quantum mechanics: description of the states of a system, the superposition principle, observables, probability of the results of a measurement, reduction of the wave packet, dynamical evolution of a quantum system, mean value and standard deviation of an observable, Robertson’s theorem, compatibility of observables, preparation of a quantum state, the time evolution operator, time evolution of the mean value of an observable, the Ehrenfest theorem, evolution of the operators in the Heisenberg picture, Bohr frequencies, the time-energy uncertainty relation.

5) Application of the postulate of quantum mechanics to model systems: two-level systems, the Bloch sphere representation, the Stern-Gerlach experiment, dynamical evolution of a spin ½ particle in a static magnetic field, Larmor frequency, dynamical evolution of a spin ½ particle in a static and in an oscillating magnetic field, Rabi oscillations, resonance condition, entangled states, Einstein-Podolsky-Rosen paradox and Bell’s inequalities, pure states and mixed states, the density matrix, computation of eigenvalues and eigenstates of the harmonic oscillator by means of the creation and annihilation operators.