. – Quantum Mechanics
PROFESSOR FAUSTO BORGONOVI
Text under revision. Not yet approved by academic staff.
COURSE AIMS
To understand the conceptual crisis and fundamental experiments that led to the
formulation of quantum mechanics. To understand its axiomatic bases. To learn to
solve nonrelativistic quantum mechanical problems using perturbative and nonperturbative methods.
COURSE CONTENT
1 - The crisis of classical physics: Photoelectric effect, specific heat of solids, black
body, atomic spectra, De Broglie hypothesis and Bohr model.
2 - Schrödinger equation: Wave–particle duality. Statistical interpretation.
Stationary state equation. Norm conservation. Current density. Free and bound
states. Position, momentum and energy observables. Properties of operators
associated with observables.
3- Uncertainty principle: Compatible and incompatible observables. Minimum
uncertainty package and relationship with the uncertainty principle. Thought
experiments.
4 – Solvable models: The free particle. Spectrum, improper eigenfunctions and
comparison with the classical case. Piecewise constant potential, potential barrier.
Reflection and transmission coefficient. Potential step. Resonance scattering. The
harmonic oscillator: eigenvalues and eigenfunctions. Creation and annihilation
operators. Coherent states: properties and classical limit. Two-body problem:
classical motion. Kepler problem. Angular equation (spherical harmonics). Radial
equation. Coulomb potential (hydrogen atom). Bound states.
5- The Stern–Gerlach experiment and spin: matrix representation of spin operators.
Commutation rules.
6 - The physical foundations and formal rudiments of quantum mechanics: The
general principles of the theory: observables and operators. States and
representations. Dirac notation. Sets of compatible observables and maximum
information on the state of a system. Position and momentum operators. The
translation operator. Discrete and continuous spectra. The time evolution operator.
Schrödinger and Heisenberg representations. Ehrenfest theorem. Conserved
quantities. Correlation amplitude and time-energy uncertainty relation.
7 - Perturbation theory: time-independent for a discrete spectrum (degenerate and
non-degenerate). Time-dependent. Interaction picture. Dyson expansion. Twostate problem. Rabi oscillations. Fermi's golden rule. Time-periodic potential.
Variational method.
8 - General theory of angular momentum: eigenfunctions and eigenvalues.
Spherical harmonics, properties. Comparison with the classical case. Addition of
angular momenta. Clebsch-Gordan coefficients.
9 – Identical particles: exchange operator. Properties. Completely symmetric wave
function and completely anti-symmetric wave function. Slater determinant. Spinstatistics connection. Bosons and fermions. Pauli exclusion principle.
READING LIST
1. L.D. LANDAU - L. LIFSHITZ, Quantum Mechanics, Dover New York, 2000.
2. C. COHEN-TANNOUDJI - B. DIU - F. LALOE, Quantum Mechanics, Vol. I, II, Wiley and Sons,
Paris, 2005.
3. P. CALDIROLA - R. CIRELLI - G.M. PROSPERI, Introduzione alla fisica teorica, Utet.
4. J. SAKURAI, Meccanica quantistica moderna, Zanichelli, Bologna, 1996.
TEACHING METHOD
Lectures (66 hours) and exercises (30 hours).
ASSESSMENT METHOD
Written and oral examination.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.
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