Quantum And Statistical Mechanics 2012/2013

Quantum Mechanics

Daniele Di Castro

1) Elements of Classical Mechanics:

    1.1: Material point, degrees of freedom, and generalized coordinates;

         1.2.1: Hamilton’s principle

         1.2.2: inertial systems, properties of space and time, relativity principle of Galileo;

         1.2.3: Lagrangian for a free particle and for a system of non-interacting and interacting     

                   particles (potential energy);

         1.2.4: Lagrangian for a system of interacting particles in generalized coordinates;

         1.2.5: particle in an external field; constraints.

    1.3: Conservation laws:

         1.3.1: First integrals from the properties of space and time; cyclic coordinates;

         1.3.2: Conservation of energy; generalized momentum; total energy in generalized

                   coordinates

         1.3.3: conservation of momentum; center of mass;

         1.3.4: conservation of angular momentum;

    1.4:

         1.4.1: Integration of the equations of motion, solution from first integrals;

         1.4.2: one-dimensional motions; two-body problem and the reduced mass;

         1.4.3: motion in a central field: the example of Coulomb field; finite and infinite motions;

         1.4.4: free, forced, and forced and damped oscillators;

    1.5: Legendre transformations, Hamiltonian, and Hamilton's equations; Hamiltonian of a free   particle and in a presence of external field.

    1.6: Poisson brackets.

2) Quantum Mechanics

     2.1: Basic concepts of quantum mechanics:

uncertainty principle; the configurations space and the wave function; physical quantities (observables), the superposition principle, operators, eigenfunctions and eigenvalues​​(discrete and continuous spectrum), expansion coefficients, the mean value of a physical quantity; transposed operator, Hermitian conjugate operator, inverse operator; Hermitian operators; commutator of two operators and the concept of operators defined simultaneously in a state; characteristics of the continuous spectrum, coordinate operator; the concept of measurement in quantum mechanics.

      2.2:

            2.2.1: Classical limit of the wave function;

            2.2.2: Wave equation and Hamiltonian operator;

            2.2.3: Derivative of operators with respect to time and conservative physical quantities;

             2.2.4: Energy and wave function of the stationary states, discrete and continuous spectrum of energy eigenvalues​​, finite (bound states) and infinite motion.

2.3:

2.3.1 Matrices: matrix elements of an operator, Hermitian matrices, the product of       matrices;

2.3.2: Momentum: translation operator, conservation of momentum,  momentum operator, eigenvalues ​​and eigenfunctions.

2.3.3: Eigenvalue equations for a physical quantity in the representation of energy; matrices in diagonal form; complete set of common eigenfunctions.

 2.3.4: Transformations of matrices

2.3.5: Uncertainty relations.

    2.4:

          2.4.1: Hamiltonian for a system of free particles, of interacting particles, and of particles in an external field;

           2.4.2: Schrödinger equation for a free particle and corresponding eigenfunctions;

           2.4.3: classical limit of the Schrödinger equation;

          2.4.4: basic properties of the Schrödinger equation and of the wave function solution of the equation; properties of the ground state;

          2.4.5: current density vector and continuity equation for the probability density.

    2.5: One-dimensional motions:

           2.5.1: Schrödinger equation and general principles;

           2.5.2: infinite potential well; potential step; coefficients of transmission and reflection (wave character of the particle); finite potential well; potential barrier and tunnel effect;

          2.5.3: definition and properties of the Dirac notation: bra and ket; discrete spectrum and the continuous spectrum, position physical quantity and wave functions; measurement process;

            2.5.4: Harmonic oscillator: solution by means of the creation and annihilation operators.

2.6. Angular momentum:

2.6.1: Rotation operator; conservation of angular momentum; angular momentum operator;

2.6.2: commutation relations;

2.6.3: angular momentum in polar coordinates; eigenvalues ​​and eigenfunctions of the angular momentum along z, ℓz; lowering and raising operators; eigenvalues ​​of the square modulus of the angular momentum ℓ2.

2.6.4: matrix elements of the raising and lowering operators and of the components of the angular momentum.

2.6.5: eigenfunctions of angular momentum: eigenfunctions of â„“z and â„“2 and spherical harmonics.

  2.7: Motion in a central field:

           2.7.1: the two-body problem: the reduced mass; reduction to a motion in a central field; Schrödinger equation for the radial part of the wave function, effective potential (centrifugal barrier);

           2.7.2: motion in Coulomb field: hydrogen atom; eigenfunctions and eigenvalues.

    2.8: Spin:

2.8.1: concept of spin; commutation relations; eigenvalues ​​of Sz and S2; integer or half-  integer spin; spinors; matrix elements of the components of the spin: Pauli matrices for the spin operator in two dimensions; eigenket of Sz as a basis;

2.8.2: total angular momentum (spin plus orbital angular momentum); composition of angular momenta.

    2.9: Identical particles:

          2.9.1: concepts of indistinguishable particles in quantum mechanics;

          2.9.2: spin states for two identical particles of spin 1/2;

          2.9.3: exchange operator, the symmetric and antisymmetric states;

          2.9.4: fermions and bosons; Pauli principle; state and wave functions of N fermions and N bosons;

Reference books:

Part 1): L. D. Landau & E. M. Lifshitz, “Mechanics”; H. Goldstein: “Classical Mechanicsa”

Part 2): L. D. Landau & E. M. Lifshitz, “Quantum Mechanics: non relativistic theory”;

             S. Gasiorowicz, “Quantum Physics”; J. J. Sakurai, “Modern Quantum Mechanics”

Statistical Mechanics

Andrei Varlamov

1. The main principles of Statistical Physics

Statistical distribution, Liuville theorem, statistical matrix and statistical distributions in quantum mechanics.

1. Gibbs distribution

Ideal gas: Boltzmann and Maxwell distributions. Quantum gases: Fermi-Dirac and Bose-Einstein distributions. Fermi surface of the ideal Fermi gas. Heat capacity of the Fermi gas.  Phenomenon of the Bose-condensation. Black radiation.

1. Solids

Lattice heat capacity. Thermal expansion. Classical treatment of the lattice vibrations. Phonons.

1. Phase transitions of the II order

Landau theory of  phase transitions. Order parameter fluctuations.

1. Superconductivity

General description and historical review.  Ginzburg-Landau phenomenology. Flux quantization. Superconductivity of the second type. Evaluation of the lower and upper critical fields (8 hours).