Physics Of Matter 2013/2014

Quantum Mechanics

Professor: 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, relativity principle of Galileo, properties of space and time;

         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: double pendulum.

    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 for a particle.

    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, coordinates 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 motion (bound states) and infinite

     2.3:

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

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

2.3.3: eigenvalue equations for a physical quantity in the representation of energy (stationary states); matrices in diagonal form, complete set of common eigenfunctions.

2.3.4: Transformations of matrices

2.3.4: 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 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 operator and continuity equation for the probability density.

    2.5: One-dimensional motions:

2.5.1: Schrödinger equation and general principles;

2.5.2: potential well with infinite walls; potential step; coefficients of transmission and      reflection (wave character of the particle); potential well with finite walls; 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 observable and wave functions; measurement process;

2.5.5: Harmonic oscillator: solution by means of the operators of creation and destruction.

    2.6. Angular momentum:

2.6.1: Operation of infinitesimal rotation; conservation of angular momentum; angular momentum operator;

2.6.2: commutation relations between the square modulus of the angular momentum, the components of the angular momentum, the coordinates and the momentum;

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 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 and 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 spin or half-  integer; 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 and orbital angular momentum); composition of angular momenta.

    2.9: Identical particles:

2.9.1: concepts of indistinguishable particles in quantum mechanics;

2.9.2: two identical particles of spin 1/2; ket with collective index;

2.9.3: exchange operator, the symmetric and antisymmetric states

2.9.4: Pauli principle; fermions and bosons; state of N fermions and N bosons; the wave function for fermions and 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”