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 a
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
2.3.4: 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 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;
2.5.5: 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 ℓz and ℓ2 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 halfinteger 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”