LEARNING OUTCOMES

At the end of this lecture course, the student will have developed a good understanding of quantum mechanics basic principles, as well as the theory's mathematical foundations. The student will be able to apply this knowledge to solve concrete problems involving physical phenomena at the atomic scale, both exactly and by using approximation methods

AIMS AND LEARNING OUTCOMES

At the end of these courses the student will be able to

1. Identify the physical contexts which require a quantum description of the relevant phenomena;
2. Compute the probabilities of simple quantum mechanical processes involving systems with finite number of states;
3. Compute the tunnelling probabilities for potential barriers in one spatial dimension; 
4. Compute the spectrum of angular momentum (spin) operators for both simple and composed systems;
5. Solve the Schrödinger equation for a system of two particles interacting with a central potential to compute the energy spectrum of hydrogen atom;
6. Compute wave functions and energy spectra of systems composed by identical particles; 
7. Relate the laws of motion of classical mechanics to those of quantum mechanics, using both the WKB method, and the variational method;
8. Calculate the time-independent perturbation to the spectrum of a known Hamiltonian;
9. Determine a transition amplitude by the theory of time-dependent perturbations;
10. Express the cross section in terms of a transition amplitude;
11. Determine the density matrix for a given statistical mixture and use it to calculate an average value.

 

PREREQUISITES

 

PREREQUISITES

Basic knowledge of classical mechanics and analytical mechanics, mathematical analysis, geometry and linear algebra.

TEACHING METHODS

The course is delivered through frontal lectures that include

• Blackboard presentation
• Problems solved by the lecturers at the blackboard
• Problem solved by the students and discussed all together

 

SYLLABUS/CONTENT

Part A:

A1 The crisis of classical physics 

1.1 Atomic models

1.2 Photoelectric effect and photons

1.3 Compton effect

1.4 Atomic absorption and emission spectra 

 

A2 The old quantum theory 

2.1 Bohr atomic model

2.2 De Broglie wavelength and wave-particle duality 

2.3 Bohr-Sommerfeld quantization rule

2.4 Davisson e Germer experiment

2.7 Particle interference

 

A3  The formalism of quantum mechanics 

3.1 Superposition principle: states and vectors

3.2 Scalar products and transition probabilities

3.3 Observables, operators and eigenvector basis

3.4 Compatible and incompatible observables

3.5 Equivalent representations and unitary transformations

3.6 Quantum systems with finite basis 

 

A4  Particle Quantum mechanics 

4.1 Uncertainty relations

4.2 Canonical relations

4.3 Continuous spectrum: generalized eigenstates and observables

4.4 Coordinates and momenta representations 

4.5 Wave packets

4.6 Schrödinger equation 

 

A5 Temporal evolution 

5.1 Schrödinger and Heisenberg pictures

5.2 Time evolution of a Gaussian packet 

5.3 Continuity equation

5.4 Collective interpretation of the wave function

 

A6 One-dimensional Schrödinger equation 

6.1 Free particle

6.2 Particle in a box

6.3 General properties of energy eigenfunctions in 1 dimension 

6.4 Step potential

6.5 Square potential well

6.6 Potential barrier: transmission and reflection coefficients 

6.7 Tunnel effect: semi-classical limit. Alpha decay

6.8 Harmonic oscillator: creation and destruction operators 

 

A7 Symmetries

7.1 Translations and rotations

7.2 Discrete translations: Bloch theorem

7.3 Angular momentum and its representations

7.4 Spin

7.5 Addition of angular momenta

7.6 Scalar and vector operators

7.7 Harmonic polynomials and spherical harmonics

7.8 Schrödinger equation in central potential

7.9 The levels and the energy eigenfunctions of the hydrogenoides

 

A8 Atoms and Molecules 

8.1 Fundamental level of Helium

8.2 Excited levels of Helium 

8.3 Identical particles: bosons and fermions

8.4 Pauli principle

8.5 Zeeman effect

8.6 Hydrocarbons

 

Part B:

 

B.I: Advanced topics in QM (3 weeks)

 

1. Symmetries in Quantum Mechanics

2. Selection rules for scalar and vector operator. Wigner-Eckhart theorem.

3. Mixed-states, density matrix

 

B.II: Approximation methods (4 weeks)

1. Time-independent perturbation theory: degenerate and non-degenerate case. 

2. Applications to hydrogenoids: fine and hyperfine structure, Zeeman effect. 

3. Variational methods

4. WKB approximation

 

B.III: Time-dependent Hamiltonians and Scattering (4 weeks)

1. Formal solution of the Schrodinger equation with time-dependent hamiltonians. Dyson series. Exact solutions: spin resonance. 

2. Time-dependent perturbation theory. Interactions with classical radiation.  The adiabatic approximation.

3. Elastic scattering: time-dependent and time-independent formalism. Lippmann Schwinger equation. Born approximation.

4. Partial waves expansion.

 

B.IV: Towards Relativistic Quantum Mechanics (1 week)

1. Klein-Gordon and Dirac equation.

RECOMMENDED READING/BIBLIOGRAPHY

 

 

•  L. E. Picasso, "Lezioni di Meccanica Quantistica", (Edizioni ETS, Pisa, 2000)
•  Richard Phillips Feynman, Robert B. Lieghton and Matthew Sands, "The Feynman Lectures on Physics", Vol 3 (Quantum Mechanics),(1966)  (edizione on-line http://www.feynmanlectures.caltech.edu)
•  L.D. Landau, E.M. Lifsits, vol. 3: "Meccanica Quantistica", Editori Riuniti
• Griffiths, Schroeter, “An introduction to Quantum Mechanics", 3rd edition
• Sakurai and Napolitano, “Modern Quantum Mechanics", 3rd  edition
• S. Weinberg, "Lectures on Quantum mechanics", ed. Cambridge