Elements of Mathematical Physics

Module MODULO II

Sample exam questions

1. Explain in what sense quantum mechanics can be regarded as an extension of classical mechanics, referring to Poisson brackets and the passage to operators.

2. What are the fundamental properties of self-adjoint operators in Hilbert space, and why do they represent physical observables?

3. Derive the Schrödinger equation from the fundamental postulates of the theory and discuss the differences between the Schrödinger and Heisenberg pictures.

4. State and interpret Heisenberg’s uncertainty principle, providing a concrete example of its application.

5. Solve the quantum harmonic oscillator problem using creation and annihilation operators, and explain the physical meaning of the discrete energy levels.

6. Describe the eigenvalue structure of the angular momentum operator and discuss the representation of the rotation group SO(3) in this context.

7. Analyze the radial equation for the hydrogen atom and explain the role of degeneracy of eigenvalues.

8. Introduce the basic ideas of quantum scattering theory and explain the significance of the cross-section.

9. Define spin from a mathematical point of view and discuss the consequences of Pauli’s exclusion principle for identical particles.

10. Explain how the Hartree–Fock method provides a mathematical model for understanding the periodicity of chemical elements.

In accordance with current regulations, face-to-face lessons will be offered.

Information for students with disabilities and/or learning disabilities (LDs).

To ensure equal opportunities and in compliance with current legislation, interested students may request a personal interview to plan any compensatory and/or dispensatory measures, based on their learning objectives and specific needs. Students may also contact the CInAP (Center for Active and Participatory Integration - Services for Disabilities and/or Learning Disabilities) contact teacher in our Department, Professor Patrizia Daniele.

Basic knowledge of analysis, linear algebra, and classical mechanics.

The course Quantum Mechanics for Mathematicians is designed to introduce the essential concepts of quantum theory. The first part develops the transition from classical to quantum mechanics, showing how observables and states can be represented mathematically by Hermitian matrices and self-adjoint operators. The mathematical formalism of states, the role of eigenvalues and eigenvectors, and the uncertainty principle will be examined in detail.

The dynamics of quantum systems will be treated through the Schrödinger equation and the two fundamental pictures, Schrödinger and Heisenberg. Position and momentum representations will be discussed, with attention to generalized eigenfunctions, and the passage to the classical limit will be illustrated, together with the study of central observables such as energy and angular momentum.

Special emphasis will be given to concrete systems. The free particle and the harmonic oscillator, analyzed via creation and annihilation operators, will serve as paradigmatic models. The three-dimensional particle will be studied with particular reference to rotations and representations of the group SO(3). The radial equation and the hydrogen atom will then be presented, highlighting the structure of energy levels and degeneracies.

In the final part, the course will cover advanced topics such as scattering theory, spin and the exclusion principle for identical particles, and an introduction to multi-electron systems through the Hartree–Fock method, providing a mathematical explanation for the periodicity of chemical elements.

• L.D. Faddeev, O.A. Yakubovskiī, "Lectures on Quantum Mechanics for Mathematics Students", American Mathematical Society
• Brian C. Hall, "Quantum Theory for Mathematicians", Springer
• Leon A. Takhtajan, "Quantum Mechanics for Mathematicians", American Mathematical Society 
• V. Moretti, "Spectral Theory and Quantum Mechanics", Springer
• D. Bohm, "Quantum Theory", Dover Publications 

The grade is expressed on a scale of 30 points according to the following criteria:

Fail (Not eligible)

• Knowledge and understanding of the subject: significant gaps and inaccuracies

• Analytical and synthesis skills: negligible, frequent generalizations

• Use of references: completely inappropriate

18–20

• Knowledge and understanding of the subject: very limited, with evident flaws

• Analytical and synthesis skills: barely sufficient

• Use of references: barely appropriate

21–23

• Knowledge and understanding of the subject: slightly more than sufficient

• Analytical and synthesis skills: fair ability to analyze and synthesize, arguments are logical and coherent

• Use of references: standard references used

24–26

• Knowledge and understanding of the subject: good knowledge

• Analytical and synthesis skills: good analytical and synthesis skills, arguments are presented coherently

• Use of references: standard references used

27–29

• Knowledge and understanding of the subject: more than good knowledge

• Analytical and synthesis skills: remarkable analytical and synthesis skills

• Use of references: has explored topics in greater depth

30–30 with honors

• Knowledge and understanding of the subject: excellent knowledge

• Analytical and synthesis skills: outstanding analytical and synthesis skills

• Use of references: significant in-depth exploration

Example questions