Quantum Mechanics P

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Quantum Mechanics

P

^ROF

. F

^AUSTO

B

^ORGONOVI

COURSE AIMS

Analysis of the pre-quantum physics. Understanding simple one dimensional quantum problems. To introduce the fundamental concepts and the mathematical formalism of quantum mechanics.

COURSE CONTENT

1 The crisis of classical physics : Photoelectric effect. Compton effect. Specific heat of solids. Black-body radiation. Atomic spectra.

Bohr model.

2 – The Schrödinger equation: Wave-particle dualism. Statistical interpretation. Stationary states. Probability density and current density. Free and bound states. Free particle. Position, momentum and energy. Operators and observables.

3 – Heisenberg uncertainty relations: Compatible and non- compatible observables. Spreading of wave packets. The minimum uncertainty wave-packet. Ideal experiments.

4 – Simple Solvable Models : Potential step. Transmission and reflections, Square well : bound states, free states and resonances.

Two-body problem. The Hydrogen atom. The harmonic oscillator.

5 - General theory: Observable and operators. Dirac notation.

Commutation rules. Compatible observables. Position and momentum. The time evolution and the translation operator.

Continuous and discrete spectrum. Schrödinger and Heisenberg

representation. Ehrenfest theorem. Constants of motion. Decay of

Correlations and Time-Energy uncertainty.

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6 – Perturbation Theory : Time-independent and time-dependent.

Fermi Golden Rule. Two-States problem, Rabi Oscillations.

Variational Method.

7 – Theory of Angular Momentum : Eigenfunctions and Eigenvalues. Spherical Harmonics. Addition of Angular Momenta.

Clebsch-Gordan coefficients.

READING LIST

1. J. S

AKURAI

, Modern Quantum Mechanics, Addison-Wesley Publishing, 1984.

2. A. M

ESSIAH

, Quantum Mechanics, Dover Publ., New York, 2000.

3. L.D.L

ANDAU

- L.L

IFSHITZ

, Quantum Mechanics: non-relativistic theory, Vol.3, Butterworth-Heinemann, third edition, 1981.

4. C. C

OHEN

-T

ANNOUDJI

- B.D

IU

- F.L

ALOE

, Quantum Mechanics, vol. I, II, Wiley and Sons, Paris, 2005.

TEACHING METHOD

Theory (60 hours) and Exercises (20 hours).

ASSESSMENT METHOD

At the end of the course there will be both a written and an oral test aiming to verify the students’ learning outcomes.

The written exam will include some simple Quantum Mechanics

exercises, aiming to verify the candidate’s acquisition of the

techniques and skills necessary to their solution. Essentially, students

will have to understand which approximation schemes are to be

employed concerning the physics of the proposed problems. The

assessment of the written test will not only take into account the

accuracy of the results but also the procedures adopted to elaborate

them, together with the logical scheme within which the problems

themselves have been systemised and reduced to mathematical

models.

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During the oral test, the results of the written part will be shortly discussed. Subsequently, the candidate will be let free to choose a topic to expose. The evaluation of the oral part will be based on the candidate’s explanatory skills, their acquisition of the formal language, and their ability to organise an articulated argument. The candidate will then be asked a series of questions aiming to verify:

their acquisition of the main topics dealt with during the course; their mastery of the concerned subjects; their ability to connect topics and concepts analysed throughout the course. The assessment of the oral test will take into consideration the following aspects: the accuracy of the illustrated procedures, their logical and methodological rigour, the explanatory efficacy and accuracy. The final evaluation will also reward the candidate’s assimilation of the concepts and their own personal elaboration.

NOTES

Further information can be found on the lecturer's webpage at http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.

Contacts : fausto.borgonovi@unicatt.it