Foundations of Quantum Mechanics (PW-FQM)5 ECTS
(englische Bezeichnung: Foundations of Quantum Mechanics)
(Prüfungsordnungsmodul: Foundations of Quantum Mechanics)

Lehrveranstaltungen:

• • Foundations of Quantum Mechanics
    (Vorlesung, 2 SWS, Florian Marquardt, Mo, 18:00 - 19:30, Raum n.V.)

Empfohlene Voraussetzungen:

The lectures require knowledge as obtained in a standard first course on Quantum Mechanics (more background will be beneficial but not absolutely needed). Master-level students and PhD students (as well as postdocs) will probably get the most out of this course.

Inhalt:

Content
This lecture series addresses questions related to the foundations of Quantum Mechanics.
Topics will include:

• Bell’s inequalities and Entanglement Measurements

• Decoherence and the quantum-to-classical crossover

• Interpretations of Quantum Mechanics (including Bohm’s pilot wave and Nelson’s Stochastic Quantization)

• Extensions of Quantum Mechanics (for example “spontaneous localization”)

• Geometric phases (Aharonov-Bohm effect and all that)

• Further aspects, e.g. concerning foundational questions in quantum electrodynamics
  

Lernziele und Kompetenzen:

Learning goals and competences:
Students

• explain the relevant topics of the lecture

• apply the methods to specific examples

Literatur:

Books on the Foundations of Quantum Mechanics

• J. S. Bell: Speakable and Unspeakable in Quantum Mechanics – fantastic book collecting many highly original papers by the inventor of Bell’s inequalities

• Joos, Zeh et al. Decoherence and the Appearance of a Classical World in Quantum Theory

• D. Bohm: Quantum Theory - introduction to quantum mechanics with emphasis on foundations

• Everett’s thesis “On the foundations of quantum mechanics” in the Princeton library catalogue and online

Some of the original papers on quantum theory

• On the Theory of Quanta, PhD thesis of Louis-Victor de Broglie (1924), English translation by A. Kracklauer: Download e-book
  

• Heisenberg’s original matrix mechanics - This is the work that created the modern theory of quantum mechanics (Heisenberg 1925). Heisenberg wanted to tackle the question of how to predict correctly the intensities of atomic transition lines, as Bohr had already clarified how to obtain the transition frequencies. Heisenberg began by noticing that, according to Bohr, the correct quantum transition frequencies do not depend just on the current state of motion (as do the frequencies of emitted radiation for a classical orbit), but rather on two states (initial and final). Likewise, in classical theory, the intensities of emitted radiation would be given by the squares of the Fourier amplitudes of the oscillating dipole moment for a given orbit. In an ingenious step, Heisenberg then postulated that instead of a set of Fourier amplitudes for a given orbit (enumerated by one index), one would have to introduce a set of amplitudes depending on two indices, one for the initial, the other for the final state. He assumed that the equations of motion for those amplitudes looked formally the same as in classical theory (Heisenberg equations of motion). The last crucial ingredient is the commutation relation. This he derived by looking at the linear response of an electron to an external perturbation (essentially deriving something like Kubo’s formula, containing the commutator) and then demanding that the short-time response would be always that of a free, classical electron. This fixes the commutator between position and momentum. Thus was born matrix mechanics. He applied this immediately to the harmonic oscillator and also dealt with the anharmonic oscillator using perturbation theory. See also Heisenberg’s Nobel Lecture from 1933 to learn more about his view on these developments, and the slightly earlier overview (Heisenberg 1928 (Naturwissenschaften)) that also includes much of the developments before matrix mechanics.
  

• The formalism of matrix mechanics - The formalism of matrix mechanics was then developed fully by Born, Jordan and Heisenberg (Born, Jordan and Heisenberg 1926). They discuss: canonical transformations, perturbation theory, angular momentum, eigenvalues and eigenvectors. In addition to the formalism, that work also contains the earliest discussion of a quantum field theory: A linear chain of masses coupled by springs is quantized and solved by going over to normal modes. As a result, they find the Planck spectrum of thermal equilibrium, as a direct consequence of the newly developed quantum mechanics!
  

• The hydrogen atom in matrix mechanics - Wolfgang Pauli (Pauli 1926) managed to apply the new matrix mechanics to the hydrogen atom. He found the correct energy spectrum, as well as the correct Stark effect corrections to the energy in an applied electric field. In this solution, he makes use of the Runge-Lenz vector which is an additional conserved quantity known from classical mechanics for the Kepler problem, denoting the orientation of the elliptical orbit in space.
  

• The Schroedinger equation - Shortly after Heisenberg’s work, Schroedinger came up with the equation that now carries his name. The essential idea was to start from the Hamilton-Jacobi equation, claim the action is the logarithm of some wave function psi (think WKB!), and derive a quadratic form of psi that is to be extremized (Schroedinger equation from the variatonal principle). This leads to the stationary Schroedinger equation, which he then solves for the hydrogen atom, as well as for the harmonic oscillator, the rotor and the nuclear motion of the di-atomic molecule (Schroedinger 1926a and Schroedinger 1926b).

Organisatorisches:

Discussion Group: https://groups.google.com/g/foundations-of-quantum-mechanics-202021

Weitere Informationen:

Verwendbarkeit des Moduls / Einpassung in den Musterstudienplan:

1. Physics (Master of Science)
   (Po-Vers. 2018w | NatFak | Physics (Master of Science) | Gesamtkonto | Physics elective courses | Foundations of Quantum Mechanics)

Studien-/Prüfungsleistungen:

Foundations of Quantum Mechanics (Prüfungsnummer: 71461)

Prüfungsleistung, Klausur, Dauer (in Minuten): 90, benotet, 5 ECTS

Anteil an der Berechnung der Modulnote: 100.0 %

Prüfungssprache: Englisch

Erstablegung: WS 2020/2021, 1. Wdh.: WS 2020/2021 (nur für Wiederholer)

1. Prüfer:

Florian Marquardt