Fachbereich Mathematik

# Seminar Topics in Quantum Mechanics

Winter semester 2024/25

• Taught by: Prof. Dr. Roderich Tumulka
• Language of presentations: English
• Studiengänge: This seminar is offered primarily for the master's program in mathematical physics but is open to all study programs.
• Prerequisites: Basic knowledge in quantum mechanics, analysis, linear algebra, and probability theory is expected.
• Talks: 90 minutes, one talk per week
• Maximum number of participants: 14
• Some possible topics (each with a suggested title and a short description of the goals):
• The mathematics of tensor products and traces
To introduce the concepts of the tensor product of 2 or N Hilbert spaces, the trace of an operator, and the partial trace in a precise manner, formulate the basic theorems about them, and possibly discuss the proofs.
• Density matrices
To introduce the concepts of statistical density matrix and reduced density matrix, explain from the main theorem about POVMs why they are relevant, discuss their properties and relations to Bohmian mechanics and collapse theories.
• The Aharonov-Bohm effect
To introduce the effect, discuss its physical meaning concerning the real existence of vector potentials, and describe it mathematically in terms of (i) potentials, (ii) vector bundles, (iii) covering spaces, and (iv) periodic boundary conditions.
• Topological factors and the fermionic line bundle
To introduce the general concept of topological factors for quantum mechanics on a multiply connected manifold in terms of (i) covering spaces and (ii) vector bundles; and to apply the general theory to N fermions (the manifold being the unordered configuration space).
• Anyons
The topological explanation of bosons and fermions (based on the unordered configuration space) suggests that if physical space were 2-dimensional, then further types of wave functions would be possible; the corresponding particles are called anyons.
• Shor's algorithm for factorizing integers on a quantum computer
To explain how Shor's algorithm works; it is a scheme faster than any known classical algorithm and makes use of quantum gates; it has been theoretically studied although current quantum computers only have few qubits. Also possibly an overview of the present status of building quantum computers.
• Arrival times and detection times
In Bohmian mechanics, even in the absence of any measurement apparatus, there is a well defined (though random) time T at which a particle arrives at a given surface S in physical 3-space. To prove that, and explain and discuss why, T is not necessarily the time at which a detector would click.
• The Shale-Stinespring theorem
The quantum Dirac field is the many-particle version ("second quantization") of the Dirac equation, where negative energies are excluded via the introduction of anti-particles. The Shale-Stinespring theorem provides a criterion for whether a given external electromagnetic field defines a unitary evolution on the Hilbert space of the quantum Dirac field; that is, for whether only finitely many particle-antiparticle pairs get created in finite time.
• The Frauchiger-Renner paradox
Frauchiger and Renner gave an extension of Schrödinger's cat experiment with people instead of cats and several iterations; their analysis leads to paradoxical conclusions. To explain the paradox and its resolution.
• Can we detect whether a wave function has collapsed?
To present theorems about the problem of how well any experimental procedure can in an individual case distinguish between a superposition and a mixture of several mutually orthogonal quantum states; the theorems provide certain limitations to knowledge.
• Empirical tests of collapse theories
Collapse theories make predictions that deviate a little from those of standard quantum mechanics. To give an overview of the literature on how these deviations could be used for testing between collapse theories and standard quantum mechanics.
• The non-relativistic limit of the Dirac equation
To derive the 1-particle Pauli equation from the Dirac equation in the limit c to infinity, along with the Bohmian trajectories; and possibly a discussion of how the Galilean invariance comes out of the Lorentz invariance.
• Field ontology
To discuss approaches of extending Bohmian mechanics to quantum field theory by introducing actual values of field operators; to discuss possible equations for these variables, their properties and problems.
• Relativistic trajectories: the Berndl argument
The argument is a variation of Bell's theorem and shows that no probability distribution over N-tuples of timelike trajectories leads on every spacelike surface S to a joint distribution of the N intersection points with S in agreement with |psi_S|^2; to discuss the physical meaning of this result.
• Interior-boundary conditions and self-adjoint extensions
Interior-boundary conditions are used for defining Hamiltonians with particle creation at a point source. To explain how their mathematical theory connects with the general theory of self-adjoint extensions of symmetric operators, and the physical meaning of different self-adjoint extensions.
• Weak measurements and weak values
Weak measurements are experiments in which the interaction between the quantum particle and the measurement apparatus is very weak, resulting in very inaccurate (noisy) outcomes whose statistics is nevertheless informative about the particle state. When connected with postselection (conditioning on outcomes of later experiments), they lead to seemingly paradoxical values known as weak values. To explain the paradox and its resolution.
• Scattering theory
To give an overview of mathematical approaches toward analyzing the unitary time evolution from t=-infinity to t=+infinity and computing the scattering cross section; possibly the flux-across-surfaces-theorem and the scattering-into-cones-theorem.
• The GAP distribution over wave functions
For any given density matrix rho, GAP(rho) distribution is the most spread-out distribution over the unit sphere of Hilbert space with density matrix rho. To explain its definition, properties, and possibly applications.
• Quantum mechanics and spacelike space-time singularities
Space-time singularities are points of infinite curvature where space-time ends; they are predicted to occur inside black holes. To introduce the Dirac equation in curved space-time and discuss, using Schwarzschild space-time as an example, how singularities can lead to a non-unitary time evolution.
• Complex charges
In Hamiltonians with particle creation (as in quantum field theory), the "charge" just means a coefficient in front of the creation term. If this coefficient is complex, the Hamiltonian can still be self-adjoint. To explain what the corresponding time evolution and eigenstates look like.
• Reaching the speed of light
Bohm's equation of motion for a single particle with wave function governed by the Dirac equation has the property that the speed is always less than or equal to c, but not necessarily less. To present, discuss, and possibly prove theorems showing that it is infinitely unlikely for a particle to ever actually reach the speed c.
• Paradoxical reflection in quantum mechanics
This means the phenomenon that a particle can get reflected when encountering a sudden drop (downward step) in the potential. To present different derivations of this phenomenon and discuss its physical meaning.
• Nelson's stochastic mechanics
This is a theory similar to Bohmian mechanics but with stochastic, non-differentiable trajectories given by a diffusion process instead of a differential equation. To introduce the theory, along with the general concept of diffusion processes, and discuss its properties.
• Superselection rules
A superselection rule says that one can assume, for a certain observable, that superpositions of different eigenstates do not occur in nature, only mixtures; they occur particularly in quantum field theory. To discuss examples of superselection rules, their physical justification, and their meaning in terms of Bohmian mechanics and collapse theories.
• Quantum mechanics on a network
A network (or graph) consists of several line segments glued together at their end points. Along each segment, we have the usual Schrödinger equation, but the vertices (joint end points of several segments) need special consideration for defining a unitary evolution of wave functions on the network.
• Global existence proofs of Bohmian trajectories
Being solutions of an ordinary differential equation, Bohmian trajectories can fail to exist for all times, for example by running to infinity in finite time. Global existence theorems say that this happens with probability 0.
• Literature: will be recommended individually by topic of the presentation.
• Register:If you are interested in taking this seminar, please email Prof. Tumulka or register on URL (opens on June 17, 2024).
• Organizing meeting: Friday July 26, 2024, at 12:15 pm in room C4H33.