Advanced Quantum Mechanics

Winter School, 4 - 8 February 2008,
Comenius University, Bratislava

Speakers and Abstracts:

Vladimír Balek (KTFDF): Particle in a Magnetic Field

In quantum mechanics, particles can interact with magnetic field in two ways, through the orbital motion (that is, motion in ordinary space) and through their spin. The first interaction is described by a modification of the kinetic term in the Hamiltonian that leads to a new kind of gauge invariance, more subtle than that in electrodynamics with classical charges. The second interaction contributes to the Hamiltonian by an additional term of the form minus the magnetic moment of the particle, inclusive of the mysterious factor known as gyromagnetic ratio, times the magnetic field. The effects falling under this topic include the manifestation of spin in the spectrum of the hydrogen atom, the equidistant spectrum of a charged particle in a homogeneous magnetic field, and the Aharonov-Bohm effect.

Vladimír Balek (KTFDF): WKB Approximation

The WKB approximation describes the transition from quantum to classical mechanics when quantities with the physical dimension of action assume values much greater than ħ. In the lecture we will discuss how this comes about and where one can utilize it.

Tomáš Blažek (KTFDF): Neutrino Oscillations

We show in a very introductory way the analogy between the quantum simple harmonic oscillations and the solution to the long-standing solar neutrino puzzle. No prior knowledge of neutrino physics is required.

Tomáš Blažek (KTFDF): Second Order Time Independent Perturbation Theory

Based on the perturbation expansion one can easily derive the formula for the second order energy correction. The problem arises when one really wants to apply it to a system at hand other than the simple harmonic oscilator (SHO) in a constant external electric field. The technical difficulty stems from the necessity to sum the infinite series in the formula. (In the standard school example, the SHO in electric field, the series reduces to a single term. ) In this lecture we shall follow the hint mentioned in the Pisut-Cerny-Gomolcak textbook and present a method designed to circumvent this technical difficulty in some specific cases. A paticularly cute example of the method is its application to the second order Stark effect for the hydrogen atom ground state.

Vladimír Černý (KTFDF): Bell Inequalities

Quantum Mechanics is a probabilistic theory. More specifically, this means that even if the state of the system is completely known one cannot predict with certainty the outcome of a single measurement. Hence one can suspect that the quantum description of the system is somehow incomplete and an information is missing. Does the system "know more about itself" then we do when we use the quantum description to determine its state? It is plausible to speculate that a more complete theory exists that would extend the information provided by the concept of a quantum state using some unknown or "hidden" parameters. J.Bell has shown that such a theory is subject to an experimental test. He has derived inequalities (the Bell inequalities) that would be violated by every sensible alternative theory. Their validity is testable in experiment and as a result one can exclude the existence of an alternative theory with hidden parameters even without the explicite formulation of such a theory.

Marián Fecko (KTFDF): Why Kähler Geometry in Quantum Mechanics

It turns out that the standard Quantum mechanics is intimately connected with a nontrivial combination of Riemannian and symplectic geometry, which is known as Kähler geometry. In the talk, intended to be completely introductory, we will briefly sketch how all this comes about.

Richard Hlubina (KEF): Introduction to Quantum Theory of Magnetism

We will start by showing that, for classical nonrelativistic electrons, the magnetization of a collection of electrons in equilibrium vanishes. Next, we briefly discuss the origin of elementary magnetic moments in solids and we explain how quantum mechanics leads to strong magnetic interactions between them. Finally we discuss a simple solvable model of many interacting spins.

Denis Kochan (KTFDF): Adiabatic Theorem in QM and Berry phase

Suppose a quantum system has a Hamiltonian depending on some set of external parameters (for example, the magnetic field in the Pauli Hamiltonian). One can ask the question: "What will happen with the state Ψ of the system under consideration if the external parameters in the Hamiltonian become mildly time-dependent?" The answer is, of course, provided by the solution to the corresponding Schrödinger equation. We will see that the weakness of the time-dependence (that is, the fact that the parameters are varying slowly) allows us to simplify the problem. Under special assumptions we will prove the celebrated Adiabatic Theorem in QM stating that in the regime of slow variation the eigenstates of the Hamiltonian stay "stationary". Applying the Adiabatic Theorem we will find that the phase of a quantum state is not such an irrelevant quantity as it looks at first glance.

Denis Kochan (KTFDF): Mixed States and Density Matrix Formalism

In an introductory Quantum Mechanics course one only encounters the notion of a pure quantum state. There are several situations in which this concept is not satisfactory, for example, if one tries to describe the properties of a subsystem of the given quantum system. In such situations the concept of the Density Matrix has to be employed. Our task here is to provide the audience with some basics about, and applications of, the Density Matrix formalism in QM.

Martin Mojžiš (KTFDF): Exponential Decay in QM

The decay of unstable systems is a well-known phenomenon, and its exponential form is usually taken for granted. There are, however, some serious subtleties involved and these are discussed in the present lecture.

Roman Martoňák (KEF): Introduction to Density Functional Theory

The lecture provides a brief introduction into basic concepts of density functional theory (DFT). An important problem in chemistry and condensed matter physics is calculation of the total energy of a system consisting of ions and electrons. While techniques aimed at solving the Schroedinger equation for the many-body problem are intrinsically complicated, in caseof the ground state DFT allows to use as basic variable the electronic density, instead of the many-electron wavefunction. Conceptually this provides a great simplification and offers a practical approximate computational scheme to find the ground state of the many-electron problem. DFT has found widespread applications in many branches of physics and chemistry and is implemented in a number of computer codes. We prove the fundamental Hohenberg-Kohn theorems and discuss the solution by means of the Kohn-Sham method. The problem of constructing approximate exchange-correlation functionals is briefly discussed and the local-density approximation (LDA) is presented.

Milan Noga (KTFDF):

Kernel of the Evolution Operators . Quantum-Mechanical Propagators When one finally gets through all the new concepts of quantum mechanics and endures the math required to find the stationary states for all the four or five exactly solvable simple systems there is usually very little time left to discuss the main issue of any physical theory, namely the system's time evolution. This should be an introductory lecture to make up for this omission.

Peter Prešnajder (KTFDF): Coherent States

E. Schrodinger introduced the notion of coherent states already in 1926. They represent an effective tool in quantum physics. They have found the most important application in quantum optics after pioneering works by R. J. Glauber in 1963 (this work brought him the Nobel Prize in 2005). In the lecture we compare three equivalent definitions of the harmonic oscillator coherent states, describe their properties, and finally, we sketch their generalization proposed by A. Perelomov.

Mário Ziman (RCQI-SAV): Quantum Cryptography

The randomness of quantum theory can be used to secure classical communication. In particular, using quantum systems we can distribute cipher keys between communicating parties that enable them to run the so-called Vernam cipher cryptographic protocol, which is the only known unconditionally secure communication cryptosystem. There are three basic protocols for the quantum key distribution labeled as BB84, B92 and Ekert91. The security of BB84 is based on the noise-disturbance trade-offs due to the measurement process. The idea of unambiguous conclusions is exploited in two-state protocol B92. The last of these protocols is based on purely quantum feature of the EPR paradox. All these protocols will be explained on experiments using polarization filters, photon sources and photon detectors. We will also report on the current status of the experiments on quantum cryptography.

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