Advanced quantum mechanics for nanotechnology

obligatory courses

The lecture is mostly addressed for students specializing in nano-science and nanotechnology, however, it can be also considered as the second course in the quantum mechanics. The main objective of the lecture is to make students familiar with the concepts of quantum mechanics that would facilitate the understanding of contemporary scientific papers in the fields relevant for nano-science, i.e., quantum chemistry, condensed matter physics, materials science, quantum optics. A particular attention will be given to solving practical problems.

Classroom

The course will contain the following groups of topics:

(i) The postulates of the quantum mechanics will be formulated using the concept of the density matrix that allows for description not only of pure quantum states but also states that are statistical mixture of pure states,

(ii) Composite systems and the theory of measurement process,

(iii) Bell inequalities and experimental confirmation of the validity of quantum mechanics,

(iv) Quantum mechanics of identical non-distinguishable particles,

(v) concept of field quantization and occupation number formalism,

(vi) Theory of open systems and connection between scattering theory and transport phenomena in nano-structures,

(v) The basics of the density functional Theory (DFT).

The course will consist of 15 lectures augmented by recitations. During the lectures the following issues will be addressed:

(1) Concept of pure quantum states, superposition of pure states, a statistical mixture of states, concept of density matrix operator, postulates of the quantum mechanics

(2) Theory of measurements in quantum mechanics, common set of commuting observables, preparation of a system in a given state

(3) Hilbert space for the COMPOSITE SYSTEMS, concept of tensor product, basis in the Hilbert space, concept of entanglement, entangled states

(4) System of two 1/2 spins as an example of a composite system with four dimensional Hilbert states, problem of addition of two spins as a particular case of addition of angular momenta, singlet and triplet states

(5) Experiments performed on the composite systems, measurements on a subsystem

(6) Einstein-Podolsky-Rosen paradox, 'verborgene' parameters, Bell inequalities, experiments that confirm validity of the quantum mechanics

(7) systems of non-distinguishable particles, permutations of particles, symmetry of their states, bosons, fermions, relation between parity of wave-function and statistics

(8) Quantum mechanics of systems of identical particles, examples - states of two spins in one quantum dot and two quantum dots, exchange energy and its consequences for magnetism

(9) Concept of field quantization, example of quantization of vibrations

(10) Fock-space and occupation number formalism (so-called 'second quantization'), creation and anihilation operators, Hamiltonian for interacting electrons

(11) Theory of electron gases (also in low dimensional nanostructures)

(12) Theory of open systems, basics of the scattering theory

(13) Relation between scattering theory and theory of charge transport in nanostructures

(14) Density Functional Theory as the basic tool for modeling of nanostructures

(15) Overview of the lecture, outline for further studies

J.J. Sakurai, "Modern Quantum Mechanics, Revised Edition", Edison Wesley Publishing Company, Inc., 1994

C. Cohen-Tannoudji, B. Diu, F. Laloe, "Quantum mechanics" Vol. 1 & 2

David K. Ferry, Stephen M. Goodnick, and Jonathan Bird, "Transport in Nanostructures", Cambridge University Press, 2009.

Eberhard Engel and Reiner M. Dreizler, "Density Functional Theory", Springer-Verlag, Berlin, 2011.

Students should learn the basis of quantum mechanics formulated in terms of density matrix operator and get acquainted with the occupation number formalism. They should acquire the ability to solve basic practical problems in these fields.

Written exam consisting of a few problems to solve and multiple-choice test concerning the material of the lecture. All materials such as lecture notices, books, computers are allowed during the exam.