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We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell.
Even for a few particles the Schrödinger equation is prohibitively difficult to solve. Hence it is important to have approximations which work in various regimes. One such approximation, which has a nice unifying theme and connects to a large area of mathematics, is the one approximating solutions of n-particle Schrödinger equations by products of n one-particle functions (i.e. functions of 3 variables)...
In this chapter we will solve the Schrödinger eigenvalue equation in a few special cases (i.e., for a few particular potentials) which not only illustrate some of the general arguments presented above, but in fact form a basis for our intuition about quantum behaviour.
The purpose of this chapter is to investigate the existence and a key property – conservation of probability – of solutions of the Schrödinger equation for a particle of mass m in a potential V. The relevant background material on linear operators is reviewed in the Mathematical Supplement Chapter 23.
In this chapter we introduce and discuss the standard model of non-relativistic quantum electrodynamics (QED). Non-relativistic QED was proposed in the early days of Quantum Mechanics (it was used by Fermi ([Fermi]) in 1932 in his review of theory of radiation). It describes quantum-mechanical par- ticle systems coupled to a quantized electromagnetic field, and appears as a quantization of the system...
The heuristic power of path integrals is that when treated as usual convergent integrals, they lead to meaningful and, as it turns out, correct answers.Thus to obtain a “quasi-classical approximation”, we apply the method ofstationary phase.
Given a quantum observable (a self-adjoint operator) A, what are the possible values A can take in various states of the system? The interpretation of 〈 Ψ,AΨ 〉 as mean value of the observable A in a state AΨ, which is validated by quantum experiments, leads to the answer. It is the spectrum of A. The most important observable is the energy – the Schrödinger operator, H, of a system. Hence the spectrum...
TWe have seen already in the first chapter that the space of quantum- 3 mechanical states of a system is a vector space with an inner-product (in fact 4 a Hilbert space). We saw also that an operator (a Schrödinger operator) on 5 this space enters the basic equation (the Schrödinger equation) governing the 6 evolution of states. In fact, the theory of operators on a Hilbert space provides 7 the basic...
In this chapter we address the issue of the information reduction in quantum mechanics. Namely, we would like to find out how to describe a subsystem of a larger system in terms of its own degrees of freedom. This leads us to the notion of an open system, whose states are described by positive, trace class operators on the L2 state space (density operators), instead of wave functions (i...
To have a consistent theory of emission and absorption of electromagnetic ra- diation by quantum systems, not only should the particle system be quantized, but the electro-magnetic field as well. Hence we have to quantize Maxwell’s equations. We do this by analogy with the quantization of Classical Mechan- ics as we have done this in Section 4.1. This analogy suggests we have to put the classical...
The calculus of variations, an extensive mathematical theory in its own right, plays a fundamental role throughout physics. This supplement contains an overview of some of the basic aspects of the variational calculus. This material will be used throughout the book, and in particular in Chapters 15 and 16 to obtain useful quantitative results about quantum systems in the regime close to the classical...
As we have seen, many basic questions of quantum dynamics can be reduced to finding and characterizing the spectrum of the appropriate Schrödinger operator. Though this task, known as spectral analysis, is much simpler than the task of analyzing the dynamics directly, it is far from trivial. The problem can be greatly simplified if the Schrödinger operator H under consideration is close to an operator...
Emission and absorption of electromagnetic radiation by systems of non- relativistic particles such as atoms and molecules is a key physical phe- nomenon, central to the existence of the world as we know it. Attempts to understand it led, at the beginning of the twentieth century, to the birth of quantum physics. In this chapter we outline the theory of this phenomenon. It addresses the following...
In this chapter, we discuss the procedure of passing from classical mechanics to quantum mechanics. This is called “quantization” of a classical theory.
Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics.
This representation, the “Feynman path integral”, will provide us with a heuristic but effective tool for 5 investigating the connection between quantum and classical mechanics. This 6 investigation will be undertaken in the next section.
In this chapter we prove existence of stationary, well localized and stable states of atoms and, in a certain approximation, molecules. These are the lowest energy states, and their existence means that our quantum systems exist as well-localized objects, and do not disintegrate into fragments under sufficiently small perturbations.
There is an extensive literature on quantum mechanics. Standard books in- clude [Bay,LL,Me, Schi]. More advanced treatments can be found in [Sa,BJ, DR].
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