Classical and Quantum Mechanics of Particle Systems

Abstract

In this book we will develop the quantum-theoretical description of systems which can be described in terms of a field theory. Starting from the framework of classical physics, the concept of a field at first might evoke ideas about macroscopic systems, for example velocity fields or temperature fields in fluids and gases, etc. Fields of this kind will not concern us, however; they can be viewed as derived quantities which arise from an averaging of microscopic particle densities. Our subjects are the fundamental fields that describe matter on a microscopic level: it is the quantum-mechanical wave function ψ(x, t) of a system which can be viewed as a field from which the observable quantities can be deduced. In quantum mechanics the wave function is introduced as an ordinary complex-valued function of space and time. In Dirac’s terminology it has the character of a “c number”. Quantum field theory goes one step further and treats the wave function itself as an object which has to undergo quantization. In this way the wave function ψ(x, t) is transmuted into a field operator EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK % aaaaa!37D2!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\hat \psi $$(x, t), which is an operator-valued quantity (a “q number”) satisfying certain commutation relations. This process, often called “second quantization”, is quite analogous to the route that in ordinary quantum mechanics leads from a set of classical coordinates to a set of quantum operators EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyCayaaja % WaaSbaaSqaaiaadMgaaeqaaaaa!3814!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\hat q_i}$$. There is one important technical difference, though, since ψ(x, t) is a field, i.e., an object which depends on the coordinate x. The latter plays the role of a “continuous-valued index”, in contrast to the discrete index i, which labels the set q _i . Field theory therefore is concerned with systems having an infinite number of degrees of freedom.