Quantum mechanics of one particle

In quantum mechanics the physical state of a particle is described in terms of a ket |Ψ〉. This ket belongs to a Hilbert space which is nothing but a vector space endowed with an inner product. The dimension of the Hilbert space is essentially fixed by our physical intuition; it is we who decide which kets are relevant for the description of the particle. For instance, if we want to describe how a laser works we can choose those energy eigenkets that get populated and depopulated and discard the rest. This selection of states leads to the well-known description of a laser in terms of a three-level system, four-level system, etc. A fundamental property following from the vector nature of the Hilbert space is that any linear superposition of kets is another ket in the Hilbert space. In other words we can make a linear superposition of physical states and the result is another physical state. In quantum mechanics, however, it is only the “direction” of the ket that matters, so |Ψ〉 and C|Ψ〉 represent the same physical state for all complex numbers C. This redundancy prompts us to work with normalized kets. What do we mean by that? We said before that there is an inner product in the Hilbert space.

Introduction

In this and the next two chapters we lay down the basis to go beyond the Hartree–Fock approximation. We develop a powerful approximation method with two main distinctive features. The first feature is that the approximations are conserving. A major difficulty in the theory of nonequilibrium processes consists in generating approximate Green's functions yielding a particle density n(x, t) and a current density J(x, t) which satisfy the continuity equation, or a total momentum P(t) and a Lorentz force F(t) which is the time-derivative of P(t), or a total energy E(t) and a total power which is the time-derivative of E(t), etc. All these fundamental relations express the conservation of some physical quantity. In equilibrium the current is zero and hence the density is conserved, i.e., it is the same at all times; the force is zero and hence the momentum is conserved; the power fed into the system is zero and hence the energy is conserved, etc. The second important feature is that every approximation has a clear physical interpretation. In Chapter 7 we showed that the two-particle Green's function in the Hartree and Hartree–Fock approximation can be represented diagrammatically as in Fig. 7.1 and Fig. 7.2 respectively. In these diagrams two particles propagate from (1′, 2′) to (1, 2) as if there was no interaction.

In this chapter we get acquainted with the one-particle Green's function G, or simply the Green's function. The chapter is divided in three parts. In the first part (Section 6.1) we illustrate what kind of physical information can be extracted from the different Keldysh components of G. The aim of this first part is to introduce some general concepts without being too formal. In the second part (Section 6.2) we calculate the noninteracting Green's function. Finally in the third part (Sections 6.3 and 6.4) we consider the interacting Green's function and derive several exact properties. We also discuss other physical (and measurable) quantities that can be calculated from G and that are relevant to the analysis of the following chapters.

What can we learn fromG?

We start our overview with a preliminary discussion on the different character of the space– spin and time dependence in G(1; 2). In the Dirac formalism the time-dependent wavefunction Ψ(x, t) of a single particle is the inner product between the position–spin ket |x〉 and the time evolved ket |Ψ(t)〉. In other words, the wavefunction Ψ(x, t) is the representation of the ket |Ψ(t)〉 in the position–spin basis.