36 Multi-particle Systems: Momentum Space Description
In chapter 9 we saw how to use symmetric or antisymmetric tensor products to describe a fixed number of identical quantum systems (for instance, free particles). From very early on in the history of quantum mechanics, it became clear that at least certain kinds of quantum particles, photons, required a formalism that could describe arbitrary numbers of particles, as well as phenomena involving their creation and annihilation. This could be accomplished by thinking of photons as quantized excitations of a classical electromagnetic field. In our modern understanding of fundamental physics all elementary particles, not just photons, are best described in this way, by quantum theories of fields. For free particles the necessary theory can be understood as the quantum theory of the harmonic oscillator, but with an infinite number of degrees of freedom, one for each possible value of the momentum (or, Fourier transforming, each possible value of the position). The symmetric (bosons) or antisymmetric (fermions) nature of multi-particle quantum states is automatic in such a description as quanta of oscillators.
Conventional textbooks on quantum field theory often begin with relativistic systems, but we’ll start instead with the non-relativistic case. This is significantly simpler, lacking the phenomenon of antiparticles that appears in the relativistic case. It is also the case of relevance to condensed matter physics, and applies equally well to bosonic or fermionic particles.
Quantum field theory is a large and complicated subject, suitable for a fullyear course at an advanced level. We’ll be giving only a very basic introduction, mostly just considering free fields, which correspond to systems of noninteracting particles. Much of the complexity of the subject only appears when one tries to construct quantum field theories of interacting particles.
For simplicity we’ll start with the case of a single spatial dimension. We’ll also begin using x to denote a spatial variable instead of the conventional when this is the coordinate variable in a finite dimensional phase space. In quantum field theory, position or momentum variables parametrize the fundamental degrees of freedom, the field variables, rather than providing such degrees of freedom themselves. In this chapter, emphasis will be on the momentum parametrization and the description of collections of free particles in terms of quanta of degrees of freedom labeled by momenta.
Chapter contents
- 36.1 Multi-particle quantum systems as quanta of a harmonic oscillator
- 36.2 Multi-particle quantum systems of free particles: finite cutof formalism
- 36.3 Continuum formalism
- 36.4 Multi-particle wavefunctions
- 36.5 Dynamics
- 36.6 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 379、380、381、382、383、384、385、386、387、388、389、390、391、392、393
来源版本：2025-10-20
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