Abstract Just as in classical mechanics, the study of single particle systems in one dimen- sion provides invaluable experience in the use of quantum mechanical concepts and techniques. To extend these ideas to more realistic applications (to be able to enter the Quantum World of Part II) we also need to develop the formalisms necessary to handle multiparticle systems. In this chapter we will discuss separ- able systems (in Section 14.2), the important special case of two-body systems (Section 14.3), notation for spin-1/2 wavefunctions (Section 14.4), and lastly, in Section 14.5, the important constraints placed on quantum wavefunctions of multiparticle systems due to indistinguishability.
Abstract Just as in classical mechanics, the study of single particle systems in one dimen- sion provides invaluable experience in the use of quantum mechanical concepts and techniques. To extend these ideas to more realistic applications (to be able to enter the Quantum World of Part II) we also need to develop the formalisms necessary to handle multiparticle systems. In this chapter we will discuss separ- able systems (in Section 14.2), the important special case of two-body systems (Section 14.3), notation for spin-1/2 wavefunctions (Section 14.4), and lastly, in Section 14.5, the important constraints placed on quantum wavefunctions of multiparticle systems due to indistinguishability.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
Abstract In the preceding chapters we have given a detailed discussion of quaternionic scattering theory for the case of a single particle incident on a fixed external scattering potential. In complex quantum mechanics the external potential problem is of course only an idealized model, obtained from the physically more realistic case of multiparticle scattering when all particles except the incoming projectile are very heavy, or, in the case of the space-translationinvariant two-body problem, as the exact formulation of the internal motion after separation of the center of mass motion.
This chapter focuses on a basic but systematic treatment of wavefunctions. By comparison with a spin 1/2 system, the concept of a wavefunction as a collection of coefficients is developed, followed by discussions of eigenstates, operators, and representations in the context of wavefunctions. Transformation from position to momentum representation is effected by a Fourier transform. This is developed next, and illustrated for a Gaussian. The discussion of phases and time evolution in earlier chapters is now applied to wavefunctions, and the time evolution of a free particle discussed. The concept of a wavepacket and its suitability for representing particles follows. A topic of some subtlety — and one which is discussed in detail — is the use of bra-ket (Dirac) notation for wavefunctions. The chapter closes with a short epilogue.
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
In this chapter, we take a deeper look at a number of interesting and important models that contain one or two particles. We focus on examples, from a variety of settings and fields, with the goal of not only finding relevant physical phenomena occurring in these systems, but also giving the reader a place where they can actually practise much of what has been said previously throughout the book. In some cases, straight application of the methods explained earlier suffices, while in some others slightly different (yet related) methods are needed. Most of the models studied in this chapter are justified in chapters 4 and 5, sometimes in full. Other times, the justification from chapters 4 and 5 may be only partial, and the interested reader is encouraged to consult more specialized literature in the subject.