Chapter 9. Phase Space Representations of Wavefunctions and Densities

Gary E. Bowman

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.

Leon W. Cohen

Hai-Woong Lee

Z. Ficek, Mohamed Ridza Wahiddin

In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) field that is based on the number states, the eigenstates of the Hamiltonian of the EM field. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We first determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce different representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM field or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to find the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the different representations. Most importantly, certain field states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.

Juha-Pekka Pellonpää

Covariant phase observables are obtained by defining simple conditions for mappings from the set of phase wavefunctions (unit vectors of the Hardy space) to the set of phase probability densities. The existence of phase probability density for any phase wavefunction, the existence of interference effects and the natural phase shift covariance are those simple conditions. The non-localizability of covariant phase observables is proved.