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Journal of the Physical Society of Japan

Probability Distribution Functions Relating to Blackbody Radiation

Yutaka Kano

Published Sep 1, 1964• 21 citations
Publisher

Abstract

Statistical properties of blackbody radiation are discussed by the techniques of second quantization. Various probability distribution functions are derived, in particular the Bose-Einstein distribution function and its generalization for an arbitrary number of momentum-spin states. The Sudarshan phase space distribution function for blackbody radiation is obtained. Although in general the Sudarshan distribution function appears to be a generalized function, it is found that for the radiation field considered here, it becomes an ordinary function and is in agreement with expressions derived recently somewhat heuristically by other authors. A certain new probability distribution function for a statistcal radiation field is introduced and is expressed in terms of the eigenvalues of the annihilation operator. Unlike the phase space distribution function of Sudarshan, this function is non-negative. Finally an expression is derived for this probability distribution function appropriate to blackbody radiation.

DOI
10.1143/jpsj.19.1555

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Statistical properties of blackbody radiation are discussed by the techniques of second quantization. Various probability distribution functions are derived, in particular the Bose-Einstein distribution function and its generalization for an arbitrary number of momentum-spin states. The Sudarshan phase space distribution function for blackbody radiation is obtained. Although in general the Sudarshan distribution function appears to be a generalized function, it is found that for the radiation field considered here, it becomes an ordinary function and is in agreement with expressions derived recently somewhat heuristically by other authors. A certain new probability distribution function for a statistcal radiation field is introduced and is expressed in terms of the eigenvalues of the annihilation operator. Unlike the phase space distribution function of Sudarshan, this function is non-negative. Finally an expression is derived for this probability distribution function appropriate to blackbody radiation.
In a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase-space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvalues z of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase-space distribution function is discussed.
Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.
We have now derived a number of general properties of a quantized electromagnatic field, and have encountered some useful formalisms for treating certain problems in quantum optics. We have introduced the correlation functions of the field, and we have seen in a general way how they are related to measurements. In selecting examples for illustration we have tended to focus our attention largely on certain idealized quantum states of the field, such as Fock states and coherent states. However, there exists an important class of optical fields with simple properties, the so-called thermal fields, which includes most fields commonly encountered in practice, that has not yet been discussed. These fields are produced by sources in thermal equilibrium, and they exhibit many features that can be treated almost exactly in our formalism. We now turn our attention to such fields. Blackbody radiation The density operator Blackbody radiation is the name given to an electromagnetic field in thermal equilibrium with a large thermal reservoir or heat bath at some temperature T . By definition, such a field is assumed to be coupled to the heat bath, and it is therefore not a strictly free field in the sense of the previous chapters. However, the coupling can be as weak as we wish, and it is well known from the general theory of statistical thermodynamics that the properties of a system with many degrees of freedom in thermal equilibrium (described by a canonical ensemble) are often similar to those of an equivalent isolated system (described by a microcanonical ensemble).
This paper is concerned with the extension of some recently reported results, especially those of Bourret (1960), relating to coherence properties of blackbody radiation. An explicit expression for the complex electric correlation tensor ${\mathcal{E}}_{\mathrm{ij}}(\mathrm{r}, \ensuremath{\tau})$ of blackbody radiation is derived, and on the basis of it spatial coherence ($\ensuremath{\tau}=0$) is discussed in detail. The behavior of diagonal as well as nondiagonal components is illustrated by contour diagrams. In particular, it is found that the nondiagonal components of the correlation tensor, even though being zero for zero-space separation (r=0), acquire, in general, nonvanishing values when r \ensuremath{\ne} 0. The magnetic and mixed correlation tensors are also discussed.