In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. In the presence of a magnetic field or some other polarizing mechanism, the form of the density matrix will change, and the expectation value, for example, may be nonzero. Although the situation that has been considered involves an electron beam, a related but somewhat more complicated approach is used for polarized light beams.
In dealing with systems whose eigenstates are known, such as ideal spin systems in a magnetic field, it is sometimes useful to introduce a formalism involving the density matrix. Knowledge of the density matrix operator permits the expectation values of other operators to be determined in a way that is straightforward in principle. In the presence of a magnetic field or some other polarizing mechanism, the form of the density matrix will change, and the expectation value, for example, may be nonzero. Although the situation that has been considered involves an electron beam, a related but somewhat more complicated approach is used for polarized light beams.
After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), § 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications. The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
A variational formalism for the determination of the spinless single-particle density matrix $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator X=Y+iZ, where $Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})$ is a translation operator, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator W employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.
In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com mon statistical phenomena, such as the effects of elastic collisions on the in duced dipole moments of atoms.
While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.