Canonical density matrix

The usual context for linear response theory is that the system is prepared in the infinite past, —> -x, to be in equilibrium witii Hamiltonian H and then is turned on. This means that pit ) is given by the canonical density matrix... 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

It is assumed that the system is in a state of thermodynamic equilibrium at temperature T prior to the application of the forces Fj. j5( —oo) must consequently be the canonical density matrix, p0,... 

To afford a basis for constructing Thomas-Fermi-like approximations to the electronic structure of atoms in intense applied fields, the canonical density matrix (or equivalently the Feynman propagator) for free electrons is first set up. This is done for both intense magnetic and intense electric fields in each case, exact results are available for arbitrary static field strengths. 

Canonical Density Matrix for Free Electrons in Uniform Magnetic Field of... 

Below we shall frequently use as a tool to treat strong magnetic and electric fields the canonical density matrix C(r, Tq, p). This is defined in terms of the eigenfunctions tj(r) and the corresponding eigenvalues Ej of the Hamiltonian Hj defined in Eq. (1) as... 

Let us turn immediately to illustrating the use of the canonical density matrix for treating free electrons in a uniform magnetic field. As follows from the definitions in Eqs. (1) and (2), the canonical density matrix satisfies the so-called Bloch equation, as is readily verified, namely... 

In a pioneering paper, Sondheimer and Wilson [13] obtained the canonical density matrix C for free electrons in a uniform magnetic field of strength B, taken along, say, the z axis. Their result, denoted below as CoB(r. To. P), is... 

To gain some insight into the shape of the solution for the canonical density matrix, let us consider first a one-dimensional problem of electrons moving in a potential V(x). Writing the Bloch equation (Eq. (6)) for the Hamiltonians and and subtracting them to remove the derivative, one obtains the so-called equation of motion of the density matrix as... 

For D = 3, and putting zq = z in Eq. (24) to obtain the Slater sum S, use of the explicit form of V in Eq. (22) readily allows one to verify that the diagonal form of Eq. (24) is indeed an exact solution of Eq. (25). Later, Amovilli and March [20] made similar progress on central field problems. It remains of interest to treat atoms in intense electric fields by direct use of the Slater sum rather than by use of the off-diagonal canonical density matrix. 

We shall return to these free-electron forms of the canonical density matrix in applied B and F fields of arbitrary strength below. This will allow us to develop semiclassical theory fully within this framework. However, before doing so, we wish to discuss semiclassical theory in a different way for the H atom in intense fields. This will be done, relatively briefly, in the following section. 

To gain orientation, let us first neglect the self-consistent field (see below for its inclusion). The TF solution for the diagonal element of the canonical density matrix or the Slater sum S(r, fi) can be written (compare Eq. (20)) as SoB(P)exp(— PF(r)) with F(r) = — Ze /r. This is readily shown to yield, via the ground-state electron density p(r, B)=L (5(r, p, B)/P where L denotes the inverse Laplace transform discussed earlier ... 

Below, therefore, the solution of the Bloch equation in Eq. (6) for the canonical density matrix C(r, Tq, P, F, co) for independent electrons in a constant electric field of strength F, with harmonic restoring force corresponding to an oscillator angular frequency to, will be presented. In Sect. 7.1 below, the electric field is taken as the z axis. Then this solution can readily be generalized to include harmonic restoring forces also in the x and y directions. 

Let us now turn to the problem switching on a model potential V(r) to the Hamiltonian used above. Denoting the canonical density matrix calculated there by = C(V =0), the simplest approximation is to follow the ideas of the Thomas-Fermi (TF) method. Then, with slowly varying V(r) for which the assumptions of this approximation are valid, one can return to the definition at Eq. (2.2), and simply move all eigenvalues a,- by the same (almost constant— ) amount F(r), the wavefunctions ( i(r) being unaffected to the same order of approximation. Hence one can write for the diagonal form of the canonical density matrix... 

In short, closed forms have been obtained for the canonical density matrix C for electrons moving in a static electric field E, and confined by a harmonic restoring force. Model potentials V(r) have then switched on to this above canonical density matrix via the TF approximation at Eq. (71). 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

Although identical in structure to equation (9) for the real time propagator, the path integral representation of the canonical density matrix involves a real-valued integrand in which different paths enter with different weights. These features are extremely useful in numerical calculations (see Section 3). 

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