Many-particle Schrodinger equation

Unlike the AT-particle picture, which in principle leads naturally to the exact solution of the many-particle Schrodinger equation, the single-particle picture has led to the development of a number of different approximation schemes designed to address particular issues in the physics of interacting quantum systems. A particularly pointed example of this state of affairs is the strict dichotomy that has set in between so-called single-particle theories and canonical many-body theory[31, 32]. Each of the two methodologies can claim a number of successful applications, which tends to reinforce the perceived formal gap between them. 

Use of Generalized Sturmian Basis Sets to Solve the Many-Particle Schrodinger Equation... 

Now suppose that we wish to solve the many-particle Schrodinger equation... 

In solving the many-particle Schrodinger equation, it is desirable to choose the approximate potential V0(x) to be as close as possible to the actual potential V(x), since this leads to rapid convergence of the expansion (18). Goscinski showed [20, 21] that for atoms, the approximate Schrodinger equation (10) can be solved exactly provided that V0(x) is chosen to be the attractive Coulomb potential of the bare atomic nucleus ... 

The generalized Sturmian method [1-22] for solving the Schrodinger equation of an /V-particle system is a direct configuration interaction method, in which the configurations are chosen to be isoenergetic solutions to the approximate many-particle Schrodinger equation... 

To obtain the generalized Sturmian secular equations, we substitute the superposition (6) into the many-particle Schrodinger equation (5) ... 

Section 3.4 The Three-Dimensional Many-Particle Schrodinger Equation 47... 

In the past few years the method of dimensional scaling [12,22,23] has become increasingly more importemt in quantum theory. Using this technique one can solve the many-particle Schrodinger equation in a space of arbitrary dimension D [17]. By taking the limit of infinite... 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The first few 4-dimensional hyperspherical harmonics K i, ,m(u) are shown in Table 5. Shibuya and Wulfman [19] extended Fock s momentum-space method to the many-center one-particle Schrodinger equation, and from their work it follows that the solutions can be found by solving the secular equation (63). If Fock s relationship, equation (67), is substituted into (65), we obtain ... 

In most present implementation of DFT, the many-body Schrodinger equation is replaced by a set of coupled effective one-particle equations, the so-called Kohn-Sham equations [21]... 

Some examples to illustrate this approach may be found in articles in ref. 81 and 154. The local approximations to the exchange and correlation part of G introduced above are discussed in further detail in refs. 121 and 122, and are encouraging for local density approximations to the many-electron part of G above. The kinetic part of G can again be treated by solving the single-particle Schrodinger equation, by a generalization of the approach described in Section 17, but now with different potentials for the two different spin directions. 

Note that at this point we have turned the original (hopeless) many-body problem into a series of effective single particle Schrodinger equations. 

It is straightforward to write down and solve the many-electron Schrodinger equation if it is assumed that the electrons do not interact, or interact only to a very small extent. Indeed, it is on this premise that the fabric of modem qualitative molecular orbital theory is based. For the two electrons in a helium atom [Z = 2] for example, this independent particle model Schrodinger equation is simply... 

However, it is the Pauli principle which prevents us from simply ignoring the existence of electron spin altogether. The trial wavefunction must be antisymmetric with respect to the exchange of the coordinates (space-spin) of any two particles. Without this constraint the solutions of the many-electron Schrodinger equation would be wrong there are many more solutions of the Schrodinger equation than there aje antisymmetric solutions of that equation. Electron spin, at this level, simply ensures that the spatial part of the wavefunction behaves properly when the electrons coordinates are permuted. Thus, notwithstanding the manipulational convenience of the use of spin functions it would be attractive to be able to deal explicitly only with a spatial trial function and solve a spatial variational problem. 

The Kohn-Sham theorem then allows to obtain a more tractable single-particle Schrodinger equation. Many-particle effects are included by an effective exchange and correlation potential, which is derived variation-ally. 

The basic concept is that instead of dealing with the many-body Schrodinger equation, Fq. (2.1), which involves the many-body wavefunction P ( r ), one deals with a formulation of the problem that involves the total density of electrons (r). This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 

Let us now consider elastic scattering of a particle from a system. It has been often emphasized that one can avoid the computation of complicated many-particle scattering states by introducing an optical wave function. This function obeys a single-particle Schrodinger equation, the potential of which is called optical potential. The optical potential and wave function are not unique. A particularly convenient choice is to relate them to the one-particle GF. Bell and Squires were the first to show that the optical potential may be identified with the self-energy of this GF. The self-energy i7(a>) connects the GF with the free GF via the Dyson equation. In matrix notation this renowned equation reads... 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

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