The Interacting System

Abstract

The calculation of the motion of one particle is, under classical or quantum mechanics, a well-defined problem with a well-defined solution. True, in the latter case we must then be careful as to how we interpret our solution and remember that we work with probabilities, but in principle the solution can be achieved to the Schrödinger equation for that particle. In the usual form we have

$$ H \downarrow = E \downarrow] $$

([1.1])

with

$$ H = - \frac{{{h^{2}}}}{{2m}}{\nabla ^{2}} + V\left( r \right)] $$

([1.2])

and, with its boundary conditions, it forms a well-defined computational problem. If we consider two or three particles, we may consider the problem soluble whether they have a mutual interaction or not. Once we get into the realm of large numbers of particles, however, sheer computational difficulty prevents a solution. The only exception is the case of a noninteracting set of particles, for then the total Hamiltonian is of the form

$$ H = \sum\limits_{i} { - \frac{{{h^{2}}}}{{2{m_{i}}}}} \nabla _{i}^{2} + V\left( {{r_{i}}} \right)] $$

([1.3])

and splits into a set of independent single-particle Hamiltonians

$$ H = \sum\limits_{i} {{H_{i}}}] $$

([1.4])