While a single particle is described by a wave function
(5.1) |
The wave function must be normalized to express that the electrons must be
somewhere:
(5.2) |
The underlying idea of increasing system size is “every possible combination:” allow for every possible
combination of state for particle 1 and state for particle 2. For
example, in one dimension, all possible
Similarly, in three dimensions the three-dimensional space of
positions
The increase in the number of dimensions when the system size
increases is a major practical problem for quantum mechanics. For
example, a single arsenic atom has 33 electrons, and each
electron has 3 position coordinates. It follows that the wave
function is a function of 99 scalar variables. (Not even counting the
nucleus, spin, etcetera.) In a brute-force numerical solution of the
wave function, maybe you could restrict each position coordinate to
only ten computational values, if no very high accuracy is desired.
Even then,
Sometimes the problem size can be reduced. In particular, the problem for a two-particle system like the proton-electron hydrogen atom can be reduced to that of a single particle using the concept of reduced mass. That is shown in addendum {A.5}.
Key Points
- To describe multiple-particle systems, just keep adding more independent variables to the wave function.
- Unfortunately, this makes many-particle problems impossible to solve by brute force.
A simple form that a six-dimensional wave function can take is a product of two three-dimensional ones, as in
Show that for a simple product wave function as in the previous question, the relative probabilities of finding particle 1 near a position
Note: This is the reason that a simple product wave function is called uncorrelated.
For particles that interact with each other, an uncorrelated wave function is often not a good approximation. For example, two electrons repel each other. All else being the same, the electrons would rather be at positions where the other electron is nowhere close. As a result, it really makes a difference for electron 1 where electron 2 is likely to be and vice-versa. To handle such situations, usually sums of product wave functions are used. However, for some cases, like for the helium atom, a single product wave function is a perfectly acceptable first approximation. Real-life electrons are crowded together around attracting nuclei and learn to live with each other.