While a single particle is described by a wave function
, a system of two particles, call them 1 and 2, 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 positions of particle 1 geometrically form an -axis. Similarly all possible positions of particle 2 form an -axis. If every possible position is separately combined with every possible position , the result is an -plane of possible positions of the combined system.
Similarly, in three dimensions the three-dimensional space of positions combines with the three-dimensional space of positions into a six-dimensional space having all possible combinations of values for with all possible values for .
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, values at 10 different combined positions must be stored, requiring maybe 10 Gigabytes of storage. To do a single multiplication on each of those those numbers within a few years would require a computer with a speed of 10 gigaflops. No need to take any of that arsenic to be long dead before an answer is obtained. (Imagine what it would take to compute a microgram of arsenic instead of an atom.) Obviously, more clever numerical procedures are needed.
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 if and are normalized, then so is .
Show that for a simple product wave function as in the previous question, the relative probabilities of finding particle 1 near a position versus finding it near another position is the same regardless where particle 2 is. (Or rather, where particle 2 is likely to be found.)
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.