5.1 Wave Function for Multiple Particles

While a single particle is described by a wave function $\Psi({\skew0\vec r};t)$, a system of two particles, call them 1 and 2, is described by a wave function

$$\Psi({\skew0\vec r}_1,{\skew0\vec r}_2;t)$$ (5.1)

depending on both particle positions. The value of $\vert\Psi({\skew0\vec r}_1,{\skew0\vec r}_2;t)\vert^2{\,\rm d}^3{\skew0\vec r}_1{\,\rm d}^3{\skew0\vec r}_2$ gives the proba­bility of simulta­neously finding particle 1 within a vicinity ${\,\rm d}^3{\skew0\vec r}_1$ of ${\skew0\vec r}_1$ and particle 2 within a vicinity ${\,\rm d}^3{\skew0\vec r}_2$ of ${\skew0\vec r}_2$.

The wave function must be normalized to express that the electrons must be somewhere:

$$\langle \Psi \vert \Psi \rangle_6 = \mathop{\int\kern-7p... ...t^2{\,\rm d}^3 {\skew0\vec r}_1{\rm d}^3 {\skew0\vec r}_2 = 1$$ (5.2)

where the subscript 6 of the inner product is just a reminder that the integr­ation is over all six scalar position coordinates of $\Psi$.

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 $x$ positions of particle 1 geometri­cally form an $x_1$-axis. Similarly all possible $x$ positions of particle 2 form an $x_2$-axis. If every possible position $x_1$ is separately combined with every possible position $x_2$, the result is an $x_1,x_2$-plane of possible positions of the combined system.

Similarly, in three dimensions the three-di­mensional space of positions ${\skew0\vec r}_1$ combines with the three-di­mensional space of positions ${\skew0\vec r}_2$ into a six-di­mensional space having all possible combin­ations of values for ${\skew0\vec r}_1$ with all possible values for ${\skew0\vec r}_2$.

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, $\Psi$ values at 10$\POW9,{99}$ different combined positions must be stored, requiring maybe 10$\POW9,{91}$ 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$\POW9,{82}$ 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

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To describe multiple-particle systems, just keep adding more independent variables to the wave function.

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Unfortunately, this makes many-particle problems impossible to solve by brute force.

5.1 Review Questions

1. A simple form that a six-di­mensional wave function can take is a product of two three-di­mensional ones, as in $\psi({\skew0\vec r}_1,{\skew0\vec r}_2)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\psi_1({\skew0\vec r}_1)\psi_2({\skew0\vec r}_2)$. Show that if $\psi_1$ and $\psi_2$ are normalized, then so is $\psi$.

   Solution complex-a

2. Show that for a simple product wave function as in the previous question, the relative proba­bilities of finding particle 1 near a position ${\skew0\vec r}_a$ versus finding it near another position ${\skew0\vec r}_b$ 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.

   Solution complex-b