9.1: Introduction

Quantum mechanics allows us to predict the results of experiments. If we conduct an experiment with indistinguishable particles a correct quantum description cannot allow anything which distinguishes between them. For example, if the wavefunctions of two particles overlap, and we detect a particle, which one is it? The answer to this is not only that we don’t know, but that we can’t know. Quantum mechanics can only tell us the probability of finding a particle in a given region. The wavefunction must therefore describe both particles. The Schrödinger equation is then:

$$\left[ − \frac{\hbar^2}{2m} (\nabla^2_1 + \nabla^2_2 ) + V ( {\bf r_1}) + V ( {\bf r_2}) \right] \Phi ( {\bf r_1}, {\bf r_2}) = E\Phi ( {\bf r_1}, {\bf r_2}) \nonumber$$

where the subscripts label each particle, and there are six coordinates, three for each particle. $\Phi$ is a wave in six dimensions which contains the information we can measure: the probability of finding particles at ${\bf r_1}$ and ${\bf r_2}$, but not what we can’t measure: which particle is which.

What basis states would be appropriate for $\Phi$? An approximation is to use a product such as $\Phi ( {\bf r_1}, {\bf r_2}) = |a( {\bf r_1})b( {\bf r_2}) \rangle$ where $a( {\bf r_1})$ and $b( {\bf r_2})$ are one-particle wavefunctions of atoms 1 and 2. This allows us to separate the two particle equation into two one particle equations:

$$[ \frac{−\hbar^2}{2m} \nabla^2_1 + V ( {\bf r_1})]|a( {\bf r_1}) \rangle = E_1|a( {\bf r_1}) \rangle; \quad [\frac{−\hbar^2}{2m} \nabla^2_2 + V ( {\bf r_2})]|b( {\bf r_2}) \rangle = E_2|b( {\bf r_2}) \rangle \nonumber$$

provided that the particles do not interact (n.b $\nabla^2_1$ does not act on $b( {\bf r_2})$).

Unfortunately, by doing this we have introduced unphysical labels to the indistinguishable particles. And this is wrong: the effect of it is that the particles do not interfere with each other because they are in different dimensions (six dimensional space - remember?). When we construct a twoparticle wavefunction out of two one-particle wavefunctions we must be ensure that the probability density (the measurable quantity $|\Phi |^2$) is independent of the artificial labels.