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Symmetric / antisymmetric wave functions

We have to construct the wave function for a system of identical particles so that it reflects the requirement that the particles are indistinguishable from each other. Mathematically, this means interchanging the particles occupying any pair of states should not change the probability density ($\vert\Psi\vert^2$) of the system. This simple statement has the enormous consequence of dividing all particles in nature into one of two classes.

An example for two non-interacting identical particles will illustrate the point. The probability density of the the two particle wave function $\Psi(\bf {r}_1, \bf {r}_2)$ must be identical to that of the the wave function $\Psi(\bf {r}_2, \bf {r}_1)$ where the particles have been interchanged.

\begin{displaymath}
\vert\Psi({\bf r}_1, {\bf r}_2)\vert^2 = \vert\Psi({\bf r}_2, {\bf r}_1)\vert^2
\end{displaymath} (55)

We can achieve this in two ways.
\begin{displaymath}
\mbox{Symmetric case : \ \ \ }\Psi({\bf r}_1, {\bf r}_2) = \Psi({\bf r}_2, {\bf r}_1)
\end{displaymath} (56)

or
\begin{displaymath}
\mbox{Anti-symmetric case : \ \ \ }\Psi({\bf r}_1, {\bf r}_2) = -\Psi({\bf r}_2, {\bf r}_1)
\end{displaymath} (57)

It turns out that particles whose wave functions which are symmetric under particle interchange have integral or zero intrinsic spin, and are termed bosons. Particles whose wave functions which are anti-symmetric under particle interchange have half-integral intrinsic spin, and are termed fermions. Experiment and quantum theory place electrons in the fermion category. Any number of bosons may occupy the same state, while no two fermions may occupy the same state. This result, which establishes the behaviour of many-electron atoms, is proved below.


next up previous
Next: Pauli's exclusion principle Up: Quantum Mechanics of Atoms Previous: Many-electron atoms
Simon Connell 2004-10-04