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Pauli's exclusion principle

Consider a two-particle non-interacting fermion system. The ``non-interacting'' qualifier implies the two-particle wave function can be written as the product of two single particle wave functions. These can be written as
\begin{displaymath}
\Psi_I({\bf r}_1,{\bf r}_2) = \psi_a({\bf r}_1)\psi_b({\bf r}_2)
\end{displaymath} (58)

or
\begin{displaymath}
\Psi_{II}({\bf r}_1,{\bf r}_2) = \psi_a({\bf r}_2)\psi_b({\bf r}_1)
\end{displaymath} (59)

where $a$ and $b$ label two different single particle states. Because we cannot distinguish between the particles, we cannot know which of $\Psi_I$ or $\Psi_{II}$ describes the system. Consequently we have to consider the system as being in some linear combination of $\Psi_I$ and $\Psi_{II}$. There are only two correctly normalised combinations possible.
\begin{displaymath}
\mbox{Symmetric case (Bosons) : \ }\Psi({\bf r}_1, {\bf r}_2...
...f r}_1)\psi_b({\bf r}_2) + \psi_a({\bf r}_2)\psi_b({\bf r}_1)]
\end{displaymath} (60)

and
\begin{displaymath}
\mbox{Anti-symmetric case (Fermions) : \ }\Psi({\bf r}_1, {\...
...f r}_1)\psi_b({\bf r}_2) - \psi_a({\bf r}_2)\psi_b({\bf r}_1)]
\end{displaymath} (61)

In the case of Fermions, if $a=b$, then $\Psi=0$, implying that no two fermions can occupy the same state. By considering the form of wave-function for a system of identical particles, we have arrived almost effortlessly at Pauli's Exclusion Principle ! This result will be shown soon to account for the structure of atoms and metals. Its tremendous significance implies that the wave-function, even though it is not directly observable, must nonetheless be an object of tremendous physical, not only mathematical, validity.

Clearly, bosons do not obey this principle, and many particles of this type may be found in the lowest (ground) state. Bosonic systems display curious properties such as super conductivity.


next up previous
Next: Understanding the periodic Table Up: Quantum Mechanics of Atoms Previous: Symmetric / antisymmetric wave
Simon Connell 2004-10-04