Wigner approached this theory in a "simple, classical conext," the vibrations of a drum. A typical musical drum is usually circular, but can be any shape. When the drum is hit, the skin vibrates and a noise is created. Different sounds occur depending on the shape of the drum. The spectrum, or he range of frequencies that the drum can produce, depends in a complex manner on the drums shape. If the drum is symmetrical, it is expected that the symmetry would show up in the spectrum, and it does, but very subtly.
Let's say we have a rectangular drum, typically its vibration patterns would divide the rectangle into a number of smaller rectangles. Here are two examples:
Here are two different virational patterns with two different frenquencies. They are each a snapshot taken at one instant. The dark areas are displaces downwards and the white ones upward.
The symmetries of the drum are connected to the pattern. Any symmetry of the drum can be applied to possible pattern of vibration to produce another possible pattern of vibration, therefore the patterns come in symmetrically related sets.
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