Wave Functions as Bases for Irreducible Representations

It will be appropriate to give first a brief discussion of the wave equation. It is not necessary that the reader have any very extensive knowledge of wave mechanics in order to follow the development in this chapter, but the information given here is essential. The wave equation for any physical system is 

/ is the Hamiltonian operator, which indicates that certain operations are to be carried out on a function written to its right. The wave equation states that, if the function is an eigenfunction, the result of performing the operations indicated by J( will yield the function itself multiplied by a constant that is called an eigenvalue. Eigenfunctions are conventionally denoted P, and the eigenvalue, which is the energy of the system, is denoted E. 

The Hamiltonian operator also commutes with any constant factor c. Thus MV = c. /rV = cEV (5.1-3) 

It has been more or less implied up to this point that for any eigenvalue Ej there is one appropriate eigenfunction P,. This is often true, but there are also many cases in which several eigenfunctions give the same eigenvalue, for example, 

In such cases we say that the eigenvalue is degenerate, and in the particular example given above we would say that the energy , is fc-fold degenerate. Now in the case of a degenerate eigenvalue, not only does the initial set of 

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