Introduction to the Basic Concepts of Modern Physics by Carlo Maria Becchi & Massimo D’Elia

Author:Carlo Maria Becchi & Massimo D’Elia
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(2.136)

Notice that, according to (2.135) and (2.136), in three dimensions several degenerate solutions can be found having the same energy, corresponding to all possible integers such that where n is a non-negative integer. The number of such solutions is .

Since we have looked for particular solutions, having the dependence on x, y and z factorized, it is natural to ask if in this way we have exhausted the possible solutions of equation (2.134). In some sense this is not true: since the Schrödinger equation is linear, we can make linear combinations (with complex coefficients) of the degenerate solutions described above, obtaining new solutions having the same energy but not writable, in general, as the product of three functions of x, y and z. However we have exhausted all the possible solutions in some other sense: indeed it is possible to demonstrate that no further solution can be found beyond all the possible linear combinations of the particular solutions in equation (2.136). In other words, all the possible solutions of equation (2.134), which are found for , form a linear space of dimension , having the particular solutions in equation (2.136) as an orthonormal basis. We have thus found a possible complete set of solutions of equation (2.134): we shall find a different complete set (i.e. a different basis) for the same problem in Sect. 2.9 (see also Problem 2.47).

A further generalization is that regarding small oscillations around equilibrium for a system with N degrees of freedom, whose energy can be separated into the sum of the contributions from N one-dimensional oscillators having, in general, different proper frequencies (). In this case the quantization formula reads

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