An Introductory Path to Quantum Theory by Stephen Bruce Sontz

Author:Stephen Bruce Sontz
Language: eng
Format: epub, pdf
ISBN: 9783030407674
Publisher: Springer International Publishing

12.3 Ladder Operators

This section concerns some very helpful, but rather technical, operators that will remind us of the raising and lowering operators of the harmonic oscillator. So, we introduce the ladder operators

Since and are symmetric operators, the adjoint of (resp., ) is (resp., ). So neither nor is a symmetric operator, and therefore neither represents a physical observable. Nonetheless, they are quite useful.

Exercise 12.3.1

Prove these identities:

Exercise 12.3.2

Suppose that for some real number m and some . (This simply says that is an eigenvector of with eigenvalue . We already know that , but that fact does not enter into this exercise.)Prove that or else is an eigenvector of with eigenvalue .

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