LINEAR OPERATORS FOR QUANTUM MECHANICS by Thomas F. Jordan

Author:Thomas F. Jordan
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2018-03-12T16:00:00+00:00

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1 See Section 12, Eigenvalues and Eigenvectors, particularly Theorem 12.7.

2 For unbounded operators these definitions have to be stated more carefully to take account of domains and to get the following theorem; see Riesz and Nagy, Sections 115–117 and Stone, Chapter IV, Section 3.

3 From Section 16, Functions of Commuting Operators, we know that if a Hermitian operator commutes with B, then it commutes with Ex. If A is a bounded non-Hermitian operator which commutes with B, then taking the adjoint of AB = BA yields BA† = A†B, so B commutes with Re A and Im A. Therefore Ex commutes with Re A and Im A, so Ex commutes with A.

4 This step is valid for a closed unbounded operator A; see Riesz and Nagy, Sections 115-117.

5 Riesz and Nagy, Sections 126, 129, and 130.

6 Naimark, Section 34.

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