Types of linear operator

Hermitian operator

A Hermitian operatior on a Hilbert space is a linear operator satisfying linalg

for all , i.e. .¹

Properties

1. The matrix exponential of times a Hermitian operator is a Unitary operator
2. A Hermitian operator has only real eigenvalues^[A more general statement holds for the Spectrum, not proved here.]
3. Eigenvectors of different eigenvalues are orthogonal.

Proof of 1–3

From ^P4 and ^P3 it follows

proving ^P1

Let be an eigenvector of Hermitian with eigenvalue , Then

hence is real, proving property ^P2

Let and with . Then

where we invoked ^P2 for the last equality; hence and since it follows , thus proving property ^P3

Continuous spectrum

Without proof, eigenvectors of continuous spectrum have the following properties

1. They are non-normalizable (‘generalized eigenfunctions’ — related to formal definition of spectrum?)
2. They are Dirac orthonormal

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Footnotes

1. A self-adjoint operator has the additional property that the domain of and are the same. 2018. Introduction to quantum mechanics, problem 3.48, p. 130 ↩