Types of linear operator

Hermitian operator

A Hermitian operatior on a Hilbert space is a linear operator satisfying #m/def/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 eigenvalues2
  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

#state/tidy | #lang/en | #SemBr

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 ↩

  2. A more general statement holds for the Spectrum, not proved here. ↩