Types of linear operator

Hermitian operator

A Hermitian operatior ˆ𝑄 on a Hilbert space 𝑉 is a linear operator satisfying #m/def/linalg

βŸ¨π‘£|Λ†π‘„π‘€βŸ©=βŸ¨Λ†π‘„π‘£|π‘€βŸ©

for all 𝑣,𝑀 βˆˆπ‘‰, i.e. ˆ𝑄† =ˆ𝑄.1

Properties

  1. The matrix exponential of 𝑖 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

(𝑒𝑖ˆ𝐻)†𝑒𝑖ˆ𝐻=π‘’βˆ’π‘–Λ†π»π‘’π‘–Λ†π»=π‘’πŽ=𝐈

proving ^P1

Let 𝑣 be an eigenvector of Hermitian ˆ𝑄 with eigenvalue πœ†, Then

πœ†=βŸ¨π‘£|Λ†π‘„π‘£βŸ©βŸ¨π‘£|π‘£βŸ©=βŸ¨Λ†π‘„π‘£|π‘£βŸ©βŸ¨π‘£|π‘£βŸ©=πœ†βˆ—

hence πœ† is real, proving property ^P2

Let ˆ𝑄𝑣 =πœ†π‘£ and ˆ𝑄𝑒 =πœ‡π‘’ with πœ† β‰ πœ‡. Then

πœ†βŸ¨π‘’|π‘£βŸ©=βŸ¨π‘’|Λ†π‘„π‘£βŸ©=βŸ¨Λ†π‘„π‘’|π‘£βŸ©=πœ‡βˆ—βŸ¨π‘’|π‘£βŸ©=πœ‡βŸ¨π‘’|π‘£βŸ©

where we invoked ^P2 for the last equality; hence (πœ† βˆ’πœ‡)βŸ¨π‘’|π‘£βŸ© =0 and since πœ† β‰ πœ‡ it follows βŸ¨π‘’|π‘£βŸ© =0, thus proving property ^P3

Continuous spectrum

Without proof, eigenvectors of continuous spectrum have the following properties

  1. They are non-normalizable (β€˜generalized eigenfunctions’ β€” 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 ˆ𝑄 and ˆ𝑄† 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. ↩