Types of linear operator

Hermitian operator

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

⟨𝑣|ˆ𝑄𝑤⟩=⟨ˆ𝑄𝑣|𝑤⟩

for all 𝑣,𝑤 ∈𝑉, 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 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

(𝑒𝑖ˆ𝐻)†𝑒𝑖ˆ𝐻=𝑒−𝑖ˆ𝐻𝑒𝑖ˆ𝐻=𝑒𝐎=𝐈

proving ^P1

Let 𝑣 be an eigenvector of Hermitian ˆ𝑄 with eigenvalue 𝜆, Then

𝜆=⟨𝑣|ˆ𝑄𝑣⟩⟨𝑣|𝑣⟩=⟨ˆ𝑄𝑣|𝑣⟩⟨𝑣|𝑣⟩=𝜆∗

hence 𝜆 is real, proving property ^P2

Let ˆ𝑄𝑣 =𝜆𝑣 and ˆ𝑄𝑢 =𝜇𝑢 with 𝜆 ≠𝜇. Then

𝜆⟨𝑢|𝑣⟩=⟨𝑢|ˆ𝑄𝑣⟩=⟨ˆ𝑄𝑢|𝑣⟩=𝜇∗⟨𝑢|𝑣⟩=𝜇⟨𝑢|𝑣⟩

where we invoked ^P2 for the last equality; hence (𝜆 −𝜇)⟨𝑢|𝑣⟩ =0 and since 𝜆 ≠𝜇 it follows ⟨𝑢|𝑣⟩ =0, 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 ˆ𝑄 and ˆ𝑄† are the same. 2018. Introduction to quantum mechanics, problem 3.48, p. 130 ↩