Quantum Mechanics · Lesson 09

Chapter practice

Bring the chapter together: matrices of operators and their products, adjoints, eigenvalues, the Hermitian and unitary operators with their special spectra, projectors, the spectral decomposition and commutators. No worked example sits above the answers.

Foundations8 min readRevision 1

Read freely. Sign in when you want to save your place.

Sign in to save progress

A^∣ ψ ⟩

A^^{†}

[A^, B^]

A^∣ a ⟩ = a ∣ a ⟩

Chapter navigation

Intuition

Bring the chapter together: matrices of operators and their products, adjoints, eigenvalues, the Hermitian and unitary operators with their special spectra, projectors, the spectral decomposition and commutators. No worked example sits above the answers.

A carpenter’s toolbox holds a few tools used over and over. The skill is reaching for the right one without being told which.

Before you practise

An operator is its matrix in a basis, with $A_{mn}=\bra{m}\hat{A}\ket{n}$ . The adjoint is the conjugate transpose. Hermitian operators have real eigenvalues and orthogonal eigenvectors and decompose into projectors; unitary ones keep every bracket and have eigenvalues on the unit circle; projectors square to themselves.

\hat{A}^{\dagger}=\hat{A},\qquad \hat{U}^{\dagger}\hat{U}=\hat{1},\qquad \hat{P}^{2}=\hat{P}

Reminders

$(\hat{A}\hat{B})^{\dagger}=\hat{B}^{\dagger}\hat{A}^{\dagger}$ , and numbers come out conjugated.
Eigenvalues of a $2\times2$ matrix: $\lambda^{2}-(\operatorname{tr}A)\lambda+\det A=0$ .
$f(\hat{A})=\sum f(a_{n})\ket{a_{n}}\bra{a_{n}}$ .
$[\hat{A},\hat{B}\hat{C}]=[\hat{A},\hat{B}]\hat{C}+\hat{B}[\hat{A},\hat{C}]$ , and .

Applications