Quantum Operators

Topics

• Linear Operators
• Commutators
• Anticommutators
• Generalized Ehrenfest Theorem
• Operator Transformation

Types

• Position Operator
• Momentum Operator
• Ladder Operators
• Kinetic Energy Operator
• Potential Energy Operator
• Spin Operator
• Angular Momentum Operator
• Translation Operator
• Parity Operator
• Rotation Operator
• Time-evolution Operator
• Heisenberg-picture Operators

Properties

• Some operators are non-commutative

$⟨A⟩=\int {\Psi }^{\ast }\hat{A}\Psi \mathrm{d}x=⟨\Psi |\hat{A}|\Psi ⟩$

$⟨Q⟩=\sum _n{q}_n|c_n{|}^2=\sum c_m^{\ast }{c}_n⟨m|\hat{Q}|n⟩$

• Average of a quantum operator equals the weighted sum of all possible eigenvalues
• $|c_n{|}^2$ is the probability of measuring $q_n$ as an eigenvalue

$$⟨Q(x,p,t)⟩=\{\begin{array}{ll}\int {\Psi }^{\ast }\hat{Q}\left(x,-i\hslash \frac{\partial }{\partial x},t\right)\Psi \mathrm{d}x,& \text{in position space;}\\ \int {\Phi }^{\ast }\hat{Q}\left(i\hslash \frac{\partial }{\partial p},p,t\right)\Phi \mathrm{d}p,& \text{in momentum space}\end{array}$$