Quantum Operators

#Physics

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

$\displaystyle {\left\langle{A}\right\rangle}=\int \Psi^{*}\hat{A}\Psi , \mathrm{d}x=\bra{\Psi}\hat{A}\ket{\Psi}$

$\displaystyle {\left\langle{Q}\right\rangle}=\sum_{n}q_{n}\lvert c_{n}\rvert^{2}=\sum c_{m}^{*}c_{n}\braket{ m |\hat{Q}|n }$

• Average of a quantum operator equals the weighted sum of all possible eigenvalues
• $\displaystyle \lvert c_{n}\rvert^{2}$ is the probability of measuring $\displaystyle q_{n}$ as an eigenvalue

$$

{\left\langle{Q(x,p,t)}\right\rangle}=\begin{cases}
\int \Psi^{}\hat{Q}\left( x,-i\hbar \frac{ \partial }{ \partial x } ,t \right)\Psi , \mathrm{d}x, & \text{in position space;} \\\int \Phi^{}\hat{Q}\left( i\hbar \frac{ \partial }{ \partial p } ,p,t \right)\Phi , \mathrm{d}p, & \text{in momentum space}
\end{cases}
$$