Quantum Operators

Quantum Operators

Kinetic Energy Operator

$$ \large \hat{T} = \frac{\hat{p}^2}{2m} $$

Hamiltonian Operator

$$ \large \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) $$

Pauli Spin Matrix

$$ \large \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$$$ \large \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$$$ \large \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
  • GRE Physics GR0877 Problem 087

Angular Momentum Operator

$$ \large [J_x, J_y] = i \hbar J_z $$$$ \large [J_y, J_z] = i \hbar J_x $$$$ \large [J_z, J_x] = i \hbar J_y $$

Cyclic order

  • GRE Physics GR0877 Problem 095

Useful Properties

Orthogonal (Perpendicular) = Independent

  • GRE Physics GR0177 Problem 028

Useful Formula

$$ \large [AB, C] = A[B,C] + [A,C]B $$
  • GRE Physics GR0177 Problem 043

Eigenvalue

$$ \large \det (A-\lambda I) = 0 $$
  • Eigenvalues of the Hamiltonian operators are always real.

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