Heisenberg Picture
Table of Contents
The Heisenberg picture (also known as the Heisenberg representation) is a fundamental time-dependent differential equation in quantum mechanics. In the Heisenberg picture, the state vectors remain fixed, and the operators evolve over time. The general form of the equation is:
$$ {d \over d t}A_H = \frac{1}{i\hbar}[A_H, H] + \left(\frac{\partial A_H}{\partial t}\right) $$
Where:
- \( \small A_H \) is an operator/observable (e.g. position, momentum, etc.) in the Heisenberg picture,
- \( \small H \) is the Hamiltonian of the system,
- \( \small \hbar \) is the reduced Planck constant,
- \( \small [·,·] \) denotes the commutator
No Time Dependence #
If the operator \( \small A_H \) does not have an explicit time dependence, the equation simplifies to:
$$ {i\hbar}{d \over d t}A_H = [A_H, H]. $$