Heisenberg picture

Author: kevinlinxc

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+.python-version
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README.md

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content/formulas/heisenberg-picture/index.md

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+---
+title: "Heisenberg Picture"
+description: "Heisenberg's equation that describes a particle in a potential"
+summary: "Heisenberg's equation that describes a particle in a potential"
+tags: ["math", "physics", "quantum mechanics"]
+date: 2025-01-20
+latex: "\\small {d \\over d t}A_H = \\frac{1}{i\\hbar}[A_H, H] + \\left(\\frac{\\partial A_H}{\\partial t}\\right)"
+---
+
+{{< katex >}}
+
+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](https://en.wikipedia.org/wiki/Commutator)
+
+
+## No Time Dependence
+
+If the operator \\( \small A_H \\) does not have an explicit time dependence, the equation simplifies to:
+
+$$ {i\hbar}{d \over d t}A_H = [A_H, H]. $$
+
+## Sources
+
+- [Wikipedia: Heisenberg Picture](https://en.wikipedia.org/wiki/Heisenberg_picture)