---
title: "Unitary Time Evolution and the Heisenberg Picture"
description: "Time evolution is a unitary operator generated by the Hamiltonian, which is what keeps total probability fixed. Putting the time dependence on the operators instead of the state gives identical predic"
canonical: https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8
source: https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Unitary Time Evolution and the Heisenberg Picture

Time evolution is a unitary operator generated by the Hamiltonian, which is what keeps total probability fixed. Putting the time dependence on the operators instead of the state gives identical predictions and often less algebra.

Subject: Science · Area: Quantum & Modern Physics · Ages 22 to 24
Page: https://lightmysky.com/learn/science/unitary-time-evolution-and-the-heisenberg-picture-mt_3JsTc9VYj8

## Ready when they can

- Writes the evolution operator for a time-independent Hamiltonian and applies it to a superposition
- Explains why unitarity is exactly what conserves total probability
- Moves a problem between the Schrödinger and Heisenberg pictures and checks the answers agree

## Lesson: Time moves states, or operators

For a Hamiltonian that does not change with time, evolution is one operator: U of t equals the exponential of minus i times H times t. Feed it the starting state and it returns the state at time t. Time travel for states is a single multiplication.

**Example.** Start with a superposition of two energy states. Each part spins its phase at its own energy rate while every probability sits still. The total stays exactly 1 at all times: phases turn, populations wait.

Unitarity is exactly what conserves total probability. A unitary operator keeps every length at 1, and lengths are probabilities waiting to be read. Because U never stretches a state, nothing leaks out of the total.

**Tip.** The Heisenberg picture freezes states and lets operators carry the time dependence instead. Every prediction matches the usual picture, often with less algebra. Ask about an observable over time: evolve the operator. Ask about the state itself: keep states moving.

**Recap.** Unitary evolution preserves probability; Heisenberg moves the clock onto operators.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx)
- [Commutators, Compatible Observables and the General Uncertainty Relation](https://lightmysky.com/learn/science/commutators-compatible-observables-and-the-general-uncertainty-relation-mt_TKoUwu74K9)
- [The Time-Independent Schrödinger Equation](https://lightmysky.com/learn/science/the-time-independent-schrodinger-equation-mt_zeBVPery5T)

## Opens up

- [The Path Integral and the Sum Over Histories](https://lightmysky.com/learn/science/the-path-integral-and-the-sum-over-histories-mt_QRLheflB1O)