--- title: "Unitary Time Evolution and the Heisenberg Picture" description: "A closed system with a self-adjoint Hamiltonian has unitary time evolution, preserving state norms and total probability. For time-independent H, U(t)=exp(-i H t/hbar). The Heisenberg picture transfer" 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: 2026-11-12 --- > **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 A closed system with a self-adjoint Hamiltonian has unitary time evolution, preserving state norms and total probability. For time-independent H, U(t)=exp(-i H t/hbar). The Heisenberg picture transfers evolution-related time dependence to observables and gives the same measurement predictions. 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 closed system with time-independent, self-adjoint Hamiltonian H, U(t)=exp(-i H t/hbar). Here i²=-1 and hbar is Planck's constant divided by 2 pi. Take the starting time as zero. The Schrödinger-picture state is |psi_S(t)>=U(t)|psi0>. The exponent is dimensionless. Omitting hbar would require a stated convention hbar=1. In an orthonormal energy eigenbasis, H|n>=E_n|n>. Each coefficient evolves as c_n(t)=c_n exp(-i E_n t/hbar). For real theta, |exp(-i theta)|=1; exp(-i pi)=-1 and exp(-i pi/2)=-i. Thus energy-basis populations |c_n|² stay fixed. For a degenerate energy, add its basis-state populations to get that energy's probability. Relative phases can change probabilities in another measurement basis. *(drawing: This preserves energy populations, not every measurement's outcome probabilities.)* Unitarity means U†U=UU†=I. The dagger denotes the adjoint and I is the identity. Then ==. Every vector keeps its original norm, which need not be one. A state's squared norm is total probability, so a normalized state stays normalized. For reversible linear evolution, preserving every vector's norm is equivalent to unitarity. For the finite-dimensional examples here, let A have no explicit time dependence. The Heisenberg state is |psi_H>=|psi0> and A_H(t)=U†(t) A U(t). Substituting |psi_S>=U|psi0> and =. Transforming measurement projectors the same way gives equal outcome probabilities in both pictures. **Example.** Let H|0>=0 and H|1>=epsilon|1>, with orthonormal states. Start in |+>=(|0>+|1>)/sqrt(2). At epsilon t/hbar=pi/2, psi(t)=(|0>-i|1>)/sqrt(2). To find P(+), square the magnitude of the overlap: P(+)=|<+|psi(t)>|²=|(1-i)/2|²=1/2. Initially it was one. Energy populations remain 1/2 each, but this measurement probability changes. For fixed H and a finite-dimensional observable A with no explicit time dependence, dA_H/dt=(i/hbar)[H,A_H]. Here [H,A_H]=H A_H-A_H H. If [H,A]=0, A commutes with U, so A_H=U† A U=U†U A=A. Its expectation is constant. This can help with conservation questions, but neither picture always uses less algebra. An explicitly time-dependent A would add another derivative term. **Recap.** U(t)=exp(-i H t/hbar) preserves norms. Relative phases can change measurement probabilities. A_H=U† A U gives the same predictions with a fixed state. *Lesson, practice questions, worked solutions and flashcards written for this dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [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) ## Optional connections - [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx) ## 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)