Time Evolution

The Hamiltonian generates the time evolution of a closed quantum system. In the Schrödinger picture, a state satisfies a first-order initial-value equation; equivalently, a unitary propagator maps the state from one time to another. Energy eigenstates, stationary states, conservation laws, and pictures of motion are consequences and reorganizations of this core rule.

This chapter states the basic formalism and the distinctions needed throughout quantum mechanics. Detailed propagator methods, time ordering, Dyson expansions, Heisenberg and interaction pictures, path integrals, phase-space formulations, and numerical evolution belong to Quantum Dynamics.

What This Chapter Owns

This chapter is the canonical home for

• the Hamiltonian as energy observable and generator of time evolution;
• the closed-system unitarity requirement at the level of Core Formalism;
• energy eigenstates and their measurement meaning;
• stationary pure and mixed states;
• the expectation-value form of conservation laws;
• a compact map of the Schrödinger, Heisenberg, and interaction pictures.

The canonical detailed treatments of the time-dependent Schrödinger equation, time-evolution operator, and time-dependent Hamiltonians live in the dynamics volume. The corresponding Core articles are concise bridges that establish the minimum vocabulary and route readers onward.

Open-system reduced dynamics is not generally unitary and belongs to Measurement and Open Quantum Systems. Solving the spatial eigenvalue problems of particular Hamiltonians belongs to Wave Mechanics and Model Systems.

Hamiltonian and Schrödinger Equation

For a closed system in the Schrödinger picture, the state vector obeys

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

The Hamiltonian $H(t)$ may have explicit time dependence. For an isolated system with a time-independent Hamiltonian, it is both the generator of time translations and the observable whose spectral values are the possible energies.

The equation is an initial-value problem: a state at $t_0$, together with the Hamiltonian and suitable domain information, determines later states. It is first order in time, so one initial state is required. In position space, a particle Hamiltonian such as

$$H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf x,t)$$

turns the abstract equation into a partial differential equation for $\psi(\mathbf x,t)$. The abstract evolution rule belongs here; boundary conditions, domains, and model-specific solutions belong to their dedicated mathematical and canonical-system pages.

For a density operator under closed-system unitary dynamics,

$$i\hbar\frac{d\rho}{dt} = [H(t),\rho].$$

This Liouville–von Neumann equation is the density-operator form of the same closed-system dynamics, not a dissipative master equation.

Propagator Structure

The time-evolution operator $U(t,t_0)$ is defined by

$$\lvert\psi(t)\rangle = U(t,t_0)\lvert\psi(t_0)\rangle.$$

It satisfies

$$i\hbar \frac{\partial}{\partial t} U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

Consistent propagation requires the composition law

$$U(t_2,t_1)U(t_1,t_0) = U(t_2,t_0),$$

and unitary closed-system evolution gives

$$U(t,t_0)^\dagger = U(t_0,t) = U(t,t_0)^{-1}.$$

For a time-independent self-adjoint Hamiltonian,

$$U(t,t_0) = \exp\!\left[ -\frac{i}{\hbar} H(t-t_0) \right].$$

If $H$ depends on time, the ordinary exponential of its time integral is valid only under additional commutation conditions. In general,

$$U(t,t_0) = \mathcal T \exp\!\left[ -\frac{i}{\hbar} \int_{t_0}^{t} H(s)\,ds \right],$$

where $\mathcal T$ orders later-time Hamiltonians to the left. If $[H(t),H(s)]=0$ for every pair of relevant times, time ordering becomes unnecessary.

Why Evolution Is Unitary

Unitary evolution preserves inner products:

$$\langle\phi(t)\rvert\psi(t)\rangle = \langle\phi(t_0)\rvert\psi(t_0)\rangle.$$

Norms, orthogonality, and transition probabilities between co-evolved closed-system states are therefore preserved. For a time-independent self-adjoint Hamiltonian, Stone’s theorem supplies the converse mathematical structure: a strongly continuous one-parameter unitary group has a self-adjoint generator.

The word “closed” matters. A subsystem entangled with an environment can evolve nonunitarily after the environment is traced out even when the larger system evolves unitarily. Effective non-Hermitian Hamiltonians can also appear in conditional or approximate descriptions; they are not counterexamples to the standard closed-system postulate because probability accounting has been restricted or modified.

Energy Basis and Stationarity

For a time-independent Hamiltonian,

$$H\lvert E_n,\alpha\rangle = E_n\lvert E_n,\alpha\rangle,$$

where $\alpha$ labels degeneracy. An initial expansion in the energy basis evolves as

$$\lvert\psi(t)\rangle = \sum_{n,\alpha} c_{n\alpha} e^{-iE_n(t-t_0)/\hbar} \lvert E_n,\alpha\rangle,$$

with sums replaced or supplemented by integrals for continuous spectral components.

A single energy eigenstate acquires only a global phase. Its ray and all expectation values of time-independent observables are stationary. A coherent superposition of distinct energies acquires changing relative phases and generally produces time-dependent interference. Superpositions within one degenerate energy eigenspace remain stationary because every component receives the same phase.

For a density operator and time-independent $H$, stationarity is expressed by

$$[H,\rho]=0.$$

This criterion includes mixtures diagonal in energy and coherences confined within degenerate energy subspaces. “Stationary” does not mean that the state vector is literally constant in the Schrödinger picture, nor that every classical-looking quantity such as momentum vanishes.

Conservation Laws

For an observable $A(t)$ that may depend explicitly on time,

$$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar} \langle[H,A]\rangle + \left\langle \frac{\partial A}{\partial t} \right\rangle.$$

If $A$ has no explicit time dependence and commutes with the Hamiltonian, its expectation value is constant in every evolving state. More strongly, its full probability distribution is conserved under the corresponding closed dynamics. For a time-independent Hamiltonian, energy is conserved. When $H(t)$ is driven explicitly, the system can exchange energy with the external agent, so the instantaneous Hamiltonian expectation need not be constant.

Commutation with $H$ is the operational bridge from conserved quantities to symmetry. The full relation between continuous symmetries, generators, and constants of motion belongs to the symmetry volume.

Pictures of Motion

Pictures redistribute time dependence between states and observables without changing predictions.

• In the Schrödinger picture, states evolve and operators are fixed unless they have explicit time dependence.
• In the Heisenberg picture, states are fixed and operators evolve through unitary conjugation.
• In the interaction picture, a chosen solvable part of the Hamiltonian defines the reference evolution while the remaining interaction drives states.

Expectation values agree when states, operators, and density matrices are transformed consistently. A picture is bookkeeping, not a different physical theory. Detailed transformations, sign conventions, and perturbative uses belong to the Pictures of Motion chapter.

Page Map

Question	Core page	Canonical scope
What operator generates evolution?	Hamiltonians	energy observable and generator
What differential equation evolves a state?	Schrödinger Equation	core bridge to the canonical dynamics treatment
Why are closed-system probabilities preserved?	Unitary Time Evolution	unitarity as the Core rule
How is evolution mapped between two times?	Time-Evolution Operator	core bridge to propagator methods
Which states have time-independent predictions?	Stationary States	ray and density-operator criteria
What does definite energy mean?	Energy Eigenstates	spectral, measurement, and phase structure
What changes under external driving?	Time-Dependent Hamiltonians	core bridge to time ordering and driven dynamics
When is a quantity conserved?	Conservation Laws	commutator and explicit-time criteria
How do pictures redistribute time dependence?	Pictures of Motion Overview	compact bridge to detailed formulations

These nine articles form the planned chapter.

Suggested Routes

First systematic pass

Read Hamiltonians, the Schrödinger equation, unitary evolution, and the time-evolution operator. Then read energy eigenstates and stationary states before applying the formalism to solvable systems.

Wave-mechanics route

After the core sequence, continue to the Time-Independent Schrödinger Equation and model-specific spectra in the canonical-systems volume.

Driven-dynamics route

Read time-dependent Hamiltonians and the pictures overview, then continue to Time Ordering, the Interaction Picture, and time-dependent perturbation theory.

Symmetry and conservation route

Pair conservation laws with Commutators and Constants of Motion.

Evolution Sanity Checks

• Verify $U(t_0,t_0)=I$, composition, and unitarity.
• Differentiate a proposed propagator to confirm its sign and Hamiltonian convention.
• Use the ordinary exponential for $H(t)$ only after checking time commutation or another valid simplification.
• Confirm norm, trace, and Hermiticity preservation for closed-system states.
• Distinguish global phases of energy eigenstates from relative phases in superpositions.
• Include explicit operator time dependence when testing conservation.
• State whether the system is closed, reduced, conditioned, or effectively non-Hermitian.
• In numerics, test convergence together with norm or trace drift and reversibility.

Common Mistakes

• Calling the Hamiltonian only an energy matrix. It also generates time translations.
• Confusing the time-dependent and time-independent Schrödinger equations. The latter is a spectral eigenvalue problem, not the general evolution equation.
• Exponentiating the integral of a noncommuting $H(t)$ without time ordering. Hamiltonians at different times need not commute.
• Calling every energy superposition stationary. Relative phases between distinct energies evolve.
• Treating a global phase as dynamical change of a pure physical state. The ray is unchanged.
• Assuming closed-system unitarity for a reduced subsystem. Tracing out an environment generally gives a channel, not a unitary operator.
• Inferring conservation from one state’s constant expectation alone. A state-specific cancellation is weaker than an operator conservation law.
• Ignoring explicit time dependence in an observable or Hamiltonian. Commutation alone may then be insufficient.
• Mixing pictures. States, operators, and density operators must be transformed consistently.

Cross-Links

• States and Representations
• Observables and Operators
• Compatibility, Commutators, and Uncertainty
• Quantum Dynamics
• Wave Mechanics and Model Systems
• Measurement and Open Quantum Systems
• Hamiltonian Mechanics Review
• Hamiltonian glossary entry

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, rev. ed., Academic Press, 1980.