Foundations of Time Evolution

How does a Hamiltonian turn an initial state into predictions at a later time? This chapter answers that question for closed systems, beginning with finite-dimensional examples and explaining the additional care required by unbounded Hamiltonians.

Helpful background. Unitary Time Evolution supplies norm-preserving evolution; the wavefunction Schrödinger equation provides its position-space example.

The central idea is that time evolution is generated by the Hamiltonian. In the Schrödinger picture,

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

and the same evolution can be represented by an operator $U(t,t_0)$:

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

This chapter explains what these equations mean, when the familiar exponential solution is valid, and why time ordering is unavoidable for general driven systems.

Core Objects

The time-evolution operator is defined by

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

A time-independent self-adjoint Hamiltonian generates a unitary group. Sufficiently regular Hermitian matrix-valued $H(t)$ also defines unitary evolution. For unbounded time-dependent Hamiltonians, self-adjointness at each time alone does not supply all the existence and domain conditions. When the unitary propagator is defined,

$$U^\dagger(t,t_0)U(t,t_0)=I.$$

Together with $UU^\dagger=I$, this expresses reversible preservation of inner products and total probability. For a local Schrödinger Hamiltonian, the corresponding local statement is the continuity equation for density and current. Global unitarity alone does not imply the familiar current formula for an arbitrary nonlocal Hamiltonian.

When $H$ is time independent,

$$U(t,t_0) = e^{-iH(t-t_0)/\hbar}.$$

When $H(t)$ depends on time and fails to commute with itself at different times,

$$[H(t_1),H(t_2)]\ne 0,$$

the ordinary exponential of the integral is generally wrong. The formal solution is instead

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(t')\,dt' \right],$$

where $\mathcal T$ places later-time operators to the left.

Reading Path

Question	Start here	What to watch
What role does time play in ordinary quantum mechanics?	Time as a Parameter	Time labels evolution; it is not a universal observable in the default formalism.
What equation moves a state in time?	Time-Dependent Schrödinger Equation	It is an equation of motion, not an energy eigenvalue problem.
Why is the Hamiltonian the generator?	Hamiltonians as Generators	Self-adjointness is what makes the generated evolution unitary.
What maps all states from $t_0$ to $t$?	Time-Evolution Operator	Composition fixes the order of factors.
Why is total probability conserved?	Unitarity and Conservation of Probability	Fixed measurement probabilities can change even while total probability is conserved.
Why do energy superpositions oscillate?	Stationary States and Phases	Single eigenspaces give global phases; different energies give relative phases.
When does the simple exponential work?	Time-Independent Hamiltonians	Energy components acquire phases; superpositions acquire relative phases.
What changes for driven systems?	Time-Dependent Hamiltonians	Time dependence is not automatically perturbative.
Why does operator order matter?	Time Ordering	Noncommuting Hamiltonians cannot be freely rearranged.
How is time-ordered evolution expanded?	Dyson Expansion as Formal Evolution	For bounded regular generators the series converges; a finite truncation still needs an error estimate.
What quantities stay fixed during evolution?	Constants of Motion	Conservation means more than one stationary expectation value.

After this chapter, the natural next step is the Schrödinger Picture and Heisenberg Picture, where the same evolution is described with different placements of time dependence.

Physical Interpretation

The Hamiltonian generates evolution. When the physical Hamiltonian is time independent, its expectation is conserved. A driven system can have a time-dependent energy observable, whose expectation need not be conserved: under the usual domain assumptions, $d\langle H\rangle/dt=\langle\partial_tH\rangle$. A time-dependent change of frame can also change the generator, so identify the physical energy in that frame before interpreting it. A driven atom, a pulsed qubit, and a particle in a time-dependent trap can all evolve unitarily while exchanging energy with the external apparatus represented by $H(t)$.

Stationary states are special because a single energy eigenstate changes only by a phase. Superpositions of different energies develop relative phases that can produce interference and beats. A superposition entirely inside one degenerate eigenspace acquires only a common phase and remains stationary. More generally, a mixed state is stationary for fixed $H$ when it commutes with the evolution.

Common Mistakes

• Treating the time-dependent Schrödinger equation as the same object as the time-independent eigenvalue equation.
• Applying $e^{-iH(t-t_0)/\hbar}$ when the Hamiltonian depends on time.
• Forgetting that $U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$ means the earliest factor acts first and appears on the right.
• Treating time ordering as notation that can be dropped after writing it once.
• Confusing closed-system unitary dynamics with effective nonunitary dynamics used after tracing out, conditioning on, or approximating other degrees of freedom.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977 — time evolution and conservation laws.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980 — self-adjoint generators and unitary groups.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — time ordering and driven systems.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — evolution operators and stationary states.

Exercises

Composition order

Suppose $t_0\lt t_1\lt t_2$. Starting from $\lvert\psi(t_1)\rangle=U(t_1,t_0)\lvert\psi(t_0)\rangle$ and $\lvert\psi(t_2)\rangle=U(t_2,t_1)\lvert\psi(t_1)\rangle$, derive the composition law for $U(t_2,t_0)$.

Solution

Substitute the first equation into the second:

$$\lvert\psi(t_2)\rangle = U(t_2,t_1)U(t_1,t_0)\lvert\psi(t_0)\rangle.$$

Since this must hold for arbitrary initial states,

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

The factor that evolves the state first is on the right.

When can time ordering be removed?

In finite dimensions, assume a continuous $H(t)$ with $[H(t_1),H(t_2)]=0$ for all $t_1,t_2$ in the interval. Explain why the time-ordered exponential reduces to the ordinary exponential of the integral.

Solution

Time ordering matters only because products at different times may not commute. If every $H(t_i)$ commutes with every $H(t_j)$, then any ordered product

$$H(t_1)H(t_2)\cdots H(t_n)$$

can be rearranged without changing its value. The ordered integration domains in the Dyson expansion combine into the full integration domain with the usual factor $1/n!$, reproducing the ordinary exponential series:

$$U(t,t_0) = \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(t')\,dt' \right].$$