Foundations of Time Evolution

This chapter fixes the basic language of closed-system quantum dynamics before the volume branches into pictures, propagators, Green functions, path integrals, phase-space methods, and field-theory bridges.

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

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

For a closed system with a self-adjoint Hamiltonian, $U(t,t_0)$ is unitary:

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

This is the Hilbert-space statement of probability conservation. In coordinate space it becomes a continuity equation for probability density and probability current; in abstract Hilbert space it is simply preservation of inner products.

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	The series is formal until a truncation and error regime are specified.
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.

Boundaries

This chapter owns the common closed-system infrastructure: the time parameter, the Hamiltonian as generator, unitary evolution, time-independent and time-dependent Hamiltonians, and the origin of time ordering.

It does not own detailed solution methods for particular potentials. Those belong to Wave Mechanics and Model Systems. It also does not own measurement update, decoherence, Lindblad dynamics, or quantum trajectories; those belong to Measurement and Open Quantum Systems. When an effective non-Hermitian Hamiltonian appears, it should be identified as an open-system, conditional, or approximation device rather than ordinary closed-system evolution.

Physical Interpretation

The Hamiltonian does two jobs that are easy to conflate. It generates time translations, and, when time-translation symmetry is present, it is the conserved energy observable. A time-dependent Hamiltonian still generates evolution, but it usually does not define a conserved system energy. 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 are not stationary in the same sense: their relative phases evolve, and those relative phases control interference, beats, and time-dependent expectation values.

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.
• Calling every Hamiltonian “energy” without checking whether time-translation symmetry is present.
• 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.

Cross-Links

• Map of Quantum Dynamics
• Time as a Parameter
• Time-Dependent Schrödinger Equation
• Hamiltonians as Generators
• Time-Evolution Operator
• Unitarity and Conservation of Probability
• Stationary States and Phases
• Time-Independent Hamiltonians
• Time-Dependent Hamiltonians
• Time Ordering
• Dyson Expansion as Formal Evolution
• Constants of Motion
• Schrödinger Picture
• Heisenberg Picture
• Unitary Time Evolution

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.

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?

Assume $[H(t_1),H(t_2)]=0$ 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:

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