Time-Evolution Operators: Definition and Composition

The time-evolution operator $U(t,t_0)$ answers a reusable question: given any initial state at $t_0$, what is its state at $t$? For a closed system, this map preserves inner products and all probability normalization. Its most useful properties are composition and reversibility.

Helpful background. Hamiltonians and the Schrödinger Equation explain what generates the map constructed here.

Evolution between Two Times

The propagator acts on the initial state as

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

Splitting an evolution interval at an intermediate time changes nothing:

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

The earliest factor is on the right, because it acts on the initial ket first. Unitarity gives the backward map:

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

This inverse reverses the mathematical evolution. Physically implementing it may require reversing a control protocol; merely waiting longer generally does not undo the motion.

For a time-independent self-adjoint Hamiltonian,

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

The spectral theorem defines this bounded unitary operator even when $H$ is unbounded. It acts on every Hilbert-space state; the strong differential Schrödinger equation additionally requires a state in $D(H)$.

For a mixed state, evolve both sides of its density operator:

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

The detailed Time-Evolution Operator treatment develops spectral formulas and transition amplitudes. For changing controls, continue to Time-Dependent Hamiltonians.

Exercise: Undoing Two Pulses

An initial state undergoes $U_1$ and then $U_2$. Which operator restores it for every initial state?

Solution

The forward operator is $U_2U_1$, so its inverse is $U_1^\dagger U_2^\dagger$. The second pulse must be undone first:

$$(U_1^\dagger U_2^\dagger)(U_2U_1)=I.$$

In general $U_2^\dagger U_1^\dagger$ gives a different result. It is the inverse of the reversed forward sequence.

References

• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 2.