Time-Evolution Operator

Purpose

The time-evolution operator $U(t,t_0)$ maps every state of a closed system from an initial time $t_0$ to a final time $t$:

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

It packages the Schrödinger initial-value problem into one linear operator. Once $U(t,t_0)$ is known, it propagates state vectors, density operators, and transition amplitudes without solving a new differential equation for each initial state.

The canonical construction is Time-Evolution Operator. This card collects the formulas and checks needed in calculations.

At a glance

Task	Formula
Propagate a ket	$\lvert\psi(t)\rangle=U(t,t_0)\lvert\psi(t_0)\rangle$
Equal-time condition	$U(t_0,t_0)=I$
Forward equation	$i\hbar\,\partial_tU(t,t_0)=H(t)U(t,t_0)$
Backward equation	$-i\hbar\,\partial_{t_0}U(t,t_0)=U(t,t_0)H(t_0)$
Composition	$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$
Inverse and adjoint	$U(t,t_0)^{-1}=U(t_0,t)=U^\dagger(t,t_0)$
Time-independent $H$	$U(t,t_0)=e^{-iH(t-t_0)/\hbar}$
Commuting driven $H(t)$	$U(t,t_0)=\exp[-(i/\hbar)\int_{t_0}^{t}H(s)\,ds]$
General driven $H(t)$	$U(t,t_0)=\mathcal T\exp[-(i/\hbar)\int_{t_0}^{t}H(s)\,ds]$
Propagate a density operator	$\rho(t)=U(t,t_0)\rho(t_0)U^\dagger(t,t_0)$

The ordinary exponential of the time integral is valid only when

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

for all relevant times, or when another argument makes time ordering unnecessary.

Definition and initial condition

The operator is defined by its action on arbitrary initial states:

$$\lvert\psi(t)\rangle = U(t,t_0)\lvert\psi_0\rangle, \qquad \lvert\psi_0\rangle \equiv \lvert\psi(t_0)\rangle.$$

It depends on the Hamiltonian and both endpoint times, not on the selected initial vector. No elapsed time has passed when the endpoints agree, so

$$U(t_0,t_0)=I.$$

For a closed system, the propagator is unitary:

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

It therefore preserves all inner products:

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

Writing only $U(t)$ is safe when $t_0=0$ is fixed or when time-translation invariance makes dependence on $t-t_0$ explicit. Otherwise it hides information needed for composition and differentiation.

Endpoint differential equations

Substitute

$$\lvert\psi(t)\rangle = U(t,t_0)\lvert\psi_0\rangle$$

into the time-dependent Schrödinger equation. Since the initial vector is arbitrary,

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

This forward equation differentiates the final endpoint while holding $t_0$ fixed. Its equivalent integral equation is

$$U(t,t_0) = I -\frac{i}{\hbar} \int_{t_0}^{t} H(s)U(s,t_0)\,ds.$$

Differentiating the initial endpoint gives the backward equation

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

The Hamiltonian multiplies on the left in the forward equation and on the right in the backward equation. That order matters when operators at different times fail to commute.

For an unbounded Hamiltonian, these equations are understood on suitable domains. A formal equality of differential expressions does not alone prove the existence of a unitary propagator.

Composition, inverse, and adjoint

Evolution through an intermediate time $t_1$ composes as

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

The earliest interval acts first and is therefore the rightmost factor:

$$\lvert\psi(t_0)\rangle \xmapsto{\,U(t_1,t_0)\,} \lvert\psi(t_1)\rangle \xmapsto{\,U(t_2,t_1)\,} \lvert\psi(t_2)\rangle.$$

Set $t_2=t_0$ in the composition law:

$$U(t_0,t)U(t,t_0)=I.$$

Hence

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

For unitary evolution,

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

Together,

$$U(t_2,t_1)^\dagger = U(t_1,t_2).$$

These identities are useful analytical checks and sensitive numerical diagnostics.

Time-independent Hamiltonian

If $H$ is self-adjoint and time independent, let

$$\tau=t-t_0.$$

Then

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

This is an operator exponential, not an entry-by-entry exponential of a matrix. For finite matrices it can be computed from diagonalization, Schur-based algorithms, or scaling-and-squaring methods. For a general self-adjoint operator it is defined by spectral calculus.

Discrete spectral form

If

$$H=\sum_n E_nP_n$$

is the discrete spectral decomposition, then

$$U(t,t_0) = \sum_n e^{-iE_n\tau/\hbar}P_n.$$

Every energy eigenspace acquires a phase. A degenerate eigenspace receives one common phase, which is why the formula uses spectral projectors rather than a preferred basis inside that space.

For

$$\lvert\psi(t_0)\rangle = \sum_n c_n\lvert E_n\rangle,$$

the evolved state is

$$\lvert\psi(t)\rangle = \sum_n c_n e^{-iE_n\tau/\hbar} \lvert E_n\rangle.$$

The magnitudes $\lvert c_n\rvert$ remain fixed, while relative phases between different energies evolve.

Continuous spectral form

The projection-valued spectral measure gives the general expression

$$H = \int_{\mathbb R}E\,dP(E),$$ $$U(t,t_0) = \int_{\mathbb R} e^{-iE\tau/\hbar}\,dP(E).$$

This includes discrete, continuous, and mixed spectra without treating generalized energy eigenvectors as normalizable Hilbert-space states.

One-parameter group

Time-independent evolution depends only on elapsed time:

$$U(t,t_0)=U(\tau).$$

Composition becomes

$$U(\tau_2)U(\tau_1) = U(\tau_1+\tau_2).$$

The order is immaterial here because both factors are functions of the same Hamiltonian. Strong continuity, unitarity, and this group law are the setting of Stone Theorem.

Shift of energy origin

Replacing

$$H\mapsto H'=H+E_{\mathrm c}I$$

gives

$$U'(t,t_0) = e^{-iE_{\mathrm c}\tau/\hbar}U(t,t_0).$$

For one isolated branch this is a global phase and leaves probabilities unchanged. If different interferometric branches experience different energy offsets, the resulting phase difference can be observable.

Time-dependent Hamiltonian

For general $H(t)$, the formal solution for $t\geq t_0$ is

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

The time-ordering operator $\mathcal T$ places later-time Hamiltonians to the left. Iterating the integral equation gives

$$\begin{aligned} U(t,t_0) &= I -\frac{i}{\hbar} \int_{t_0}^{t}dt_1\,H(t_1) \\ &\quad+ \left(-\frac{i}{\hbar}\right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, H(t_1)H(t_2) \\ &\quad+\cdots . \end{aligned}$$

The nested limits enforce chronological order. The complete expansion and its interaction-picture use belong to the planned Dyson-series card and Time Ordering.

If

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

for every pair of relevant times, then ordering has no effect:

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

Time dependence alone is not the obstruction. Noncommutation at distinct times is.

For backward propagation $t<t_0$, use

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

or an equivalent anti-time-ordered expression. Do not use the forward time-ordering convention blindly with reversed limits.

Short-time and piecewise evolution

For a sufficiently regular Hamiltonian and a short positive interval $\delta t$,

$$U(t+\delta t,t) = I-\frac{i}{\hbar}H(t)\delta t +O(\delta t^2).$$

The first-order term is not exactly unitary by itself; its unitarity defect is of order $\delta t^2$. Exponential or structure-preserving integrators are often preferable for long numerical propagation.

Suppose

$$H(t) = \begin{cases} H_1,&t_0\leq t<t_1,\\ H_2,&t_1\leq t\leq t_2. \end{cases}$$

Then

$$\begin{aligned} U(t_2,t_0) &= e^{-iH_2(t_2-t_1)/\hbar} \\ &\quad\times e^{-iH_1(t_1-t_0)/\hbar}. \end{aligned}$$

The earlier Hamiltonian appears on the right. If $[H_1,H_2]\ne0$, reversing the factors changes the prediction.

Transition amplitudes

In orthonormal bases, the matrix elements

$$U_{mn}(t,t_0) \equiv \langle m\rvert U(t,t_0)\lvert n\rangle$$

are transition amplitudes. Coefficients evolve as

$$c_m(t) = \sum_n U_{mn}(t,t_0)c_n(t_0).$$

Unitarity gives

$$\sum_m U_{mn}^*(t,t_0) U_{mk}(t,t_0) = \delta_{nk}.$$

Composition becomes a sum over a complete set of intermediate states:

$$U_{mn}(t_2,t_0) = \sum_k U_{mk}(t_2,t_1) U_{kn}(t_1,t_0).$$

This is a coherent sum over unobserved alternatives. If a measurement records the intermediate state, state update and classical conditioning define a different process.

In a continuous position basis,

$$K(x,t;x_0,t_0) \equiv \langle x\rvert U(t,t_0)\lvert x_0\rangle$$

is the propagator kernel. Its integral composition law is collected in Propagator Composition.

Density-operator evolution

For a closed system,

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

Differentiation gives the von Neumann equation:

$$i\hbar\dot\rho(t) = [H(t),\rho(t)].$$

Unitary conjugation preserves:

• trace and positivity;
• the spectrum of $\rho$;
• purity $\operatorname{Tr}(\rho^2)$;
• von Neumann entropy;
• rank and all finite-dimensional spectral moments.

The formula is not the most general evolution of a subsystem. Open-system dynamics, selective measurement, and effective non-Hermitian propagation require channels, instruments, or conditional normalization.

Two-level example

For

$$H = \frac{\hbar\Omega}{2} \mathbf n\cdot\boldsymbol\sigma, \qquad \lVert\mathbf n\rVert=1,$$

use

$$(\mathbf n\cdot\boldsymbol\sigma)^2=I$$

to sum the exponential:

$$\begin{aligned} U(\tau) &= \exp\!\left( -\frac{i\Omega\tau}{2} \mathbf n\cdot\boldsymbol\sigma \right) \\ &= \cos\!\left(\frac{\Omega\tau}{2}\right)I \\ &\quad- i\sin\!\left(\frac{\Omega\tau}{2}\right) \mathbf n\cdot\boldsymbol\sigma. \end{aligned}$$

For $\mathbf n=\hat{\mathbf x}$ and initial state $\lvert0\rangle$,

$$\lvert\psi(t)\rangle = \cos\!\left(\frac{\Omega\tau}{2}\right) \lvert0\rangle -i \sin\!\left(\frac{\Omega\tau}{2}\right) \lvert1\rangle.$$

Thus

$$p_1(t) = \sin^2\!\left( \frac{\Omega\tau}{2} \right).$$

The operator exponential turns the Hamiltonian directly into a Bloch-sphere rotation.

Assumptions and validity

The standard formulas assume:

• a closed quantum system between any stated interventions;
• a self-adjoint Hamiltonian, not merely a formally symmetric differential expression;
• a well-posed Schrödinger initial-value problem;
• one consistent picture and sign convention;
• a dimensionless exponent, including the factor $1/\hbar$ unless $\hbar=1$ has been declared.

In finite dimensions, a piecewise continuous Hermitian matrix $H(t)$ gives a well-defined unitary propagator. For time-dependent unbounded Hamiltonians, self-adjointness at each instant is not by itself a complete existence theorem; domains, regularity, and stability of the family matter.

A non-Hermitian effective Hamiltonian may generate useful no-jump or resonance evolution, but the resulting operator is not unitary. Its norm loss must be interpreted within the model rather than silently renormalized as closed-system evolution.

Numerical checks

For a computed approximation $\widetilde U(t,t_0)$, inspect:

$$\epsilon_{\mathrm{init}} = \lVert \widetilde U(t_0,t_0)-I \rVert,$$ $$\epsilon_{\mathrm{unit}} = \lVert \widetilde U^\dagger\widetilde U-I \rVert,$$

and the composition defect

$$\begin{aligned} \epsilon_{\mathrm{comp}} &= \bigl\lVert \widetilde U(t_2,t_0) \\ &\quad- \widetilde U(t_2,t_1) \widetilde U(t_1,t_0) \bigr\rVert. \end{aligned}$$

Also monitor conserved quantities implied by the Hamiltonian and compare step sizes or integrator orders. Norm preservation alone does not prove that the phase or dynamics is accurate.

Calculation workflow

1. Specify both endpoint times and the picture.
2. Decide whether $H$ is time independent, commuting at distinct times, piecewise constant, or genuinely time ordered.
3. Use spectral projectors or a stable matrix-exponential routine for time-independent $H$.
4. Put earlier intervals on the right in every product.
5. Apply the resulting operator to all desired states or density operators.
6. Check the equal-time condition, composition, unitarity, and relevant conserved quantities.
7. Separate unitary propagation from measurement update or reduced open-system evolution.

Common mistakes

• Writing $U(t)=e^{-iHt/\hbar}$ without fixing $t_0=0$ and assuming time-independent $H$.
• Exponentiating matrix entries separately rather than computing the matrix exponential.
• Reversing the order of factors in composition or piecewise evolution.
• Omitting time ordering because $H(t)$ is Hermitian; Hermiticity does not imply $[H(t_1),H(t_2)]=0$.
• Putting $H(t_0)$ on the wrong side in the backward endpoint equation.
• Treating a truncated first-order short-time operator as exactly unitary.
• Treating the phase of one energy eigenstate as an observable change while ignoring that only relative phases matter.
• Calling $U$ an observable; a generic unitary is not self-adjoint.
• Applying closed-system conjugation to postselection or a noisy subsystem.
• Ignoring domains for unbounded Hamiltonians.
• Using forward time ordering unchanged for backward propagation.

Cross-links

• Canonical Time-Evolution Operator
• Core Formalism Calculation Guide
• Schrödinger Equation Formula Card
• Unitary Time Evolution
• Time-Dependent Hamiltonians
• Time Ordering
• Propagator Composition
• Unitary Identities
• Baker–Campbell–Hausdorff
• Stone Theorem

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958, Sections 26–28.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977, Volume 1, Chapter III.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Chapters 4 and 11.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, Chapter 2.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, Chapters 5–6.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.

Exercises

1. Verify both endpoint differential equations for

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

with time-independent $H$.

Solution

Differentiating the final endpoint gives

$$\frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU,$$

so

$$i\hbar\frac{\partial U}{\partial t}=HU.$$

Differentiating the initial endpoint changes the sign:

$$\frac{\partial U}{\partial t_0} = \frac{i}{\hbar}HU.$$

Because $H$ commutes with every function of itself, $HU=UH$. Therefore

$$-i\hbar\frac{\partial U}{\partial t_0}=UH.$$

2. A system evolves with $H_1$ for a duration $\tau_1$ and then with $H_2$ for a duration $\tau_2$. Write the total propagator. Under what condition can it be written as $\exp[-i(H_1\tau_1+H_2\tau_2)/\hbar]$?

Solution

Chronological composition gives

$$U = e^{-iH_2\tau_2/\hbar} e^{-iH_1\tau_1/\hbar}.$$

The earlier interval is on the right. The two exponentials combine into the ordinary exponential of their sum when

$$[H_1,H_2]=0.$$

Without this condition, BCH produces commutator corrections.

3. Suppose $H(t)=f(t)A$, where $A=A^\dagger$ is fixed and $f(t)$ is real. Find $U(t,t_0)$.

Solution

At any two times,

$$[H(t_1),H(t_2)] = f(t_1)f(t_2)[A,A] = 0.$$

Time ordering is unnecessary, so

$$U(t,t_0) = \exp\!\left[ -\frac{iA}{\hbar} \int_{t_0}^{t}f(s)\,ds \right].$$

The exponent is anti-Hermitian because $A$ is self-adjoint and the integral is real; hence $U$ is unitary.

4. Prove from the composition law and unitarity that

$$U(t_2,t_1)^\dagger=U(t_1,t_2).$$

Solution

Composition through $t_2$ gives

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

Thus

$$U(t_1,t_2) = U(t_2,t_1)^{-1}.$$

Unitarity identifies the inverse with the adjoint:

$$U(t_2,t_1)^{-1} = U(t_2,t_1)^\dagger.$$

Combining the two equations yields the result.