Unitary Evolution and the Propagator

Postulate 5 gives the Schrödinger equation; its solution is a unitary time-evolution operator $\hat{U}(t,t_0)$. In the position representation the kernel of this operator is the propagator $K(x,t;x^{\prime },t^{\prime })$ — the probability amplitude to travel from one spacetime point to another. The propagator packages all dynamics, connects the operator and wave formulations, and is the object the path integral computes directly. This page assembles these pieces.

1. The Time-Evolution Operator

For a time-independent Hamiltonian, integrating $i\hslash {\partial }_t|\psi ⟩=\hat{H}|\psi ⟩$ gives

$$|\psi (t)⟩=\hat{U}(t,t_0)|\psi (t_0)⟩,\hat{U}(t,t_0)=e^{-i\hat{H}(t-t_0)/\hslash }.$$

$\hat{U}$ is unitary (${\hat{U}}^{†}\hat{U}=\hat{1}$, guaranteed by self-adjoint $\hat{H}$ via Stone's theorem), composes as $\hat{U}(t_2,t_1)\hat{U}(t_1,t_0)=\hat{U}(t_2,t_0)$, and reduces to the identity at $t=t_0$. Unitarity is exactly conservation of total probability.

2. Time-Dependent $\hat{H}$ and Time Ordering

When $\hat{H}(t)$ depends on time and does not commute with itself at different times, the exponential must be time-ordered:

$$\hat{U}(t,t_0)=\mathcal{T}\exp [-\frac{i}{\hslash }{\int }_{t_0}^t\hat{H}(t^{\prime })d{t}^{\prime }]=\sum _{n=0}^{\infty }(\frac{-i}{\hslash }{)}^n{\int }_{t_0}^{t}d{t}_1\cdots {\int }_{t_0}^{t_{n-1}}d{t}_n\hat{H}(t_1)\cdots \hat{H}(t_n),$$

the Dyson series, with later times ordered to the left. This is the same series that, in the interaction picture, generates time-dependent perturbation theory and, in QFT, the $S$-matrix expansion.

3. The Propagator

Projecting evolution onto position eigenstates defines the propagator (or kernel):

$$\boxed{K(x,t;x^{\prime },t^{\prime })\equiv ⟨x|\hat{U}(t,t^{\prime })|x^{\prime }⟩,t>t^{\prime }.}$$

It evolves the wavefunction by convolution,

$$\psi (x,t)=\int d{x}^{\prime }K(x,t;x^{\prime },t^{\prime })\psi (x^{\prime },t^{\prime }),$$

so $K$ is the Green's function of the Schrödinger equation: $(i\hslash {\partial }_t-\hat{H})K=i\hslash \delta (x-x^{\prime })\delta (t-t^{\prime })$, with $K\to \delta (x-x^{\prime })$ as $t\to t^{\prime +}$. Knowing $K$ is knowing the dynamics completely.

4. Spectral Representation

Inserting a complete set of energy eigenstates $\hat{H}|n⟩=E_n|n⟩$ gives the propagator as a sum over stationary states:

$$K(x,t;x^{\prime },t^{\prime })=\sum _n{\varphi }_n(x){\varphi }_n^{\ast }(x^{\prime })e^{-i{E}_n(t-t^{\prime })/\hslash }.$$

Each eigenstate contributes a phase rotating at its own frequency $E_n/\hslash$. For a free particle the sum is a Gaussian integral,

$$K_{\text{free}}(x,t;x^{\prime },0)=\sqrt{\frac{m}{2\pi i\hslash t}}\exp \left[\frac{im(x-x^{\prime }{)}^2}{2\hslash t}\right],$$

whose phase is exactly the classical action $S_{\text{cl}}/\hslash$ of the straight-line path — the seed of the path-integral form.

5. Bridge to the Path Integral

Splitting the evolution into $N$ short steps and inserting position-completeness at each gives the propagator as a multiple integral over intermediate positions; in the $N\to \infty$ limit this becomes Feynman's sum over paths,

$$K(x,t;x^{\prime },t^{\prime })=\int \mathcal{D}[x(\cdot )]e^{iS[x(\cdot )]/\hslash },$$

with $S$ the classical action. This is derived in full on the path integral page; the propagator is the object that formulation computes, and its stationary-phase (large-action) limit recovers the classical trajectory, matching the classical limit of wave mechanics. Wick rotation $t\to -i\tau$ turns $K$ into the statistical-mechanical partition function $Z=\mathrm{Tr}{e}^{-\beta \hat{H}}$, the link between quantum dynamics and thermal equilibrium.

See also

• Postulates — Postulate 5 (unitary evolution).
• Path Integral Formulation — the sum-over-paths form of $K$.
• Spectral theorem — self-adjointness and Stone's theorem.
• Time-dependent perturbation theory — the Dyson series in practice.