Math Infinitum

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Intuition

Every solution of the Schrödinger equation is the initial state acted on by one operator, the evolution operator. For a Hamiltonian that does not change in time it is $e^{-i\hat{H}t/\hbar}$ . It is unitary, which is the Schrödinger equation’s promise that no information is lost, and evolving for two stretches of time is evolving for their sum.

A film projector moves every frame forward by the same rule. The evolution operator is that rule: give it a frame and a time, and it returns the frame that far ahead — or behind, since the film runs both ways.

Time evolution as an operator

The evolution operator $\hat{U}(t)$ maps the state at time 0 to the state at time $t$ . For a time-independent Hamiltonian it is an exponential, computed through the spectral decomposition of $\hat{H}$ : each energy eigenstate is multiplied by its own phase.

\begin{aligned}\ket{\psi(t)}&=\hat{U}(t)\ket{\psi(0)},\qquad \hat{U}(t)=e^{-i\hat{H}t/\hbar}\\ \hat{U}(t)&=\sum_{n}e^{-iE_{n}t/\hbar}\ket{E_{n}}\bra{E_{n}}\end{aligned}

Properties

$\hat{U}(0)=\hat{1}$ and $\hat{U}(t_{1})\hat{U}(t_{2})=\hat{U}(t_{1}+t_{2})$ : evolution in stages composes.

The evolution operator is unitary

\hat{U}(t)^{\dagger}\hat{U}(t)=\hat{1}

Use the spectral decomposition. The adjoint conjugates each phase, and the product of the two sums collapses, because the projectors onto different eigenstates multiply to zero, into the sum of the projectors — which is the identity.

Proof steps

\hat{U}(t)^{\dagger}=\sum_{n}e^{+iE_{n}t/\hbar}\ket{E_{n}}\bra{E_{n}}

The energies are real, so taking the adjoint conjugates each phase.

\hat{U}^{\dagger}\hat{U}=\sum_{m,n}e^{iE_{m}t/\hbar}e^{-iE_{n}t/\hbar}\ket{E_{m}}\braket{E_{m}|E_{n}}\bra{E_{n}}

Multiply the two sums.

=\sum_{n}\ket{E_{n}}\bra{E_{n}}

$\braket{E_{m}|E_{n}}=\delta_{mn}$ and the phases then cancel.

=\hat{1}

The completeness relation of the energy eigenbasis.

Applications

Practice

An Exponential of the Hamiltonian

For a Hamiltonian that does not depend on time, the evolution operator is $e^{-i\hat{H}t/\hbar}$ . In the energy basis it multiplies each eigenstate by its own phase.

\hat{U}(t)=e^{-i\hat{H}t/\hbar}

Try it

What is the evolution operator for a time-independent Hamiltonian?