Time reversal · Quantum Mechanics

Intuition

Film a quantum system and run the film backwards: is the reversed film also possible? For a particle in an ordinary potential it is, and the operation that reverses it — keeping positions and flipping momenta — is complex conjugation of the wavefunction. It is not a unitary operator but an antiunitary one, which conjugates the numbers in front of states, and it has to be: a unitary reversal would turn every energy $E$ into $-E$ , and energies bounded below have no such mirror.

A film of a ball thrown upward, run backwards, shows a ball thrown upward from where it landed: a perfectly possible motion. The film of an egg breaking, run backwards, is not; but that irreversibility belongs to the many particles involved, not to the laws for each.

The momentum density of a packet moving to the right, and of its time reverse $\hat{\Theta}\psi=\psi^{*}$ : the same shape, centred at $-p$ where the original was at $+p$ . At the moment of reversal the position density does not change at all.

The time-reversal operator

For a particle without spin, time reversal is complex conjugation of the wavefunction. It keeps $\hat{x}$ , reverses $\hat{p}$ and $\hat{\mathbf{L}}$ , and is antiunitary: it conjugates coefficients and inner products while keeping every probability.

\begin{aligned}\hat{\Theta}\psi(x)&=\psi(x)^{*},\qquad \hat{\Theta}(\alpha\psi)=\alpha^{*}\hat{\Theta}\psi\\ \hat{\Theta}\hat{x}\hat{\Theta}^{-1}&=\hat{x},\qquad \hat{\Theta}\hat{p}\hat{\Theta}^{-1}=-\hat{p}\end{aligned}

Properties

$\braket{\hat{\Theta}\phi|\hat{\Theta}\psi}=\braket{\phi|\psi}^{*}$ : every probability is kept, as Wigner’s theorem allows.

Time reversal cannot be unitary

\begin{aligned}&\hat{\Theta}e^{-i\hat{H}t/\hbar}=e^{i\hat{H}t/\hbar}\hat{\Theta},\ \ \hat{\Theta}\hat{H}=\hat{H}\hat{\Theta}\\ &\Longrightarrow\ \hat{\Theta}\ \text{antilinear}\end{aligned}

Reversing the film means that evolving forward and reversing is reversing and evolving backward. For a short time this says $\hat{\Theta}$ turns $-i\hat{H}$ into $+i\hat{H}$ . A linear operator would have to anticommute with , pairing every energy with ; an antilinear one turns into by itself and can commute with .

Proof steps

\hat{\Theta}e^{-i\hat{H}t/\hbar}=e^{i\hat{H}t/\hbar}\hat{\Theta}

Evolving forward then reversing equals reversing then evolving backward.

\hat{\Theta}\,(-i\hat{H})=(i\hat{H})\,\hat{\Theta}

Compare the first-order terms in $t$ .

\hat{\Theta}\ \text{linear}\ \Rightarrow\ \hat{\Theta}\hat{H}=-\hat{H}\hat{\Theta}

A linear operator lets $-i$ pass through unchanged.

\hat{H}\ket{E}=E\ket{E}\ \Rightarrow\ \hat{H}\hat{\Theta}\ket{E}=-E\,\hat{\Theta}\ket{E}

Then every energy $E$ has a partner $-E$ : impossible for a free particle, whose energies are all positive.

\hat{\Theta}\ \text{antilinear}\ \Rightarrow\ \hat{\Theta}(-i\hat{H})=i\,\hat{\Theta}\hat{H}=i\hat{H}\hat{\Theta}

An antilinear $\hat{\Theta}$ conjugates $-i$ itself, and commuting with $\hat{H}$ is then consistent.

Applications

Practice

Reverse the Motion

For a particle without spin, time reversal takes a wavefunction to its complex conjugate. It keeps the position density and reverses the momentum.

\hat{\Theta}\psi(x)=\psi(x)^{*}

Try it

What does time reversal do to the plane wave $e^{ikx}$ ?