Time reversal operator from a mathematical viewpoint

Question

I am a math student who is reading Sakurai's Modern Quantum Mechanics Chapter 4 on the the time reversal operator.

First, Sakurai derived that for a given Hamiltonian $H$ of a system, the time reversal operator $\Theta$ is an anti-unitary operator such that $[H,\Theta] = 0$.

Suppose the existence of $\Theta$. It seems that the condition $[H,\Theta] = 0$ alone is not enough to uniquely determine $\Theta$ (up to a constant with unit length).

For example, let's consider that for a spinless system, $H = -p^2/2m + V$ for a real potential function $V$. If we further require $\Theta p \Theta^{-1} =-p$ and $\Theta x \Theta^{-1} =x$, then we uniquely (up to a constant) determine $\Theta = K$. Here, $K$ is "taking complex conjugate".

My question is that given a Hamiltonian $H$, how to determine its time reversal operator $\Theta$ (if exists)? Should we have to make more (probably natural) assumptions and try to find (probably unique) $\Theta$? Like in the previous example, we further assume $\Theta p \Theta^{-1} =-p$ and $\Theta x \Theta^{-1} =x$.

On the other hand, how should we prove that a Hamiltonian does not have time symmetry? Does it mean we have to show that (under those additional natural assumptions) there does not exist any $\Theta$ such that $[H,\Theta]=0$?

Or, for a spinless system with a Hamiltonian $H$, we "choose" the time reversal operator $\Theta = K$ and then examine whether $[H,\Theta]=0$ holds or not, while for a $1/2$-spin system, we "choose" $\Theta = \sigma_yK$?

Answer 1

In the book the time reversal operator $\Theta$ (that should be called motion reversal operator) is physically defined as the operator that when applied to a state reverses the kinematic variables of that state.

It is established that $\Theta$ must be anti-unitary to make sense. Further it is argued (on physical grounds) that we must have

• $\Theta p \Theta^{-1} =- p$
• $\Theta x \Theta^{-1}=x$
• $\Theta S \Theta^{-1} = -S$ (where $S$ is the spin operator).

As a mathematician you would probably like to define $\Theta$ to be the (unique up to phase) anti-linear map that satisfies these properties.

Note that all of these properties are independent of the Hamiltonian. We dont really need a Hamiltonian to reverse the kinematic variables of a state. The Hamiltonian enters through the question whether it has motion reversal symmetry ($[\Theta , H]=0$) or not.

It is then shown that it follows from the properties in the above list that:

• $\Theta = K$ (up to phase) when acting on the spatial part of a wave function.
• $\Theta$ acting on the spin part of a wave function must satisfy $\Theta = \exp(-i \pi S_y/\hbar)K$ (up to a phase).

We can of course combine the spatial part and the spin part using the tensor product to obtain $\Theta = \Theta_{\mathrm{spatial}} \otimes \Theta_{\mathrm{spin}}$ acting on an arbitrary state. And so $\Theta$ is uniquely determined (up to phase) by the above properties (independent of the Hamiltonian).

So to summarize: The motion reversal operator $\Theta$ of a system is uniquely determined (up to phase) by the axiomatic description of being anti-unitary and having the properties in the first list. In particular $\Theta$ does not depend on the Hamiltonian.

We can then use $\Theta$ to check for motion reversal symmetry of the Hamiltonian ($[H, \Theta ] = 0$ or not). If we have motion reversal symmetry we can use the usual techniques to obtain extra information from the symmetry.