How to generalize the momentum reversal operator?

Question

In non-relativistic quantum mechanics, in position space, the complex conjugate operator $C$ flips the sign of the momentum operator, $CpC=-p$ (and thus also flips the sign of the orbital angular momentum).

The operator $T=C$ is usually called the time reversal operator, but this only works in certain cases. For example, it does not work in the presence of spins (where $T=Ci\sigma_y$). So $C$ may not always be called the (full) time reversal operator.

Would it be safe to say that $C$ is the momentum reversal operator? I think this holds always. Also, does $C$ do anything else that is not associated to momentum directly? Is the general behavior of $C$ well-studied? References would be much appreciated.

Edit: after writing this, I realized that it also flips the sign of $i\partial_t$ (some kind of energy reversal?), but I am not sure how helpful it can be to talk of non-stationary states (or of relativistic quantum mechanics).

Edit: I now realize that this question is ill defined, I did not want people to focus on the basis issue. A better question would be to ask specifically how to generalize the $C$ operator to other basis in relation to time and momentum.

Answer 1

The "complex conjugation" operator is basis dependent and so ill defined.

A vector that has real components in one basis may be have complex components in another. For example the $x\leftrightarrow p$ change-of-basis formula for the components is $$ \tilde \psi(p) = \langle p|\psi\rangle = \int dx \langle p|x\rangle\langle x|\psi\rangle= \int dx e^{ipx}\psi(x) $$ and the $i$ in $e^{-ipx}$ means that a state $|\psi\rangle$ that has real components $\psi(x)=\langle x|\psi\rangle$ in the $|x\rangle$ basis can have complex components $\tilde \psi(p)=\langle p|\psi\rangle$ in the $|p\rangle$ basis.

Correspondingly the $\hat p$ opertor acts as $-i\partial _x$ in the $|x\rangle$ basis and so flips sign under congugation, while $\hat x$ which acts by multiplication by the real number $x$ does not. But in the $|p\rangle$ basis $\hat p$ is just multiplication by the real number $p$ while $\hat x$ acts as $i\partial_p$, so the "flips" are reversed.

As consequence of this, "complex conjugation" can only be defined as an operator if you also specify a set of basis vectors that are declared "real"

Thus, when defining antilinear operators such as time reversal it is dangerous to factor the operator as "$T=Ci\sigma_y$" -- although this is common practice. It is confusing because this formula is only correct when acting on states in the $\sigma_x$-diagonal or $\sigma_x$-diagonal bases which are connected by change-of-basis matrices with real entries. If you have need to use a different spin basis, say the general ${\bf n}\cdot {\boldsymbol \sigma}$-diagonal basis, the formula for $T$ is different.