[NOTES/QM-11004] Time Reversal Symmetry

Consider classical motion of a particle under influence of a force field. If at some instant $t_0$ the direction of velocity is reversed, the particle will retrace its path. For a charged particle in magnetic field, the path will be retraced if the magnetic field is also reversed. To see the time reversal symmetry of the classical equations in another way consider a thought experiment of motion of a particle thrown up in a gravitational field. Let the motion of the particle be captured on a film while going up and on a second film while going down. Now let one of the films be run backwards, what you see cannot be distinguished from what you see in the second film running forward. Assuming that the effect of air friction to be negligible, no measurement on the motions seen in the two films, one running backwards and the other film running forward, will be able to distinguish between them. We say that the classical Newton's laws retain their form under time reversal $t \to t^\prime = -t$. This means that one cannot distinguish the motion of a particle in a force field with time reversed motion. What about quantum mechanics? In quantum mechanics the time evolution is described by the Schr\"{o}dinger equation. \begin{equation} i \hbar \dd[\psi(\vec{r},t)]{t} = -\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r},t) + V(\vec{r}) \psi(\vec{r},t) \label{eq01}. \end{equation} It is easy to see that the form of the Schr\"{o}dinger equation does not change if we take take complex conjugate of the Schrodinger equation and then make a replacement $t \to -t\equiv t^\prime$, we would successively get  \begin{eqnarray} - i \hbar \dd[\psi^*(\vec{r},t)]{t} = -\frac{\hbar^2}{2m} \nabla^2 \psi^*(\vec{r},t) + V(\vec{r}) \psi^*(\vec{r},t), \label{eq02A}\\ i \hbar \dd[\psi^*(\vec{r},t^\prime)]{t^\prime} = -\frac{\hbar^2}{2m} \nabla^2 \psi^*(\vec{r},t^\prime) + V(\vec{r}) \psi^*(\vec{r},t^\prime). \label{eq02} \end{eqnarray}
The wave function $\psi^*(\vec{r},t^\prime)$ describes the time reversed motion. As a simple example, $\exp(ikx-iE t/\hbar)$ represents a (free) particle moving to the right on $x$ axis, and the complex conjugate wave function, $\exp(ikx+E t^\prime/\hbar)$, represents a particle moving to the left. The Schrodinger equation for a charged particle in presence of magnetic field has time reversal symmetry, if the sign of the magnetic field is also reversed.