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Unformatted text preview: Physics 221A Fall 2005 Notes 16 Time Reversal 16.1. Introduction We have now considered the space-time symmetries of translations, proper rotations, and spatial inversions (that is, improper rotations) and the operators that implement these symmetries on a quantum mechanical system. We now turn to the last of the space-time symmetries, namely, time reversal. As we shall see, time reversal is different from all the others, in that it is implemented by means of antiunitary transformations. 16.2. Time Reversal in Classical Mechanics Consider the classical motion of a single particle in three-dimensional space. Its tra- jectory r ( t ) is a solution of the equations of motion, F = m a . We define the time-reversed classical motion as r (- t ). It is the motion we would see if we took a movie of the original motion and ran it backwards. Is the time-reversed motion also physically allowed (that is, does it also satisfy the classical equations of motion)? The answer depends on the nature of the forces. Consider, for example, the motion of a charged particle of charge q in an electric field E =- , for which the equations of motion are m d 2 r dt 2 = q E ( r ) . (16 . 1) If r ( t ) is a solution of these equations, then so is r (- t ), as follows easily from the fact that the equations are second order in time, so that the two changes of sign coming from t - t cancel. However, this property does not hold for magnetic forces, for which the equations of motion include first order time derivatives: m d 2 r dt 2 = q c d r dt B ( r ) . (16 . 2) In this equation, the left-hand side is invariant under t - t , while the right-hand side changes sign. For example, in a constant magnetic field, the sense of the circular motion of a charged particle (clockwise or counterclockwise) is determined by the charge of the particle, not the initial conditions, and the time-reversed motion r (- t ) has the wrong sense. 2 We see that motion in a given electric field is time-reversal invariant, while in a magnetic field it is not. We must add, however, that whether a system is time-reversal invariant depends on the definition of the system. In the examples above, we were thinking of the system as consisting of a single charged particle, moving in given fields. But if we enlarge the system to include the charges that produce the fields (electric and magnetic), then we will find that time-reversal invariance is restored, even in the presence of magnetic fields. This is because when we set t - t , the velocities of all the particles change sign, so the current does also. But this change does nothing to the charges of the particles, so the charge density is left invariant. Thus, the rules for transforming charges and currents under time reversal is , J - J . (16 . 3) But according to Maxwells equations, this implies the transformation laws E E , B - B , (16 . 4) for the electromagnetic field under time reversal. With these rules, we see that time-reversalfor the electromagnetic field under time reversal....
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