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Unformatted text preview: Physics 221A Fall 2005 Notes 16 Time Reversal 16.1. Introduction We have now considered the spacetime 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 spacetime 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 threedimensional space. Its tra jectory r ( t ) is a solution of the equations of motion, F = m a . We define the timereversed 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 timereversed 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 =∇ φ , 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 lefthand side is invariant under t →  t , while the righthand 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 timereversed motion r ( t ) has the wrong sense. – 2 – We see that motion in a given electric field is timereversal invariant, while in a magnetic field it is not. We must add, however, that whether a system is timereversal 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 timereversal 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 ρ → ρ, J →  J . (16 . 3) But according to Maxwell’s 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 timereversalfor the electromagnetic field under time reversal....
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This document was uploaded on 10/31/2011.
 Spring '09
 Physics

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