Consider two Cartesian coordinate systems rotated with respect to each other in the x-y plane as shown

Question:

Consider two Cartesian coordinate systems rotated with respect to each other in the x-y plane as shown in Fig. 1.4.


Fig. 1.4.


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(a) Show that the rotation matrix that takes the barred basis vectors to the unbarred basis vectors is


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and show that the inverse of this rotation matrix is, indeed, its transpose, as it must be if this is to represent a rotation.


(b) Verify that the two coordinate systems are related by Eq. (1.13c).


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(c) Let Aj be the components of the electromagnetic vector potential that lies in the x-y plane, so that Az = 0. The two nonzero components Ax and Ay can be regarded as describing the two polarizations of an electromagnetic wave propagating in the z direction. Show that A + iA = (Ax + iAy)e−i∅. One can show (cf. Sec. 27.3.3) that the factor e−i∅ implies that the quantum particle associated with the wave—the photon—has spin one [i.e., spin angular momentum ℏ = (Planck’s constant)/2π].


(d) Let hjk be the components of a symmetric tensor that is trace-free (its contraction hjj vanishes) and is confined to the x-y plane (so hzk = hkz = 0 for all k). Then the only nonzero components of this tensor are hxx = −hyy and hxy = hyx. As we shall see in Sec. 27.3.1, this tensor can be regarded as describing the two polarizations of a gravitational wave propagating in the z direction. Show that hx̅x̅ + ihx̅y̅ = (hxx + ihxy)e−2i∅. The factor e−2i∅ implies that the quantum particle associated with the gravitational wave (the graviton) has spin two (spin angular momentum 2ℏ); cf. Eq. (27.31).


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