Exercise [22.17]

deant · 13 Oct 2012, 06:34

I'm not exactly sure what's being asked here, but it can be pointed out that the "answer given by projection" is the only one that preserves angular momentum (spin). Unless the photon can impart or recieve angular momentum from the mirror during the reflection process (and it can't, presumably), the reflected photon must be in the $|\rho-\!\!\!>$ state, because that's the only reflected state that has the same angular momentum as the initially emitted photon.

More generally, we can regard a wavefunction $\psi$ undergoing a measurement Q as being composed of several mutually incompatible alternatives $\phi_1\mathellipsis\phi_n$ , where $\psi = \phi_1 + \cdots + \phi_n$ and where each of the $\phi_i$ represent a different (eigen-)value for Q. The projection postulate is equivalent to saying that making the measurement "selects" one of these alternatives from $\psi$ , and drops the others; making the measurement doesn't "add anything new", however, according to this postulate.

In the case of this example and a "null" measurement, this is particularly important, because (a) the photon has not iteracted with the measuring device, so it can't have been changed by that, and (b) any "new" reflected components in the final wavefunction would alter the overall angular momentum, which is a conserved quantity... so there can't be any.