Hydrogen wave function in terms of m_z after m_y measurement

[sapphire_glow]

Homework Statement

Given the following wave function for hydrogen:

psi(r, t=0) = (1/sqrt(10))*(2*psi_100 - psi_210 + sqrt(2)*psi_211 + sqrt(3)*psi_21(-1))

where the subscripts show n, l, m_z, respectively, and the psi_nlm_z are already normalized.

- At t=0, we measure and find l = 1 and m_y = +1. Now what is the normalized wave function immediately after the measurement, in terms of the psi_nlm_z from the original expression? Also, what are the possible values of an energy measurement?

Homework Equations

- psi_21(my=+1) = 1/(sqrt(10))*(C1*2*psi_100 - C2*psi_210 + C3*sqrt(2)*psi_211 + C4*sqrt(3)*psi_21(-1))

- Ly |l, z> = (-i/2)*(L+ - L-) |l, z>, with known eigenvalues for L+ and L- (if useful)

The Attempt at a Solution

As shown in relevant equation #1, the result of the measurement must be a linear combination of the original states with mz, as that's all we have to start with (the measurement collapses the original wave function). I'm not really sure what to do next, though... anyone have any pointers? Is it true that the result of m_y = 1 implies that the operator applied here must have been Ly? (hence the second equation tentatively given in part 2)

 

[member 553083]

Or is the result of m_y = +1 simply from a measurement of psi(r, t=0), and no operator has been applied as of yet? (in which case, what do I do?)As for the second part, I'm assuming that the energy measurement would be one of the pre-defined energy levels for hydrogen, but I'm not really sure which one. Would it be the same as the original energy of the wave function prior to the measurement? (i.e., the sum of the energies of its individual components) Or would it be some combination of the original energy levels, based on the collapsed wave function's new coefficients? (if so, how would I figure out which one?) Any help would be greatly appreciated!