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Title: GR8677.048
Post by: conantee on January 04, 2010, 04:16:05 AM
48. A transition in which one photon is radiated by the electron in a hydrogen atom when the electron's wave function changes from \psi_1 to \psi_2 is forbidden if \psi_1 and \psi_2
(A) have opposite parity
(B) are orthogonal to each other
(C) are zero at the center of the atomic nucleus
(D) are both spherically symmetrical
(E) are associated with different angular momentum

Title: Re: GR8677.048
Post by: conantee on January 06, 2010, 02:24:39 PM
This problem involves with selection rules. If you still remember Fermi's golden rule and perturbation theory, it would be great. If not, just make guess as follows:
(B) is clearly wrong since any eigenstates are orthogonal to each other.
(E) is wrong since photon can also carry angular momentum. So \psi_1 and \psi_2 are possibly associated with different angular momenta
Now you have (A), (C) and (D) to guess. To make a further step, let's go to the origin of selection rules.

Well, the selection rule comes from the evaluation of \int \psi^*_1 \mu \psi_2 d \tau. This is called transition moment integral. This term appears in Fermi's golden rule in bra-ket notation where \mu comes from the perturbed Hamiltonian. In case of spontaneous decay, the moment \mu comes down to be same as one for electrical dipole transition, where \mu is x, y or z.

Now, one can see that only (D) makes the integral vanishes since weighting x,y or z over the sphere makes the one side of evaluation becomes positive and other side to be negative. It just means that transition never happens in such a case.

Hence, the correct answer is (D). 21 of 100 got the correct answer.