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Title: GR8677.028
Post by: conantee on December 31, 2009, 01:49:25 PM
28. Eigenfunctions for a rigid dumbbell rotating about its center have a \phi dependence of the form \psi (\phi) = Ae^{im \phi}, where m is a quantum number and A is a constant. Which of the following values of A will properly normalize the eigenfunction?
(A) \sqrt{2 \pi}
(B)  2 \pi
(C)  (2 \pi)^2
(D) \frac{1}{\sqrt{2 \pi}}
(E) \frac{1}{2 \pi}


Title: Re: GR8677.028
Post by: conantee on December 31, 2009, 01:55:03 PM
Since we can foresee the simplification from exponential of imaginary number. Let's do this problem properly.

The normalization condition : \int_{0}^{2 \pi}\psi ^* \psi d \phi = 1

Then we have \int_{0}^{2 \pi}A ^* A e^{-im \phi} d^{im \phi} d \phi = 1

|A|^2 2 \pi = 1

The only A satisfying this condition is A from (D)

The correct answer is (D) and 71 of 100 people answer correctly.