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Unformatted text preview: the wavefunction. Some possibly useful data: the form of the two-dimensional Laplacian in polar coordi-nates is given in problem (5). It might also be useful to recall that for a single SHO, the unnormalized eigenstates take the form: n ( x ) = H n ( r m ~ x ) e-mx 2 / 2 ~ where H n ( x ) is a (Hermite) polynomial of order n in x . 3 . Show that the quantity J = X m Y * lm ( 1 , 1 ) Y lm ( 2 , 2 ) 1 is rotationally invariant. Use the above result to prove the spherical harmonic addition theorem P l (cos ) = 4 2 l + 1 X m Y * lm ( 1 , 1 ) Y lm ( 2 , 2 ) where is the angle between the directions specied by ( 1 , 1 ) and ( 2 , 2 ). 4 . Show that U = e i~ n = cos( ) + i~ n sin( ) where n is a unit vector. 2...
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