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Unformatted text preview: the wavefunction. Some possibly useful data: the form of the twodimensional Laplacian in polar coordinates is given in problem (5). It might also be useful to recall that for a single SHO, the unnormalized eigenstates take the form: ψ n ( x ) = H n ( r mω ~ x ) emωx 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 ( θ 1 ,φ 1 ) Y lm ( θ 2 ,φ 2 ) 1 is rotationally invariant. Use the above result to prove the spherical harmonic addition theorem P l (cos θ ) = 4 π 2 l + 1 X m Y * lm ( θ 1 ,φ 1 ) Y lm ( θ 2 ,φ 2 ) where θ is the angle between the directions speciﬁed by ( θ 1 ,φ 1 ) and ( θ 2 ,φ 2 ). 4 . Show that U = e i~σ · ˆ nφ = cos( φ ) + i~σ · ˆ n sin( φ ) where ˆ n is a unit vector. 2...
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 Spring '10
 Sghy
 Physics, mechanics, Polar Coordinates, Angular Momentum, Hn

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