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HW3n - a is the Bohr radius(a Find the momentum space...

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Physics 731 Assignment #3, due Monday, September 28 1. Prove that the Legendre operator L = - (1 - x 2 ) d 2 dx 2 + 2 x d dx (1) is self-adjoint on the interval [ - 1 , 1] under the boundary conditions that f ( ± 1) = finite. 2. Sakurai, Problem 21, Chapter 1. 3. Sakurai, Problem 26, Chapter 1. 4. Sakurai, Problem 31, Chapter 1. 5. Sakurai, Problem 32, Chapter 1. 6. Using the relation x | p = 1 2 π ¯ h e ipx/ ¯ h , (2) find the expressions x | XP | α and x | PX | α in terms of ψ α ( x ) . Can these results be found directly by using the fact that in the | x representation, P acts like - i ¯ h d dx ? 7. The 1 s wave function of hydrogen is ψ 1 s ( x ) = x | 1 s = 1 πa 3 0 e - r/a 0 , (3) in which r is the usual spherical radial coordinate and
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Unformatted text preview: a is the Bohr radius. (a) Find the momentum space wavefunction ˜ ψ 1 s ( ~ p ) . Note: it is useful first to prove that if f ( ~x ) = f ( r ) , then F ( ~ q ) = Z e-i~ q · ~x f ( ~x ) d 3 x = 4 π q Z ∞ f ( r ) sin( qr ) rdr. (4) (b) Starting from ˜ ψ 1 s ( ~ p ) , determine the probability distributions for (i) a measurement of the mag-nitude of the momentum without any reference to direction and (ii) a measurement of one of the components of the momentum without reference to the other two components. 1...
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