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Unformatted text preview: Physics 731
Assignment #8, due Monday, November 16 1. Let J ≡ J (1) · n/¯ for some ﬁxed n, where J (1) are the angular momentum matrices for j = 1. (a) ˆh ˆ Using the fact that the eigenvalues of J are 1, 0, −1, prove that J 3 = J . (b) Prove that D(1) (Rn (φ)) = 1 − iJ sin φ − J 2 (1 − cos φ). ˆ (c) Derive d(1) (β ). 2. Sakurai, Chapter 3, Problem 15. 3. Sakurai, Chapter 3, Problem 17. 4. Sakurai, Chapter 3, Problem 18. 5. Consider a state with position space wavefunction ψ (x) = cze−r /a . (a) Express the state in terms of the eigenstates of L2 and Lz . (b) Use the formalism of rotation matrices to compute the wavefunction of the state obtained by rotating the original state by θ about the x axis. (c) Show that your result for (b) is equivalent to ψ (R−1 x), as expected. 6. Calculate all nonvanishing matrix elements of x between the n = 2 states and the ground state of the hydrogen atom. 7. Consider the attractive spherical well potential, V (r) = −V0 , r < r0 , 0, r > r0 .
2 2 (a) Show that the quantization condition for l = 0 bound states is k = − tan(kr0 ), κ where k = (2m(E + V0 )/¯ 2 )1/2 and κ = (−2mE/¯ 2 )1/2 . h h 2 ¯ 2 /(8mr 2 ). (b) Show that there are no solutions for V0 < π h 0 2 2h (c) Show there is one solution for V0 = 9π 2 ¯ 2 /(32mr0 ), and ﬁnd the numerical value of (−2mEr0 /¯ 2 )1/2 . h 1 ...
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 Fall '09
 Everett

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