Hw7 - stationary state energies and their degeneracies The columns of the table should be as shown below l n k nl 2 Ma 2 E nl ~ 2 degeneracy 2

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PHY 4604 Fall 2010 – Homework 7 Due at 5:00 p.m. on Friday, December 3. Please turn in your homework in class before the deadline, or else bring it to NPB 2162. (You may push it under the door if NPB 2162 is unoccupied.) No credit will be available for homework submitted after the start of class on Monday, December 6. Answer all six questions. Please write neatly and include your name on the front page of your answers. You must also clearly identify all your collaborators on this assignment. To gain maximum credit you should explain your reasoning and show all working. 1. The infinite spherical well. Apart ic leo fmass M moves in the potential V ( r )= ( 0 for | r | <a , for | r | >a . As shown in class (and in Example 4.1 of Griffiths), the bound-state energies are E nl = ~ 2 k 2 nl / (2 M ), where k nl is the n th solution (in increasing order) of the equation j l ( k nl a )=0 . The smallest values of the product k nl a
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Unformatted text preview: stationary state energies and their degeneracies. The columns of the table should be as shown below: l n k nl 2 Ma 2 E nl / ~ 2 degeneracy 2. Griffiths Problem 4.9. Note that you should be able to relate this problem to Question 1 of Homework 5, and hence to quote a pair of equations whose simultaneous solution determine the ground-state energy. 3. Griffiths Problem 4.14. 4. Griffiths Problem 4.15. 5. The hydrogen atom. (a) Show that h x i = h y i = h z i = 0 in any stationary state ψ nlm of the hydrogen atom. Hint: Write x , y , and z in spherical coordinates and focus on the angular integrals. You may find helpful the relation Y m l ( π-θ, φ ) = (-1) l + m Y m l ( θ, φ ). (b) Calculate h r i , h r 2 i , h x 2 i , and h z 2 i in the ψ 210 state of hydrogen. Hint: Integrals over θ are often simplified by the identity Z π f (cos θ ) sin θ dθ = Z 1-1 f ( μ ) dμ for any function f ( μ ). 6. Griffiths Problem 4.45....
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This note was uploaded on 12/15/2011 for the course PHY 4604 taught by Professor Field during the Fall '07 term at University of Florida.

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