ps13 - Homework for Physics 316, Modern Physics I

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Unformatted text preview: Homework for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 05/06/03 - 9:55 in 132 Rockefeller Hall Exercise 1: Compute the radial part Rnl (r) of the Hydrogen wave function for n < 3 for all possible l. Show by direct integration that the hydrogen state |n = 1, l = 0, m = 0 is orthogonal to the state |n = 2, l = 0, m = 0 . ^ Exercise 2: For Hydrogen, an eigenstate of the total angular momentum J 2 and its z-component ^ Jz is expanded as a linear combination of orbital angular momentum eigenstates: 1 1 1 1 |nljsmj = A|n, l, ml = mj - , s, ms = + B|n, l, ml = mj + , s, ms = - . 2 2 2 2 ^ ^ ^ Using J = L + S, argue that the linear combination cannot contain any other states. Exercise 3: (a) Use the Schroedinger Equation for the radial part of the wave function for hydrogen, and verify that a solution of the form Rnl = (c0 - c1 r)rl+1 e-br (2) (1) corresponds to the energy eigenvalue -1/(l + 2)2 in dimensionless units. (b) Putting l = 1, sketch (i) the form of the wave function; and (ii) the effective potential-energy curve with the energy eigenvalue superimposed as a horizontal line. (from [1,12-3]) Exercise 4: Consider the spherically symmetric hydrogen-atom state for n=2. The (unnormalized radial part of the wave function is r2 = u2 (r) = re r - 2a 0 (1 - r ). 2a0 (3) Sketch the radial probability distribution w(r) r 2 2 for this state and find the value of r for which it has a maximum. (from [1,12-4]) Exercise 5: Prepare for the final exam: Review all lecture notes. Suggested practice questions: [11-16], [12-6], [12-8], [12-14] [1] An Introduction to Quantum Physics, French and Taylor, Norton, W. W. & Company, Inc. (1990) ...
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell University (Engineering School).

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