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hw3246 - 8 Libo± problem 5.23 p 132 9(5 points Libo±...

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Physics 246, Spring 2007 Homework #3 Due in class, Wednesday, February 21, 2007 Feel free to discuss the problems with me and/or each other. Each student must write up his/her own solutions separately. Each problem is worth 10 points unless otherwise indicated. 1. Liboff, problem 4.15, p. 108. 2. Liboff, problem 4.28, p. 111. 3. Liboff, problem 4.35, p. 114. 4. Liboff, problem 4.36, p. 114. 5. (15 points) Consider a particle of mass m in N dimensions with H = 1 2 m n X i =1 p 2 i + V ( x i ) where V ( x i ) = 0 for all x i for 0 x i a and V ( x i ) = otherwise. (a) What are the energy eigenfunctions and their eigenvalues? (b) What are the six lowest energy eigenvalues (assume N 5)? (c) How many distinct states are there for each of these energies? 6. Liboff, problem 5.2, p. 128. 7. Liboff, problem 5.3, p. 128.

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Unformatted text preview: 8. Libo±, problem 5.23, p. 132. 9. (5 points) Libo±, problem 5.30, p. 143. 10. (5 points) Libo±, problem 5.52, p. 150. 11. Libo±, problem 5.53, p. 151. 12. (20 points) A harmonic oscillator in two dimensions can be written H = H 1 + H 2 where H i = p 2 i 2 m + 1 2 mω 2 x 2 i . The angular momentum L = r × p is L = x 1 p 2-x 2 p 1 . Defne A = 1 2 ω ( H 1-H 2 ) (1) B = 1 2 L (2) C =-i [ A, B ] / ¯ h. (3) (a) Give the explicit Form oF the operator C . (b) Show that A , B , C , H are Hermitian operators. (c) Show that the set A, B, C are closed under commutation, i.e., [ A, C ] = λ 1 B [ B, C ] = λ 2 A where λ 1 and λ 2 are constants. (d) Show that A, B, C each commute with H ....
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hw3246 - 8 Libo± problem 5.23 p 132 9(5 points Libo±...

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