4604_Exam2_solutions_fa07

4604_Exam2_solutions_fa07 - PHY4604 Fall 2007 R. D. Field...

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Unformatted text preview: PHY4604 Fall 2007 R. D. Field Department of Physics Page 1 of 8 Exam 2 Solutions PHY 4604 Exam 2 Solutions (Total Points = 100) Problem 1 (20 points): Circle the correct answer for following (4 points each). (1a) An electron is in a one-dimensional infinite square well with zero potential energy in the interior and infinite potential energy at the walls. A graph of its wave function (x) versus x is shown. What is the value of the quantum number n ? ( circle the correct answer ) (a) 0 (b) 2 (c) 4 (d) 6 (e) 8 (1b) An atom is in a state with orbital quantum number l = 2 . What are the possible values of the magnetic quantum number m . ( circle the correct answer ) (a) 1, 2 (b) 0, 1, 2 (c) 0, 1 (d) 1, 0, -1 (e) 2, 1, 0, -1, -2 (1c) An electron is in a quantum state for which the magnitude of the orbital momentum is h 10 3 . How many allowed values of the z-component of the angular momentum are there? ( circle the correct answer ) (a) 9 (b) 10 (c) 17 (d) 18 (e) 19 (1d) An electron is in a quantum state for which there are six allowed values of the z-component of the angular momentum. What is the magnitude of the angular momentum vector? ( circle the correct answer ) (a) h 6 (b) 2 / 5 h (c) h 6 (d) h 10 3 (e) 2 / 35 h (1e) What is 2 2 2 2 in SU(2)? ( circle the correct answer ) (a) 5 + 3+ 3+ 3 + 1 +1 (b) 16 (c) 9 + 5 + 2 (d) 6 + 5 + 3 + 2 (e) 10 + 6 Problem 2 (40 points): Consider a spin system described by the Hamiltonian: = 1 1 i i H where and 1 are real positive constants. (a) (6 points) Find the energy levels of the system. How many energy levels are there? What is the ground state energy, E , and the first excited state energy, E 1 ? Answer: There are two energy levels: E = 1 , E 1 = + 1 . Solution: The energy levels are the solution of 1 1 = i i PHY4604 Fall 2007 R. D. Field Department of Physics Page 2 of 8 Exam 2 Solutions which yields ) ( 2 1 2 = , which implies that 1 = and hence 1 = . There are two energy levels, E = 1 and E 1 = + 1 . (b) (6 points) What are the ( normalized ) eigenkets corresponding to the ground state, | E > , and the first excited state , |E 1 > ? Answer: >= 1 2 1 | i E >= 1 2 1 | 1 i E . Solution: For the ground state we see that = + + = b a b a i b i a b a i i ) ( 1 1 1 1 1 and hence a +i 1 b = ( - 1 )a, which implies that a = -ib and hence if b = -1 then a = i and >= 1 2 1 | i E ....
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4604_Exam2_solutions_fa07 - PHY4604 Fall 2007 R. D. Field...

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