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Unformatted text preview: PHY4604 Exam 2 Solutions Department of Physics Page 1 of 9 PHY 4604 Exam 2 Solutions (Total Points = 100) Problem 1 (20 points) Problem 1a (2 points): The wave function for an electron in a state with zero angular momentum: ( circle the correct answer ) (a) is zero everywhere (b) is spherically symmetric (c) depends on the angle from the z axis (d) depends on the angle from the x axis (e) is spherically symmetric for some shells and depends on the angle from the z axis for others Problem 1b (2 points): The magnitude of the orbital angular momentum of an electron in an atom is what multiple of h ? ( l = 0, 1, 2, ) ( circle the correct answer ) (a) 1 (b) (c) ) 1 ( + l l (d) 1 2 + l (e) 2 l Problem 1c (2 points): An electron is in a quantum state for which the magnitude of the orbital momentum is h 2 6 . How many allowed values of the z-component of the angular momentum are there? ( circle the correct answer ) (a) 4 (b) 5 (c) 7 (d) 8 (e) 9 I goofed up this problem. The correct answer is 17. Problem 1d (2 points): If the wave function is spherically symmetric then the radial probability density is given by: ( circle the correct answer ) (a) 2 4 r (b) 2 | | (c) 2 2 | | 4 r (d) 2 | | 4 (e) 2 | | 4 r Problem 1e (3 points): An electron in an atom is in a state with l = 3 and m l = 2 . The angle between L r and the z-axis is given by: ( circle the correct answer ) (a) 48.2 (b) 60 (c) 30 (d) 35.3 (e) 54.7 Problem 1f (3 points): SU(2) is the group of 22 matrices, U , where: (a) 1 = = UU U U (b) = U U (c) det(U)=1 (d) = U U 1 (e) T U U = * (Note: circle all the correct answers) Problem 1g (3 points): If H is an Hermitian operator then: (a) 1 = = HH H H (b) = H H (c) <H> 0 (c) <H 2 > 0 (e) its eigenvalues are real (Note: circle all the correct answers) Problem 1h (3 points): If L v is the orbital angular momentum operator then: (a) = y x L L (b) | | 2 > < z y x L L L h (c) 2 h y x L L (d) 2 L L z (Note: circle all the correct answers) PHY4604 Exam 2 Solutions Department of Physics Page 2 of 9 Problem 2 (30 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 sustem. 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 which yields ) ( 2 1 2 = , which implies that 1 = and hence 1 = . There are two energy levels, E = 1 and E 1 = + 1 ....
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