4604_Exam1_fa06 - PHY4604 Fall 2006 Exam 1 PHY 4604 Exam 1...

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PHY4604 Fall 2006 Exam 1 Department of Physics Page 1 of 5 PHY 4604 Exam 1 (Total Points = 100) Problem 1 (20 points): Circle true or false for following (2 point each). (a) (True or False) One of the “breakthroughs” that lead to quantum mechanics was the idea of associating differential operators with the dynamical variables. (b) (True or False) Solutions of Schrödinger’s equation of the form ) ( ) ( ) , ( t x t x φ ψ = Ψ correspond to states with definite energy E . (c) (True or False) Solutions of Schrödinger’s equation of the form ) ( ) ( ) , ( t x t x = Ψ correspond to states in which the probability density 2 | ) , ( | ) , ( t x t x Ψ = ρ is independent of time. (d) (True or False) The wave function Ψ (x,t) must vanish in a region of infinite potential. (e) (True or False) It is possible for a free particle to have a definite energy. (f) (True or False) In quantum mechanics particles can enter the “classically forbidden” region where V 0 > E ( i.e. KE < 0). (g) (True or False) The operator A op A op is hermitian. (h) (True or False) If A op and B op are hermitian then A op B op is also hermitian. (i) (True or False) The commutator operator [(p x ) op ,(x 2 ) op ] is equal to h i 2 . (j) (True or False) In position-space the commutator operator [(p x ) op ,sin(kx)] is equal to ) cos( kx k i h .
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PHY4604 Fall 2006 Exam 1 Department of Physics Page 2 of 5 Problem 2 (30 points): Consider an electron with mass m e confined within an infinite square well defined by V(x) = 0 for 0 < x < L , V(x) = + otherwise. (a) (2 points) Using Schrödinger’s equation calculate the allowed stationary state eigenfunctions ψ n (x) , where the complete wavefunctions are given by h / ) ( ) , ( t iE n n n e x t x = Ψ ψ , and normalize the eigenfunctions so that the probability of finding the electron somewhere in the box is one. (b) (2 points) Show that the wavefunctions Ψ n (x,t) correspond to states with definite energy ( i.e.
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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4604_Exam1_fa06 - PHY4604 Fall 2006 Exam 1 PHY 4604 Exam 1...

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