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Unformatted text preview: NAME____________________________________________________________ PHYS 360 Quantum Mechanics Spring 2010 Monday, Feb 15 Mid‐Term Exam 1 This is a closed‐book exam. You should have your own hand‐written page of equations and notes. Before starting, make sure you have signed your name above. This exam is worth 100 points. Each question has the point total indicated. Write everything on these pages. Partial credit may be awarded for incorrect or incomplete answers if you have shown sufficient understanding. It is very important that your work is legible. In some cases no points will be awarded for a correct answer if you have not shown your work in solving the problem. Note that the last page of this exam should display some integrals that may be of use. Solutions will be posted on the course web site within one day. ALL exams must be turned in no later than 2:20 pm, the end of the scheduled class period. 1. (5 points) The wave function for a particle in an infinite square well (0<x<a) at 2 3π x sin( ). Which one of the following is the wave t=0 is given by: Ψ ( x, 0 ) = a a function at time t? (Clearly circle your choice.) 2 3π x sin( ) cos( E3t / ) (a) Ψ ( x, t ) = a a 2 3π x sin( ) exp(−iE3t / ) (b) Ψ ( x, t ) = a a (c) Both (a) and (b) above are correct. (d) None of the above. 2. (5 points) The wave function for a particle in an infinite square well (0<x<a) at 2 3π x 2 sin( ). What is the probability density Ψ ( x, t ) at t=0 is given by: Ψ ( x, 0 ) = a a time t? 2 3π x 2 ) cos 2 ( E3t / ) (a) Ψ ( x, t ) = sin 2 ( a a 2 3π x 2 ) exp(−i 2 E3t / ) (b) Ψ ( x, t ) = sin 2 ( a a 2 3π x 2 ) (c) Ψ ( x, t ) = sin 2 ( a a (d) None of the above. 2 3. (5 points) The wave function for a particle in an infinite square well (0<x<a) at πx t=0 is given by: Ψ ( x, 0 ) = A sin 3 ( ) where A is a suitable normalization constant. a Which one of the following is the wave function at time t? πx (a) Ψ ( x, t ) = A sin 3 ( ) cos( E3t / ) a πx (b) Ψ ( x, t ) = A sin 3 ( ) exp(−iE3t / ) a (c) Both (a) and (b) above are correct. (d) None of the above. 4. (5 points) The wave function for a particle in an infinite square well (0<x<a) at πx t=0 is given by: Ψ ( x, 0 ) = A sin 3 ( ) where A is a suitable normalization constant. a What is the probability density Ψ ( x, t ) after time t? πx 2 2 (a) Ψ ( x, t ) = A sin 6 ( ) cos 2 ( E3t / ) a 3π x 2 2 ) exp(−i 2 E3t / ) (b) Ψ ( x, t ) = A sin 6 ( a πx 2 2 (c) Ψ ( x, t ) = A sin 6 ( ) a (d) None of the above. 2 3 Problem 5: Infinite Square Well (55 points total) An infinite square well is described by the potential Denote any wave function in region I ( x < − a ) as ψ I ( x ) , in region II ( − a ≤ x ≤ + a ) as ψ II ( x ) , and in region III ( x > + a ) as ψ III ( x ) . Consider a particle of mass m. A. (10 points) Starting with the time‐independent Schrödinger equation just in region II, derive the general solution to this eigenvalue equation for the eigenfunctions ψ II ( x ) . (That is, give the general form of the solution before applying boundary conditions.) B. (5 points) Briefly explain why ψ I ( x ) = ψ III ( x ) = 0 . 4 C. (5 points) Write down the boundary condition equations for x=‐a and for x=+a. D. (5 points) From your knowledge of the infinite square well, sketch the ground I state and the first excited state wave functions ( ψ nI ( x ) ): 5 E. (10 points) By applying the boundary conditions to your general solution, solve for the ground state eigenfunction. Make sure it agrees with your sketch. (You do not need to normalize this function.) F. (10 points) By applying the boundary conditions to your general solution, solve for the first excited state eigenfunction. Make sure it agrees with your sketch. (You do not need to normalize this function.) 6 G. (5 points) What are the energy eigenvalues of the first four lowest energy states? H. (5 points) Normalize your ground state wave function. 7 Problem 6: More Infinite Square Well (25 points) Consider a particle of mass m inside an infinite square well as shown: The initial (normalized) wave function is Ψ ( x, 0 ) = 3 x: a3 What is the probability that a measurement yields the ground state energy π 22 ? That is, how much of the total wave function is due to the n=1 state? E1 = 2 ma 2 8 Problem 6, continued. 9 2 ! sin (bx )dx = x sin(2bx ) " 2 4b ! sin (bx )dx = 3 cos( 3bx ) " 9 cos(bx ) 12b sin 4 (bx ) ! sin (bx ) sin(2bx)dx = 2b 2 2 cos3 (bx )( 3 cos(2bx ) " 7) ! sin(bx) sin (2bx)dx = 15b 2 " x sin(bx ) dx = 2 sin(bx ) ! bx cos(bx ) b2 x sin(2bx ) 4b ! cos (bx ) dx = 2 + 10 ...
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