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Unformatted text preview: NAME____________________________________________________________ PHYS 360 Quantum Mechanics Spring 2010 Friday, May 7, 2010 Final Exam This is a closed‐book exam. You should have your own hand‐written page of equations and notes. You may also have the hand‐written pages you used for the two mid‐term exams. Before starting, make sure you have signed your name above. This exam is worth 200 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. A solution set will be posted on the course web site (but maybe not today). 1 Problem One: Infinite Square Well A particle of mass m in a one‐dimensional infinite square well (of width a) at time t=0 is equally likely to be found at any position inside the well. Question 1: (10 points) What is the initial (normalized) wave function? 2 Problem One: Infinite Square Well (cont.) Question 2: (20 points) What is the probability that a measurement of the energy π 22 would yield the value ? 2 ma 2 3 Problem Two: One­dimensional Potential “Cliff” A one‐dimensional potential is given by: Region I: VI ( x ) = 0 x<0 \$# !" # II! I! Region II: VII ( x ) = −Vo 0 ≤ x Question 1: (5 points) Write down the time‐independent Schrödinger equations for region I and region II. Question 2: (5 points) Write down the general solution for the (traveling wave) eigenfunctions ψ I ( x ) in region I, and for ψ II ( x ) in region II. That means, don’t worry yet about boundary conditions, normalizations, etc. 4 Problem Two: One­dimensional Potential “Cliff” (cont.) Now we consider a specific situation. Assume a traveling wave of amplitude A exists in region I and is moving in the +x direction, towards the cliff. Question 3: (10 points) What are the boundary conditions at x=0? Apply these to your functions from the previous question. 5 Problem Two: One­dimensional Potential “Cliff” (cont.) Question 4: (10 points) What is the intensity of the reflected wave, as a fraction of the intensity of the incident wave? (Express this as a function of the energy E.) Question 5: (5 points) For what energy E will the reflectivity be a maximum? 6 Problem Three: Spin­½ Particles A single spin‐½ particle is in the state χ = 1⎛3⎞ . As usual, we are using the 10 ⎜ −1 ⎟ ⎝ ⎠ ⎛1⎞ ⎛0⎞ ( (z) convention that χ +z ) = ⎜ ⎟ , and χ − = ⎜ 1 ⎟ . ⎝0⎠ ⎝ ⎠ Question 1: (10 points) Prove that this wave function is properly normalized. Question 2: (10 points) What is the probability of getting + if you measure Sz? 2 7 Problem Three: Spin­½ Particles (cont.) Question 3: (10 points) What is the probability of getting + if you measure Sx? 2 8 Problem 4: The Hydrogen Atom The ground state wave function for the hydrogen atom is: 1 ψ 100 (r,θ , φ ) = e− r / a . 3 πa Question: (25 points) Find r 2 for an electron in this state. 9 Problem Five: Labeling the Spin States (30 points) On the drawing below, fill in the quantum numbers for the states shown as well as the values of S2 and |S|. !"# S 2 = ________! 2 +2! S = ________! +1! sm = ______ sm = _______ 0! !\$# sm = _______ -1! !%# -2! sm = ______ 10 Problem 6: Multiple Choice Questions Question 1: (10 points) Bell’s Inequality proved that it is possible to experimentally distinguish between the predictions of (orthodox) quantum mechanics and the “realistic” view that quantum mechanics is an incomplete theory, perhaps missing some hidden variable. Suppose measurements were done on two spin‐½ particles in a singlet state (e.g. π 0 → e+ + e− ). Which of the following statements is NOT supported by the subsequent experimental results? [ ] A. When the electron is measured to be spin up, prior to that the positron did not have a definite spin. [ ] B. As Einstein explained, the positron would have a definite spin even before the electron spin is measured since, after all, that’s no different from saying that the Moon must be in its orbit even if you are not looking at it. [ ] C. Despite the laws of special relativity, the collapse of the positron wave function caused by the electron measurement is instantaneous, or at least ~10,000 faster than the speed of light. [ ] D. Although the measurement on the electron seems to “force” the positron into a definite spin state, it is not clear that any “useful” information has been transmitted between the two particles. Question 2: (10 points) Consider the orbital angular momentum for a system with a central potential (e.g. the hydrogen atom). Indicate (by putting a check‐mark) which of the following commutators is zero. (There can be more than one correct choice.) [ ] [Lx,Ly] [ ] [Ly,Lz] [ ] [Lz,Lx] [ ] [L2,Lx] [ ] [L2,Ly] [ ] [L2,Lz] 11 Question 3: (10 points) The Pauli Exclusion Principle for identical fermions is responsible for which of the following phenomena? (There may be more than one correct choice.) [ ] It is very difficult to compress a rock into a smaller volume. [ ] The radiation spectrum of a black body at high temperature is drastically different than predicted by classical physics. [ ] In metallic materials, some the electrons can flow and carry a current. [ ] Some of the electrons in a carbon atom are in orbitals with non‐zero angular momentum. [ ] When some stars run out of fuel and stop burning, they collapse into neutron stars. Question 4: (10 points) The wave function for a particle in an infinite square well 2 3π x sin( ). Which one of the following is the (0<x<a) at t=0 is given by: Ψ ( x, 0 ) = a a wave 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. 12 Question 5: (10 points) The wave function for a particle in an infinite square well πx (0<x<a) at t=0 is given by: Ψ ( x, 0 ) = A sin 3 ( ) where A is a suitable normalization a constant. 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. 13 14 ...
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## This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue University-West Lafayette.

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