FinalExam_SolutionKey

FinalExam_SolutionKey - . if" . l 0 Physics 3220,...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . if" . l 0 Physics 3220, Final Exam NAME: 5. land EQQMI Student ID # Final Exam: Closed book, three 8.5x11" cheat sheets, and calculator. Test time: 2 hrs. and 30 minutes Total points: 100 Please do not open the exam until you are asked to. 0 Your exam should have 13 pages, numbered FE.1 thru FE. 13. o This exam consists of long-answer problems. 0 Write your answers in the space provided On each page for the written question. Use the back of the page if needed. SHOW YOUR WORK ON THE WRITTEN QUESTIONS. Full credit will be given only for the correct answer accompanied by your reasoning. PLEASE!! 1. Print your name on each page in the space provided. 2. Print your student Identification Number on the page above. I recommend: Don’t waste time erasing on the written problems. Just put a line through defective reasoning. Ask for more paper if needed. At the end of the exam, check that you have completed all the questions. By handing in this exam, you agree to the following statement: "On my honor, as a University of Colorado Student, I have neither given nor received unauthorized assistance on this work" Signature Good luck! Fall 2010 FE.1 Physics 3220, Final Exam NAME: QQ liglzm Problem 1. Spin 1/2 systems: Nuclear magnetic resonance. (30 pts total) Possibly useful information for spin l/z systems: » ha 0 I 0 —i 1 0 32—0", 0,: ; 0' = _ ; 0'2: ; 2 1 0 y z 0 0 -—1 One of the more important applications of spin physics is in nuclear magnetic resonance and the associated Magnetic Resonance Imaging (MRI. Marketing removed ‘nuclear’ from the name. . .). In this problem, we will consider the behavior of protons in a constant magnetic field. As you know, the proton carries an intrinsic angular momentum of 1/2 and an associated magnetic moment, it. The two quantities are related by: e 2m P S" ii: A magnetic moment in a magnetic field has a Hamiltonian: H = 11.3 In this problem, we suppose that E = 302 , where the z—hat is just the z-direction unit vector. a) There are four important operators in this problem. They are SI, S y, 32, H . The three spin operators are listed above for the representation where the eigenvectors are simple column vectors, with the eigenvectors of the z-component of spin forming the basis. Write the Hamiltonian operator in this representation. (5 pts) 'e"._§_~’..-_Cihio H:/M°B 2 " OM‘ b) There are four important operators in this problem. They are Sx,Sy,SZ, H . List a air of these 0 erators that commute. (5 pts) SE El H CWM “0ng Fall 2010 . FE.2 Physics 3220, Final Exam NAME—gaM’ Problem 1. Sgin "/5 systems: Nuclear magnetic resonance. (continued) c) For the air of 0 erators that commute list two facts that follow immediatel from their commutator bein zero: (5 points): Cornmuting Operator fact #1: M6 CM SVwa Wké' * ll ' ' 2 2 Commuting operator fact #2: M 130/ 9 él H M (2:? E d) Usin the column vector notatio find the ei en vectors for the two 0 erators that commute and find the associated ei en values. (5 points). Eigenvectors, eigenvalues Eigenvectors, eigenvalues £4.62: ' at 11;:er rang ' 0 ) 9c 0 )9?me o “L O m or+§th I 3 9‘ I ) L9 Ming): Bo . 7 2: 71%” J WAw/fl 1% yawlw Fall 2010 FE.3 Physics 3220, Final Exam ' NAME: 3) L a 231,312 5 Problem 1. Spin 1/2 systems: Nuclear magnetic resonance. (continued) e) At t=0, you measure SK and find a value of + h/ 2 . According to the discussion in Griffiths, this eigen value is associated with an eigenstate of SK and we can write that state as a linear superposition of the usual eigen column vectors of S2 as: :[1] 1m 4: 7L; tam az-Mséw‘ Z:(t=0)=%[ilm7§ 0 +E1 . a 25:0): '1‘" i) It mPiBo)% O “(ails-ii) ta ( o 1" - f) If you calculate the expectation value (SI) you would find that it oscillates in time. List the oscillation frequency and calculate its magnitude in SI units assuming that the magnetic field is of 1 Tesla strength. (5 points): Z s 2: 01 .23 F3 r'\ Pun-a. f—x \_/ ll Sill-— x ‘6 s: Fall 2010 FE.4 ) i ' Physics 3220, Final Exam NAME: g0 oJ‘wM Problem 2. Angular momentum representations: The ease of [=1 (20 points total) a) Let’s consider the case of integer angular momentum with l = 1, and use a matrix representation for the operators, just as you did in HW12. In the representation where we are using the eigenvector column vectors of Lz as the basis, the Ly operator is: h0—i0 L=—~i0—1 y JiOi 0 b) Find the eigen values for this operator. (5 points): ,. 1 -: a A L " L L H; 7k “' L F} [ y 1 7:! . O+L'/\ Fall 2010 FE.5 Physics 3220, Final Exam NAME: % 0 {JW Problem 2. Angglar momentum regresentations: The case of [=1 (continued) 0 —i 0 Again, we consider the operator: Ly = :73 i 0 —i 0 i 0 c) How many eigen vectors are there for this operator and list at least one useful property of those eigen vectors Explain your reasoning. (5 points) 77% M 32/3 MW Mu rep} 8 vac/1473 35%“, Ly 44a. MM mwyé “V M W M W] +07% Fall 2010 FE.6 Physics 3220, Final Exam NAME: Problem 3. Quantum mechanics in 2-dimensions. (20 points total) One of the nice things about solid state materials is that you can use them to make all manner of interesting quantum mechanical systems. For example, it is quite common in a crystal to find that the electrons can move like free particles, but that the electrons appear ' to have a mass that is different from the value in flee space. Not only that, often the mass depends upon the direction of motion! ‘ Let’s assume that we have a 2-d Cartesian system, where there is a different mass for motion in the X- and y—directions. Call the masses mx and my. Then, the Hamiltonian for a fine electron in the material would be: a2 "2 H: p" +p” 2nn 2n5 a) Find the wave functions for this Hamiltonian in terms of a k-Vector with components kx and [9,. You don’t need to normalize the wave function (10 points). z *2. . H4): iii—t—ifi—fiilflt Z amxaxz' a. 3 A r (Posy) M e‘wJ'Wl at; W tire b) Find an expression for the total energy for the wave function above, in terms of components kx and ky. (5 points) ' Z Z 2— 1 m' flifl'i)’ :2: E 61M” 51m), Fall 2010 FE.7 Physics 3220, Final Exam NAB/[EM Problem 3. Quantum mechanics in 2-d (continued) c) Suppose that you have a 2-d square piece of this material with each side of length a. What would the energy spectrum look like then (assume that the sides are represented by an infinite potential). Comment in particular on the quantum numbers and whether the spectrum shows degeneracy. (5 pts) m,MgJ Wflgé/ W 3% zi/or %: M 421 375:. (by) all/.1 ' W%%W Fall 2010 FE.8 Physics 3220, Final Exam NAME: ’31!“ law a Problem 3. The infinite square well. (30 points total) For this problem, you are asked to think about the I—d infinite square well potential shown below. V is infinite Well is here. V is infinite here V=0 here ‘P is zero LP is zero here here a) Use the Heisenberg Uncertainty Principle to estimate the ground state energy of a particleinthis well. (5 points) Itk "’ ""' 7% we. ac P 5L 1 AXES A’X’VQ. mark 1) “'31:: Fall 2010 FEB NAME: Eaigjzm ‘ A The infinite square well. (continued) Physics 3220, Final Exam Problem 3. b) The actual normalized wave functions for this potential are given by sine waves of the following form: aha/Estate) Write down the actual ound state ener .(5 points) a 9- Ht- 3“ : we , Elwin J QMZRZ’ 51M Q I r f“ ‘2’ Z S}. (4; :35. I. l gm 6L c) Assume that you prepare a more general state that is a superposition of the first and the second stationary state so that I/l(x, 0) = $9, (x) + u2 . Sketch the probability density for this state at i=0 (5 points) Fall 2010 FE. 10 .2 ~ NAME: 1“ Land The infinite square well. (continued) Physics 3220, Final Exam Problem 3. (1) Again assume that gar (x,0) = fl}, (x) + u2 . Explain what happens to the robabili densi for t>0. In articular ex lain Whether the robabili densit is time inde endth or not. Ifso Wh . If not what is the actual time (16 endence. (5 points) Fall 2010 FE.11 Physics 3220, Final Exam NAME: 9.0 Problem 3. The infinite square well. (continued) e) Again assume that {1/ (x,0) 2 %[u, (x) + u2 . You allow some time to pass and then you measure the position of the particle. In what ran e will measured osition? Is the s ectrum of ossible ositions discrete 0 continuous circle one of these choices ? Describe what the wave function will be like immediatel followin this measurement. (5 points) TL; smarbegvfi 0§XECL Eli/M Pox/fig sea 2,95 lli’VSh-xcgmb Fall 2010 FE.12 Wig». Problem 3. The infinite square well. (continued) Physics 3220, Final Exam 1') Now suppose that the potential well is changed so that it is only infinite on the left, and is finite on the right as shown in the figure below. V is infinite Well is here. V is finite here V=0 here ‘1‘ is zero here W e function for this well. Is the ground state energy W! an (circle one) the ground state energy of the original infinite we . Explain. (5 points) Fall 2010 FE. 1 3 ...
View Full Document

This note was uploaded on 03/03/2012 for the course PHYS 3220 at Colorado.

Page1 / 13

FinalExam_SolutionKey - . if" . l 0 Physics 3220,...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online