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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 longanswer 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 zdirection 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 zcomponent 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 azMséw‘ Z:(t=0)=%[ilm7§ 0 +E1 . a
25:0): '1‘" i) It mPiBo)% O “(ailsii)
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'\
Puna.
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 2dimensions. (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 2d 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 kVector 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 2d (continued) c) Suppose that you have a 2d 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’VShxcgmb 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 ...
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