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**Unformatted text preview: **Quantum Mechanics 357: Mid—Term Examination Keshav Dasguptal, Alex Maloney2, Rebecca Danosg, Racha Cheaib4 Rutherford Physics Building, McGill University, Montreal, QC HSA 2T8, Canada Notes
This question paper has six pages including the cover page. Please attempt four out of the
given eight questions. If you attempt more than four questions, you will be given credit
for the best set of four questions. Please read carefully all the questions before you start
writing the solutions. Make sure you write clearly and legibly. Mention all the necessary
steps required to reach a particular solution. This is a closed book examination, but
calculators, regular dictionary and translation dictionary are allowed. Total time allowed two hours. Total credits: 100 marks. Venue: Room 26, Leacock Building. Friday, 5th November 2009, 6:00 pm l Examiner 2 Associate Examiner
3 Teaching Assistant
4 Teaching Assistant Note: Answer any four out of the following eight questions. All questions carry an equal
credit of 25 marks. You are advised not to spend more than twenty ﬁve minutes on each question5. Some useful formulae are listed at the end of this question paper. 1. A n—dimensional Hilbert space is parametrised by basis vectors [A + i), z' m 0, n that
are eigenvectors of an operator A with eigenvalues /\ + i. I deﬁne another operator B in this space that satisﬁes the following commutation relation with the operator A:
[B, A] : aB where a is a given integer with 0 g a g n. A third operator Q could be used to deﬁne a Hamiltonian H in the space as:
H - B : -z’Q where the only thing known about Q is that it is Hermitian. Find the matrix representation of the Hamiltonian for an arbitrary choice of or with a 7£ 0. What happens when a = O? 1x25225 2. Give short answers to the following questions: (a) Imagine due to some reason the Planck’s constant it doubles it value i.e huew 2 2h. This
would mean, in terms of the original it, all particles would effectively behave as integer spin
particles. What would you expect to see in a Stern—Gerlach experiment if I now send silver atoms through magnetic ﬁeld oriented along z direction? Give reason for your answer. (b) Show that the eigenvalues of a Hermitian operator are real. Argue that two eigenvectors
of a Hermitian operator corresponding to two different eigenvalues are orthogonal. Also
show that if two observables A and B commute, and if W11) and W) are two eigenvectors
of A with different eigenvalues, the matrix element (wllBWu) is zero. Finally show that
if two Hermitian operators are diagonisable by the some unitary operator U, then these two operators have to commute. 5+4X5m25 5 Spend at least ﬁfteen to twenty minutes reading the questions carefully. Many of the
questions themselves have choices, so choose those questions that you may ﬁnd easy to do in the given time. Do as many as you can to maximise you chances7 because the best four will be chosen. 1 3. One of the most important set of matrices that we have been using throughout this course is called the Pauli matrices. They are given by: _01 _0—z' _10
01‘10’0y“10’02“0—1 Verify that (a) these matrices are all Hermitian with eigenvalues :i:1, (b) e‘” m (S 891 > and (c) em“ 2 11 cos a +2271 sin a where a is a constant and ll is the 2 X 2 identity
matrix Prove that for any two arbitrary vectors A and B, with components (Ax, Ay, AZ) and (Bag, By, 3;) respectively, we have the following identity:
(0-A)(0~B) :A-Bll+z'0-(A><B) where we have deﬁned 0 - A 2 03:14.7: + ayAy + UZAZ with a similar defination for 0 . B. Argue that the above relation also implies the following identity:
UjUk 2 éjk101+ 6jk11
l where the epsilon tensor 6ij is anti—symmetric wrt interchange of any two of its coordinates,
and is (a) zero if the indices j, k,l are not all different, (b) 1 if j, k,l is an even permutation
of CE, y, z, and (c) -1 if j, k,l is an odd permutation of x, y, 2:. Finally show that any
arbitrary Hermitian 2 X 2 matrix M can be written as a linear combination of the identity matrix and the three Pauli matrices 01', in the following way: iTr(M)llI + 3 {Tm/wan -a M 2 2 l
2 3+4+4+5+4+5= 25 4. I construct a molecule with three atoms located at the vertices of an equilateral tri-
angle. An electron in this molecule can reside near any of the three atoms forming three
orthonormal states given by with 2' 2 17 2, 3. When we neglect the possibility of elec—
trons jumping from one atom to another, the three states lqﬁi) are the eigenstates of a
Hamiltonian Hg with equal eigenvalues E0. However there is a minor perturbation W coming from couplings between the states that act in the following way: W M51) m" —al¢2>a W W3) 2 _ai¢2>v W |¢2> : —al¢i>-al¢3> 2 where a is a real positive constant. Find the location of the electron at any subsequent time t, if the initial state of the electron at t : 0 is localised near atom 1. 1x25225 5. We have two operators A and B, both of which commute with their commutator [A, B].
Argue that for a generic function F (B) of the operator B the following commutation relation holds:
[A9F(B)l : [AiBlF’(B) where F’ (B) denotes the derivative of F wrt B in the same sense as F’ denotes the derivative of F wrt the variable z. Now using the above result show that: l
eAeB 2 8A+B€2[A,B] where the above result is restrictive to the case when the operators commute with their commutator. Finally show that when the operators do not commute with their commuta- tor, then: eABe“A : B + [A,B] + —[A,[A,BH + —[A,[A,[A,Bm +.... 5+10+10225 6. Give short answers to the following questions: (a) Consider any observable A associated with the state of a system in quantum mechanics. If A is intrinsically time independent, then show that there is always an uncertainity of AA h
AE<ld(A)/dtl> 3 ‘2” (b) A non—Hermitian operator A has eigenvalues Ai and corresponding eigenvectors the form: The adjoint operator AJf has the same set of eigenvalues but dzﬁerent corresponding eigen—
vectors M). Show that the eigenvectors form a biorthogonal set, i.e 2 O for all
choices of A: # Aj. (c) A Hilbert space is spanned by n—states given by lui) such that they form eigenstates of an operator A with eigenvalues A1; The eigenvalue A1, corresponding to the eigenstate 3 lul) in this space, is the largest of all the eigenvalues. Let us consider a state |u> which is a small deformation of the eigenstate Jul) in the following way: 77. I“) = WI) + Zéiluzl i=2 where << 1 so that we remain close to the eigenvector [2“). Show that we can have the following inequality:
<uiAiu> 3 A1
Mu) allowing an error in /\1 to be of the order leilg.
8 + 7 + 10 : 25
7. If az denote the third Pauli matrix, verify the following identity: _Q 02 0 _ (Q l 0 _ , (Q 02 0
eXp [ 2 ( 0 -az)l ‘ [C0811 2) (0 1 smh 2) 0 -0z
Where 77 is a constant parameter. Using the above relation, show that for a state deﬁned _ a c w cosh 77 .
as K) A (b) and (d) » (sinh 77), we have. _ a: a
U00) - [x/c+d(———J~1 2" )+\/c-d(1+2 (b)
l0 [m(1+20§)+m(1—201)] U HI where (a, b, c, d) are all constant parameters. This relation is very useful in understanding certain important properties of fermions. 12.5 + 12.5 2 25 8. Consider two parallel quantum system, we call them system 1 and system 2, such that
one of them, say system 1, is governed by a Hamiltonian H1 and the system 2 is governed
by a magnetic ﬁeld vector B that has components Bx, By and Bz. The Hamiltonian H2
governing system 2 can be determined by the coupling of spin % particles with a magnetic
ﬁeld B described above. The magnetic moment it of system 2 is given in terms of J as n : VJ. The Hamiltonian H1 governing system 1, which we will call as the twalevel 4 system, has no spin % particles but is shifted slightly from another Hamiltonian H by choosing the energy origin to be %(H11 + H22). H is deﬁned with eigenvalues Ed: to be: H 11 H12)
H :
H21 H22
Using the above analysis, show that system 1 and 2 are exactly similar to each other if we make the following map between the variables of system 1 and 2: SYSTEM 1 SYSTEM Z_'HJ
lwi> __ l+z> ﬂ
|¢z> #_ [—2)
lw+> l“)
lw-> l-N>
E+ —- E — hm
ANGLES 9 AND (9 POLAR ANGLES or: B _ﬂ
H“ — H22 - m B d
[H24 r —m B_L/2 where [9013) and |$i> are the eigenvectors of the Hamiltonians H (without the H12, H21
pieces) and H1 in system 1 respectively. On the other hand, for system 2, I :1: z) are the
usual eigenvectors of J z and I j: N > are eigenvectors of J N where N is a unit vector in the
three dimensions parametrised by angles (6’, 90). Furthermore B L E le +iByl in system 2.
Show that this way any two level system (like say ammonia molecule) is exactly identical to
a spin 3% system. From here argue that this would imply that the tunelling between two levels in system 1 is equivalent to spin precession in system 2 with a frequency w E 1 x 25 : 25
Useful formulae: - Uncertainity relation for [A, B] : iC: AA AB 2 5%
o The commutator relation [JmJy] 2 ﬁt J Z and cyclic permutations between them, implies: m “ m(mi1) where j is the highest eigenvalue of J 2 in a given Hilbert space and Ji are the raising and lowering operators respectively. ...

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