PHYS 622 (Fall 2019):Quantum Mechanics

Take-home final exam: assigned on December 9(Monday) around noon, due December 16 (Monday)

by 12 pm., in a folder near Rm. 3118 of PSC

Name:

Student ID:

Read the instructions/guidelines below

It is necessary to show the details of the derivation and not just the final answer for allproblems (this especially applies to questions whose answers are sort of already suggested).

Having said this, you can simply use “standard” formulae (as appropriate) obtained duringlecture or given in Sakurai (for example, energies or eigenfunctions of simple harmonic os-cillator or one-electron atom), i.e., you do not have to derive them again, although it willbe nice if you (briefly) indicate where you got this formula from (for example, give equationnumber from Sakurai).

This is an open book and notes exam.

Please write clearly and if you use the backside of a page, then please indicate so.

Check that there are total of 6 problems.

Each problem has multiple parts and hints given, so read them carefully.

You should avoid discussing with other students specifically about these problems.

Before you ask questions specifically about these problems, please read the statements ofthe problems (and especially hints) very carefully. Again, there are no trick questions andno new concepts involved here. So, many of your questions might be answered if you thinkmore about them. Also, if needed, you should make reasonable assumptions while solvingthese problems.

If you still have questions or clarifications, I prefer that you ask them during the usual officehours on Tuesday, December 10 from 3.30 to 4.30 pm. by the TA Yixu Wang in Rm. 3264of PSC; by the other TA, Majid Ekhterachian, on Thursday, December 12 from 2.30 to 3.30pm. or by me on Wednesday, December 11 from 3.30 to 4.30 pm. (usual day, but postponedby half an hour), in Rm. 3118 of PSC (my office) or a special office hour 2.30 to 3.30 pm. onFriday, December 13 by me in Rm. 3118 of PSC.

If you are unable to have all your questions answered as above, then you can send emails tothe instructor and the TA.

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Please try to use the notation (for parameters in potential, masses of particles etc.) whichis specified in the statement of the problems even if it is for intermediate steps.

Feel free to look up (purely) mathematical formulae in reference tables (including online)and to use computer programs (such as mathematica) for doing the (purely) mathematicalparts (for example, for computing integrals or digaonalization of matrices). However, if youtake such a step, please indicate what exactly you did here, for example, give the reference(this will help us grade appropriately). Having said this, most such parts can be done byhand in this exam!

The points assigned to the problems are subject to minor changes.

The problems are of varying levels of difficulty, so please plan to work on them accordingly!

Some of the problems can be tedious; so, needless to say, get started working on them assoon as possible!

Problem # Points scored Maximum points

1 262 203 214 165 156 27

Total 125

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1. (This problem is about Landau levels and the quantum Hall effect.) Let us consider aparticle confined to two dimensions (x, y) and subject to a uniform constant magnetic fieldB in the perpendicular direction z. Let us describe the magnetic field using the Landaugauge:

Ax = 0, Ay = Bx. (1)

(a). [10 points] (i). Substituting the vector potential Eq. (1) into the Schrodinger equa-tion (2.7.29) of Sakurai, show that the solutions of the stationary equation can be obtained

in the factorized form ψ(n)py (x, y) = eipyy/~ψ

(n)py (x), where ψ

(n)py (x) satisfies the equation[

− ~2

2m

d2

dx2+

1

2m

(py −

e

cBx

)2]ψ(n)py (x) = εnψ

(n)py (x). (2)

Argue that Eq. (2) is the same as the equation for a harmonic oscillator with a shifted origindue to py. Explain (appropriately) the relevant steps.

(ii). Using the analogy with harmonic oscillator, obtain (without much calculations) theenergy levels εn, called the Landau levels. Do the eigenenergies εn depend on py? Is theredegeneracy of the energy levels?

(iii). Write down and sketch schematically an eigenfunction ψ(0)py (x, y) of the lowest energy

level (do not bother to normalize and it suffices to show only the real – or imaginary – partof the wavefunction). What can you say about localization of the wave function in the xand y directions? What is the characteristic localization length in the x direction, called themagnetic length `B? What happens to the wave function when we change the parameter py?

(b). [3 points] Suppose the size of the system is limited by the lengths Lx and Ly in thex and y directions. Both lengths are much greater than the characteristic width `B of thewave functions. Calculate the degeneracy N of each energy eigenvalue εn.

For simplicity, let us assume the periodic boundary condition in the y direction:

ψ(x, y) = ψ(x, y + Ly). (3)

The condition in Eq. (3) permits only discrete values of py, as in Eq. (2.5.6) of Sakurai. Findthe spacing ∆py between the discrete values of py.

Then count the number N of available values of py, given that the size of the system in the xdirection is limited by Lx. [Hint: in part (a) above, you determined how the wave functionchanges with py.]

Show that the degeneracy N is proportional to the area of the system LxLy and the mag-netic field B, and express N in terms of the magnetic flux Φ = BLxLy. Given that Nis dimensionless and Φ is dimensional, what is the dimensional constant that balances theequation?

(c). [3 points] Suppose, each eigenstate can be occupied by only one electron. (The electronsare fermions, and let us ignore the spin for simplicity.) Suppose the electrons completely fillν lowest Landau levels with n = 0, 1, 2, . . . , ν.

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Using the solution of Part (b), calculate the number of the electrons per unit area νN/LxLy.Calculate the magnetic flux per one electron and compare with the quantum of magneticflux Φ0 = hc/e.

(d). [5 points] Suppose a constant uniform electric field Ex is applied along the x direction,Ex B. Now we need to add the scalar potential eφ = −eExx to Eq. (2). Show thatthe modified equation still reduces to a displaced harmonic oscillator and obtain the energylevels εn,py . Do they depend on py now? Are they still degenerate? Interpret the resultphysically. What is the difference between the eigenfunctions for Ex = 0 and Ex 6= 0?

(e). [3 points] Find the electric current carried by each eigenstate in the presence of Ex.Does this current flow in the x or y direction?

Hint: This part does not require a long calculation. Similarly to Eq. (2.7.34) of Sakurai,the electric current is related to the expectation value of py − eAy/c. Compare your resultwith part (d) in HW 6.3. Be careful to distinguish between current and current density, andcheck dimensionality.

The occurrence of an electric current in the direction perpendicular to both magnetic andelectric fields is called the Hall effect. It was discovered by E. H. Hall in 1879 at the JohnsHopkins University in Baltimore.

(f). [2 points] Suppose ν Landau levels are completely filled by the electrons, as in Part (c).Calculate the total current Jy carried by all these electrons in the presence of the electricfield Ex. Then obtain the Hall resistance, which is the ratio of the voltage Vx = ExLx to thecurrent Jy

RH =VxJy. (4)

Express the result in terms of ν and the quantum of resistance:

R0 =h

e2= 25.8 kΩ. (5)

2. A system of three nonidentical (distinguishable) particles of spin 1/2, whose spin operatorsare s1, s2, and s3, is governed by the Hamiltonian

H = A (s1 · s2)/~2 +B [(s1 + s2) · s3]/~2. (6)

(a). Find the energy levels, their quantum numbers and degeneracies. [(2 + 2 + 2)× 3 points]

(Please try to show them in a tabular form.)

(b). Check that the sum of all degeneracies is equal to the dimensionality of Hilbert space.[2 points]

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Hint: You might find it useful to re-write the Hamiltonian in terms of (combinations of)operators whose eigenvalues are well-known.

3. Consider a hydrogen atom in “semispace”, i.e., the proton is located at the origin x = y =z = 0, and the electron (neglect its spin) can move only in the subspace z > 0, but cannotpenetrate z < 0.

(a). [1 point] What must happen to the wave function of the electron at z = 0?

(b). [4 points] Based on part (a) above, what is the condition on angular momentum (l,m) values of the allowed bound state?

(c). [4 + 6 points] Find the energies and quantum numbers of the (i) ground state and the(ii) first excited state. Make sure to list the complete set of possibilities.

(d). [2 points] For a general (given) l, how many values of m are permitted?

(e). [2 points] For a general (given) principal quantum number (n), what is the range ofvalues of l allowed?

(f). [2 points] Based on the above arguments, what is the degeneracy of a general energylevel (as usual, labeled by n)?

Directions: You do not need to solve any differential equations in this problem. Just exploitthe symmetry of the problem in order to figure out the answer in terms of the already knownenergies and wave functions of a regular hydrogen atom. In order to satisfy the boundarycondition in part (a) above, you just need to find a suitable subset of the standard hydrogenatom wave functions i.e., satisfying a specific symmetry property under the (mirror reflection)operation (x, y, z)→ (x, y,−z) (Can this be considered as a combination of operations whichmight be more familiar, i.e., studied in lecture/Sakurai?). So, find out how the hydrogenatom wave functions behave under this transformation.

4. Consider a double-slit interference experiment illustrated in the figure. Electricallycharged quantum particles propagate from the source C to the detector D through eitherthe left or right slit (L or R). The source is located symmetrically between the slits, but thedetector has the variable coordinate x. The distance between the slits is w, and the distancefrom the slits plane to the detection plane is `. Assume that ` w, x (the figure is notdrawn to scale). The source emits particles of the fixed energy E, which corresponds to themomentum p =

√2mE.

5

u

u

-x0

w

6

?

`

@@@@@@R

@@@@@@@@@R

Source C

Detector D

Left slit L Right slit R

Solenoid

First, ignore the solenoid and consider the standard double-slit problem.

(a). [6 points] Calculate the probability1 P (x) of detection of the particles as a function ofthe position x of the detector (no need to normalize the total probability). Show that P (x)is a periodic function of x and find the period. Does P (x) have a maximum, a minimum, ornone of the above at x = 0?

Hint: P (x) = |ψ(x)|2, where the (unnormalized) wave function (defined up to an overallfactor) has two contributions originating from the two slits:

ψ(x) = eiprLD/~ + eiprRD/hbar. (7)

Here rLD and rRD are the distances from the left and right slits to the detector (whereas thesource-slit distances are equal rCL = rCR, so they can be omitted). The difference in thedistances causes an interference effect. Under the condition ` (w, x), show geometricallythat rLD − rRD ≈ w sinα ≈ wα, where α ≈ tanα = x/` is a suitably defined view angle.

(b). [6 points] Now, suppose a solenoid with a magnetic field B inside (drawn as ) and asmall cross-section area S is placed between the two slits perpendicularly to the plane of thepage, as shown in the figure by the -decorated circle, so the total magnetic flux is Φ = BS.Magnetic field outside the solenoid is negligible, but there is vector potential A(x) createdby the solenoid.

In the presence of the vector potential A(x), we can show that the two term in Eq. (7)acquire the additional phase factors eiϕL and eiϕR . Find expressions for these phase factorsin terms of vector potential and the path.

Express the phase difference ϕL − ϕR in terms of the magnetic flux Φ produced by thesolenoid.

(c). [4 points] Recalculate the probability P (x,Φ) of detection of the particles at thedetector, now as a function of both the coordinate x and the magnetic flux Φ.

Is P (x,Φ) a periodic function of Φ for a fixed x? If yes, then find the period. Describe howthe coordinate xmax of a probability maximum changes with the increase of Φ, starting from

1Strictly speaking, it is the intensity of the beam

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xmax = 0 at Φ = 0. Is it possible to distinguish between P (x) without a solenoid (i.e. Φ = 0)and P (x,Φ = nΦ0) when the flux is an integer multiple n of the flux quantum Φ0 = hc/e,where h = 2π~?

Note: in problems # 5 and # 6 below, do not assume parity invariance, just use (as indicatedin the statement of the problems) angular momentum conservation (and its addition) andspin-statistics theorem (Fermi-Dirac for # 5 and Bose-Einstein for # 6).

(Again, as indicated in HW 11.5, angular momentum conservation and spin-statistics arevalid always, but parity could be violated in some interactions.)

Also, note that the wave function of the initial (and final) state in # 5 and the final state in# 6 consists of two “parts/sub-spaces”, i.e., orbital/position and spin.

5. The idea in this problem is to determine selection rules for reactions based on (a) Fermi-Dirac statistics and (b) angular momentum conservation. Consider an electrically neutralfermion (i.e., spin-1/2 particle), χ0 which is its own anti-particle (such a fermion is called“Majorana”). Suppose it annihilates with another χ0 (i.e., pair-annihilation) into an electron(e−) and positron (e+). Furthermore, assume that electron has helicity −1/2 (left-handed)and positron has helicity +1/2 (right-handed), where helicity (denoted by h) is defined tobe the component of spin (S) of a particle along its direction of motion (p):

h ≡ p.S (8)

The goal is to prove (by contradiction) that (for such a reaction) the initial state χ0 paircannot be in s-wave, i.e., it cannot have orbital angular momentum, Linitial = 0.

(a). [4 points] Suppose the initial state χ0 pair is in s-wave (again, Linitial = 0). Then,using Fermi-Dirac statistics [note that we have two identical fermions in the initial state, asituation which was encountered in part of HW 9.3 (for example)], determine the allowedvalue(s) of total spin (Sinitial) of this pair.

Note: you might find Eqs. (3.8.15) of Sakurai useful here.

Note: the wave function of the initial state consists of two “parts/sub-spaces”, i.e., or-bital/position and spin.

(b). [1 point] Using the above result, determine the allowed value(s) of total angularmomentum (Jinitial) of the χ0 pair.

(c). [3 points] Now on to the final state. In center-of-mass frame of the initial (and thereforethe final) state, the e−e+ are moving in opposite directions (i.e., anti-parallel) (see Fig. 1).

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χ0 χ0

e−

e+

-

Figure 1: The arrows denote directions of motion of the various particles

Denote this direction by z. Given the above helicities of e− (h = −1/2) and e+ (h = +1/2),determine the component of their total spin along their direction of motion (Sz final).

(d). [2 points] What is the value of the component of the orbital angular momentum alongthe same direction, i.e., Lz final? Explain your answer, noting that the two final state particlesneed not be exactly back-to-back, even if they are moving in opposite directions, i.e., therecould have non-zero orbital angular momentum (see Fig. 1).

(e). [1 point] Using the above results, determine the total angular momentum along thedirection of motion of e−e+, i.e., Jz final.

(f). [2 points] Using the above result, what are the allowed values of total angular momen-tum Jfinal of e−e+ (i.e., not just its z-component)?

(g). [2 points] Using all of the above information and angular momentum conservation, iss-wave for χ0 pair then allowed?

Commentary The χ0 pair thus has to be in a higher-partial wave, for example p-wave (i.e.,Linitial = 1), resulting in a suppression of the annihilation cross-section (relative to if it hadbeen s-wave). A specific application of this general result is in the context of supersymmetricextensions of the Standard Model of particle physics. Here, a fermionic, electrically neutralsupersymmetric partner of the photon or Z boson (called a “neutralino”) is a potentialcandidate for dark matter of the universe. It can be shown that in order for the neutralinoto serve this purpose, i.e., end up with the correct relic density, it has to (in turn) have theappropriate cross-section for annihilation. However, in explicit models, it turns out that theabove p-wave suppression of the cross-section then becomes a “bottleneck”.

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6. The purpose of this problem is to work through a proof of Yang’s theorem. This theoremstates that a massive spin-1 particle cannot decay into two identical massless spin-1 particles(for example, photons). As an aside, the experimental observation of the decay π0 → γγeliminated the possibility of π0 spin being 1 based on this theorem.

(a). [2 points] Determine the possible values of the total spin (S) of the final state particles,based simply on addition of the individual spins.

(b). [4 + 4 points] Figure out (by a rather involved process!) the combinations of thetotal spin S [that you obtained in part (a) above] and the orbital angular momentum (L) ofthe final state particles which are allowed by (a) angular momentum conservation and (b)Bose-Einstein statistics [note that we have two identical bosons in the final state, as in partof HW 11.5 (for example)].

Note: At this stage, do not think about the components of S and L (for example, alongdirection of motion of final state particles), which will be the topic of the next part(s).

Note: If you wish, you can go to the rest frame of the massive, decaying particle. Youshould have worked out a similar process– Eq. (37) – in HW 11.5.

Note: You might find useful the results of HW 9.2 here.

Note: Explain your answers, making sure you exhaust all the possibilities for S and L valueshere.

Note: the wave function of the final state consists of two “parts/sub-spaces”, i.e., or-bital/position and spin.

Next (as indicated above), we consider the components of the various angular momentaalong the direction of the motions of the final state particles. For this purpose, it might beconvenient to go the rest frame of the decaying/massive particle and to draw a figure of thisdecay process.

(c). [1 point] What are the relative directions of the motions of final state particles in thisframe?

(d). [2 points] What is the value of the component of the orbital angular momentum (L) ofthe final state along the direction of motion of the final state particles? Explain your answer,noting that the two final state particles need not be exactly back-to-back.

(e). [3 points] What are the possible values of the component of the total spin (S) of the finalstate particles along the same direction? You should use conservation of angular momentumhere. Again, make sure you exhaust all the possibilities.

Recall that light has two independent (transverse) polarization states. Accordingly the spin-1 photon has two rather than three allowed components of its spin. As it happens, this isa general result for massless spin-1 particles. The allowed spin components are Sz = +1and Sz = −1 where z is the direction of propagation; the Sz = 0 component (longitudinalpolarization) is forbidden as a consequence of relativity.

(f). [1 point] Using the above two results, what is the component of the total angularmomentum of the final state particles along their direction of motion?

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(g). [4 points] Using all of the above information and the (appropriate) Clebsch-Gordancoefficients (from the table given in Fig. 2), is the decay in question allowed?As usual, ex-plain your answer [for example, clearly indicating the relevant Clebsch-Gordan coefficient(s)],making sure you exhaust all the possibilities.

Note: In case you are unfamiliar with such a table of Clebsch-Gordan coefficients, thenotation is given on the upper right side. Also (as is written at the top of this figure), a“square-root” sign is to be understood over every coefficient. The simplest case is additionof two spin 1/2’s, which is given on the upper left side: you can check that these coefficientsmatch Eqs. (3.8.15) of Sakurai. Similarly, 4th row on right side is addition of two angularmomenta (spin or orbital) each being 1: you obtained these coefficients in HW 9.2.

(h). [(2 + 2) points] Does an argument analogous to this one apply if the two spin-1particles in final state are(i) are massive (note that for a massive spin-1 particle, the longitudinal polarization, i.e.,Sz = 0 component is allowed)and(ii) massless, but not identical, say photon and gluon (the carrier of strong nuclear force)?Explain (briefly) your answers.

(i). [2 points] Finally, will this argument apply if there are three, still identical and massless,particles in final state? Explain (briefly) your answer.

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Figure 2: Table of Clebsch-Gordon coefficients

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