Multiple-Choice Problems
014 qmult 00100 1 4 5 easy deducto-memory: spin defined Extra keywords: mathematical physics
1. “Let’s play Jeopardy! For $100, the answer is: It is the intrinsic angular momentum of a fundamental (or fundamental-for-most-purposes) particle. It is invariant and its quantum number s is always an integer or half-integer.
What is , Alex?
a) rotation b) quantum number c) magnetic moment d) orbital angular momentum e) spin
014 qmult 00110 1 4 1 easy deducto-memory: Goudsmit and Uhlenbeck, spin Extra keywords: Don’t abbreviate: it ruins the joke
2. “Let’s play Jeopardy! For $100, the answer is: Goudsmit and Ulhenbeck.” a) Who are the original proposers of electron spin in 1925, Alex? b) Who performed the Stern-Gerlach experiment, Alex?
c) Who are Wolfgang Pauli’s evil triplet brothers, Alex? d) What are two delightful Dutch cheeses, Alex?
e) What were Rosencrantz and Gildenstern’s first names, Alex? 014 qmult 00120 1 1 1 easy memory: spin magnitude
3. A spin s particle’s angular momentum vector magnitude (in the vector model picture) is a)ps(s + 1)h−. b) s h− c) ps(s − 1)h− d) −sh− e) s(s + 1) h−2 014 qmult 00130 1 1 5 easy memory: eigenvalues of spin 1/2 particle
4. The eigenvalues of a COMPONENT of the spin of a spin 1/2 particle are always: a) ±h−. b) ±−h3. c) ±−h4. d) ±−h5. e) ±−h2.
014 qmult 00130 1 1 2 easy memory: eigenvalues of spin s particle
5. The quantum numbers for the component of the spin of a spin s particle are always: a) ±1. b) s, s − 1, s − 2, . . . , −s + 1, −s. c) ±12. d) ±2. e) ±14. 014 qmult 00140 1 4 2 easy deducto-memory: spin and environment
6. Is the spin (not spin component) of an electron dependent on the electron’s environment? a) Always.
b) No. Spin is an intrinsic, unchanging property of a particle. c) In atomic systems, no, but when free, yes.
d) Both yes and no.
e) It depends on a recount in Palm Beach.
Chapt. 14 Spin 89 014 qmult 00400 1 4 5 easy deducto-memory: spin commutation relation
7. “Let’s play Jeopardy! For $100, the answer is:
[Si, Sj] = i h−εijkSk .”
What is , Alex?
a) the spin anticommutator relation b) an implicit equation for εijk
c) an impostulate d) an inobservable e) the fundamental spin commutation relation
014 qmult 00500 1 4 2 easy deducto-memory: Pauli spin matrices 8. “Let’s play Jeopardy! For $100, the answer is:
σx= 0 1 1 0 , σy= 0 −i i 0 , σz= 1 0 0 −1 .”
What are , Alex?
a) dimensioned spin 1/2 matrices b) the Pauli spin matrices c) the Pauli principle matrices d) non-Hermitian matrices e) matrix-look-alikes, not matrices
014 qmult 00600 1 1 1 easy memory: spin anticommutator relation 9. The expression
{σi, σj} = 2δij1
is
a) the Pauli spin matrix anticommutator relation. b) the Pauli spin matrix commutator relation. c) the fundamental spin commutator relation.
d) the covariance of two standard deviations. e) an oddish relation. 014 qmult 00700 1 1 2 easy memory: spin rotation DE
10. The expression d(~S · ˆn) dα = − i h −[~S · ˆα, ~S · ˆn] is a differential equation for
a) α.
b) the spin operator ~S · ˆn as a function of rotation angle ~α. c) ˆn.
d) the translation of the spin operator ~S. e) none of the above.
014 qmult 00800 1 4 5 easy deducto-memory: spin rotation operator 11. “Let’s play Jeopardy! For $100, the answer is:
U (α) = e−i~S·~α/ h−= e−i~σ·~α/2 .” a) What the Hermitian conjugate of U (−2α), Alex?
b) What is a bra, Alex?
c) What is a spin 1/2 eigenstate, Alex?
d) What is the NON-UNITARY operator for the right-hand rule rotation of a spin 1/2 state by an angle α about the axis in the direction ˆα, Alex?
90 Chapt. 14 Spin
e) What is the UNITARY operator for the right-hand rule rotation of a spin 1/2 state by an angle α about the axis in the direction ˆα, Alex?
014 qmult 00900 1 1 3 easy memory: space and spin operators commute 12. A spatial operator and a spin operator commute:
a) never. b) sometimes. c) always. d) always and never. e) to the office. 014 qmult 01000 2 1 4 moderate memory: joint spatial-spin rotation
13. The operator
U (α) = e−i ~J·~α/ h− a) creates a spin 1/2 particle.
b) annihilates a spin 1/2 particle
c) left-hand-rule rotates both the space and spin parts of states by an angle α about an axis in the ˆα direction.
d) right-hand-rule rotates both the space and spin parts of states by an angle α about an axis in the ˆα direction.
e) turns a state into a U-turn state.
014 qmult 01100 1 4 5 easy deducto-memory: spin-magnetic interaction 14. “Let’s play Jeopardy! For $100, the answer is:
~ µ = g q
2mJ ,~ F = ∇(~µ · ~~ B) , ~τ = ~µ × ~B , H = −~µ · ~B .” a) What are Maxwell’s equations, Alex?
b) What are incorrect formulae, Alex?
c) What are classical formulae sans any quantum mechanical analogs, Alex? d) What are quantum mechanical formulae sans any classical analogs, Alex?
e) What are formulae needed to treat the interaction of angular momentum of a particle and magnetic field in classical and quantum mechanics, Alex?
014 qmult 01200 1 1 2 easy memory: Bohr magneton 15. What is µB = e h− 2me = 9.27400915(23) × 10 −24 J/T = 5.7883817555(79) × 10−5eV/T ?
a) The nuclear magneton, the characteristic magnetic moment of nuclear systems. b) The Bohr magneton, the characteristic magnetic moment of electronic systems.
c) The intrinsic magnetic dipole moment of an electron. d) The coefficient of sliding friction.
e) The zero-point energy of an electron. 014 qmult 01210 1 1 3 easy memory: g factor g-factor
16. The g factor in quantum mechanics is the dimensionless factor for some system that multiplied by the appropriate magneton (e.g., Bohr magneton for electron systems) times the angular momentum of the system divided by h− gives the magnetic moment of the system. Sometimes the sign of the magnetic moment is included in the g factor and sometimes it is just shown explicitly. The modern way seems to be to include it, but yours truly finds that awkward and so for now yours truly doesn’t do it. For the electron, the intrinsic magnetic moment operator associated with intrinsic spin is given by
~ µop= −gµB ~ Sop h − ,
Chapt. 14 Spin 91 where µB is the Bohr magneton and Sopis the spin vector operator. What is g for the intrinsic
magnetic moment operator of an electron to modern accuracy?
a) 1. b) 2. c) 2.0023193043622(15). d) 1/137. e) 137. 014 qmult 01210 1 1 4 easy memory: magnetic moment precession
Extra keywords: The precession is also called Larmor precession (En-114)
17. An object in a uniform magnetic field with magnetic moment due to the object’s angular momentum will both classically and quantum mechanically:
a) Lancy progress. b) Lorenzo regress. c) London recess. d) Larmor precess. e) Lamermoor transgress.
014 qmult 01300 1 1 3 easy memory: Zeeman effect Extra keywords: See Ba-312 and Ba-466–468
18. What is an effect that lifts the angular momentum component energy degeneracy of atoms? a) The spin-orbit effect.
b) The Paschen-Back effect or, for strong fields, the Zeman effect. c) The Zeeman effect or, for strong fields, the Paschen-Back effect. d) The Zimmermann effect.
e) The Zimmermann telegram.
014 qmult 01500 1 4 1 easy deducto-memory: spin resonance Extra keywords: See Ba-317
19. “Let’s play Jeopardy! For $100, the answer is: the effect in which a weak sinusoidal radio frequency magnetic field causes a particle with spin to precess about a direction perpendicular to strong uniform magnetic field that separates the spin component energy levels of the particle in energy.”
a) What is spin magnetic resonance, Alex? b) What is spin magnetic presence, Alex?
c) What is the preferred spin effect, Alex? d) What is the dishonored spin effect, Alex?
e) What is the Zeeman effect, Alex?
014 qmult 01600 1 1 3 easy memory: spin resonance field
20. In spin magnetic resonance you can replace a rotating magnetic field by a sinusoidal one if you can neglect or filter:
a) magnetic fields altogether. b) precession altogether.
c) the very HIGH frequency effects of the sinusoidal field. d) spin altogether.
e) the very LOW frequency effects of the sinusoidal field. 014 qmult 01700 1 4 5 easy deducto-memory: atomic clock
21. “Let’s play Jeopardy! For $100, the answer is: The simplest of these consists of a beam of spinned particles that passes through two cavities each with crossed constant and sinusoidal magnetic fields.”
What is a/an , Alex?
a) Stern-Gerlach experiment b) Gentle-Gerlach experiment c) quartz-crystal d) nuclear magnetic resonance machine e) atomic clock
92 Chapt. 14 Spin
22. The spin state energy level separation of a 133Ce atom used in an atomic clock to define the
second corresponds by definition to a frequency of: a) 9 192 631 770 Hz. b) 3.141 592 65 Hz. c) 2.718 28 Hz. d) 0.577 215 66 Hz. e) 299 792 458 Hz.
Full-Answer Problems
014 qfull 00100 2 5 0 moderate thinking: Pauli matrices in detail 1. The Pauli spin matrices are
σx= 0 11 0 , σy = 0 −ii 0 , and σz= 1 0 0 −1 .
a) Are the Pauli matrices Hermitian?
b) What is the result when Pauli matrices act on general vector a
b
?
c) Diagonalize the Pauli matrices: i.e., solve for their eigenvalues and NORMALIZED eigenvectors. NOTE: The verb ‘diagonalize’ takes its name from the fact that a matrix transformed to the representation of its own eigenvectors is diagonal with the eigenvalues being the diagonal elements. One often doesn’t actually write the diagonal matrix explicitly. d) Prove that
σiσj= δij1 + iεijkσk ,
where i, j, and k stand for any of x, y, and z, 1 is the unit matrix (which can often be left as understood), δij is the Kronecker delta, εijkis the Levi-Civita symbol, and Einstein
summation is used. HINT: I rather think by exhaustion is the only way: i.e., extreme tiredness.
e) Prove
[σi, σj] = 2iεijkσk and {σi, σj} = 2δij ,
where {σi, σj} = σiσj+ σjσi is the anticommutator of Pauli matrices. HINT: You should
make use of the part (d) expression.
f) Show that a general 2 × 2 matrix can be expanded in the Pauli spin matrices plus the unit matrix: i.e.,
a b c d
= α1 + ~β · ~σ ,
where ~σ = (σx, σy, σz) is the vector of the Pauli matrices. HINT: Find expressions for the
expansion coefficients α, βx, βy, and βz.
g) Let ~A and ~B be vectors of operators in general and let the components of ~B commute with the Pauli matrices. Prove
Chapt. 14 Spin 93 HINT: Make use of the part (d) expression.
014 qfull 00110 1 3 0 easy math: diagonalization of y Pauli spin matrix
Extra keywords: (CDL-203:2), but it corresponds to only part of that problem
2. The y-component Pauli matrix (just the y-spin matrix sans the h−/2 factor) expressed in terms of the z-component orthonormal basis (i.e., the standard z-basis with eigenvectors |+i and |−i) is:
σy= 0 −ii 0
.
Diagonalize this matrix: i.e., solve for its eigenvalues and NORMALIZED eigenvectors written in terms of the standard z-basis eigenvector kets or, if your prefer, in column vector form for the z-basis. One doesn’t have to literally do the basis transformation of the matrix to the diagonal form since, if one has the eigenvalues, one already knows what that form is. In quantum mechanics, literally doing the diagonalization of the matrix is often not intended by a diagonalization.
014 qfull 00200 2 3 0 mod math: spin 1/2, spin Sx+ Sy
Extra keywords: (Ga-241:9), spin 1/2, spin Sx+ Sy, diagonalization
3. Consider a spin 1/2 system. Find the eigenvectors and eigenvalues for operator Sx+ Sy. Say
the system is in one of the eigen-states for this operator. What are the probabilities that an Sz
measurement will give h−/2?
014 qfull 00250 2 5 0 moderate thinking: Euler formula for matrices Extra keywords: Reference Ba-306, but I’ve generalized the result
4. Say that A is any matrix with the property that A2 = 1, where 1 is the unit matrix. If we define eixA by eixA= ∞ X ℓ=0 (ixA)ℓ ℓ! (where x is a scalar), show that
eixA= 1 cos(x) + iA sin(x) . This last expression is a generalization of Euler’s formula (Ar-299). 014 qfull 00300 3 5 0 tough thinking: rotation parameters
Extra keywords: (Ba-330:1b), but there is much more to this problem 5. The unitary spin 1/2 rotation operator is
U (~α) = e−i~S·~α/ h−= e−i~σ·~α/2 ,
where ~α is the vector rotation angle: ~α = α ˆα with α being the angle of a right-hand rule rotation about the axis aligned by ˆα. To rotate the spin component operator Sz from the z direction to
the ˆn direction one uses
~
S · ˆn = U(~α)SzU (−~α)
(Ba-305) To rotate a z basis eigenstate into an ˆn basis eigenstate one uses |n±i = U(~α)|z±i
(Ba-306).
a) Expand U (~α) into an explicit 2 × 2 matrix that can be used directly. Let the components of ˆα be written ˆαx, etc.
94 Chapt. 14 Spin
b) Now write out U (~α)|z±i explicitly.
c) You are given a general normalized spin vector |ˆn+i = a + ibc + id
.
Find expressions for α and ~α that yield this vector following a rotation of |z+i. Then for those α and ~α written in terms of a, b, c, and d, find the rotated |z−i state |ˆn−i. Show explicitly that |ˆn−i is normalized and orthogonal to |ˆn+i.
d) We gone so far: why quit now. Using our explicit matrix version of U (~α) find explicit expressions for the components of ˆn for a rotation from ˆz in terms of α and the components of ˆα. One has to solve for the components from
~σ · ˆn = U(~α)σzU (−~α) .
HINT: Write U (~α) in simplified symbols until it’s convenient to switch back to the proper variables: e.g.,
U (~α) = a + ib −c + id c + id a − ib
,
where a, b, c, and d have the same meanings as you should have found in the part (c) answer.
014 qfull 00400 1 5 0 easy thinking: electron spin in B-field Hamiltonian Extra keywords: electron spin in magnetic field Hamiltonian
6. What is the Hamiltonian fragment (piece, part) that describes the energy of an electron spin magnetic moment in a magnetic field? This fragment in a Schr¨odinger equation can sometimes be separated from the rest of the equation and solved as separate eigenvalue problem. Solve this separated problem. The intrinsic angular momentum operator is ~S and assume the magnetic field points in the z direction. HINTS: Think of the classical energy of a magnetic dipole in a magnetic field and use the correspondence principle. This is not a long question.
014 qfull 00500 2 5 0 moderate thinking: classical Larmor precession 7. Let’s tackle the classical Larmor precession.
a) What is Newton’s 2nd law in rotational form?
b) What is the torque on a magnetic dipole moment ~µ in a magnetic field ~B? HINT: Any first-year text will tell you.
c) Say that the magnetic moment of a system is given by ~µ = γ~L, where γ is the gyromagnetic ratio and ~L is the system’s angular momemtum. Say also that there is a magnetic field
~
B = (0, 0, Bz). Solve for the time evolution of ~L using Newton’s 2nd law in rotational form
assuming the INITIAL CONDITION ~L(t = 0) = (Lx,0, 0, Lz,0). HINTS: You should
get coupled differential equations for two components of ~L. They are not so hard to solve. For niceness you should define an appropriate Larmor frequency ω.
014 qfull 00600 3 5 0 tough thinking: quantum mech. Larmor spin precession Extra keywords: (Ba-330:1a), but there is much more to this problem
8. Consider a spin 1/2 particle with magnetic moment ~M = γ ~S. We put a uniform magnetic field in z direction: thus ~B = Bzz. As usual we take the z-basis as the standard basis for theˆ
problem.
a) Determine the normalized eigenstates for the Sx, Sy, and Szoperators in the z-basis. What are
Chapt. 14 Spin 95 b) Now expand the eigenvectors for Sz in the bases for Sxand Sy. You will need the expansions
below.
c) If we consider only the spin degree of freedom, the Hamiltonian for the system is H = −~m · ~B = −γ ~S · ~B ,
where γ is a constant that could be negative or positive. Sometimes γ is called the gyromagnetic ratio (CDL-389), but the expression gyromagnetic ratio is also used for the Land´e g factor which itself has multiple related meanings. What are the eigenvalues and eigenvectors of H in the present case? HINT: Defining an appropriate Larmor frequency ω would be a boon further on.
d) The time-dependent Schr¨odinger equation in general is i h− ∂
∂t|Ψi = H|Ψi .
What is the formal solution for |Ψ(t)i in terms of H and a given |Ψ(0)i. HINT: Expand |Ψ(0)i in a the eigenstates of H which you are allowed to assume you know.
e) For our system you are given
|Ψ(0)i = a+|z+i + a−|z−i .
What is |Ψ(t)i? What are the probabilities for measuring spin up and down in the z direction and what is hSzi?
f) What are the probabilities for measuring spin up and down in the x direction and what is hSxi?
Try to get nice looking expressions.
g) What are the probabilities for measuring spin up and down in the y direction and what is hSyi?
Try to get nice looking expressions.
h) What can you say about the vector of spin expectation values given the answers to parts (f) and (g)?
i) Now given the initial state as |x+i, what are hSxi, hSyi, and hSzi in this special case?
014 qfull 00700 2 5 0 tough thinking: spin algebra generalized Extra keywords: (Ba-331:3)
9. Spin algebra can be used usefully for situations not involving spin. Say we have an atom or molecule with two isolated stationary states: i.e., there can be perturbation coupling and transitions between the two states, but no coupling to or transitions to or from anywhere else. Let the states be |+i and |−i with unperturbed energies ǫ+ and ǫ−; let ǫ+ ≥ ǫ−. The states
are orthonormal.
a) Write the Hamiltonian for the states in matrix form and then decompose it into a linear combination of Pauli spin matrices and the unit matrix. What are the eigenstates |+i and |−i in column vector form? Note there is no spin necessarily in this problem: we are just using the Pauli matrices and all the tricks we have learned with them.
b) Now we add a perturbation electric field in in the z direction: E = E0cos(ωt). You are
given that the diagonal elements of the dipole moment matrix are zero and that the off diagonal elements are both equal to the real constant µ: i.e., µ = h+|ez|−i = h−|ez|+i. Note µ can be positive or negative. Write down the Hamiltonian now.
c) Write the Schr¨odinger equation for the perturbed system and then make a transformation that eliminates the unit matrix term from the problem. Do we ever really need to transform the state expressions back?
96 Chapt. 14 Spin
d) Show that a pretty explicit, approximate solution for the (transformed) Schr¨odinger equation is
|Ψi = e−iωtσz/2e−iΩˆσt/2|Ψ(0)i ,
where Ω = q (ω0− ω)2+ ω21 and σ =ˆ (ω0− ω) Ω σz− ω1 Ωσx.
The solution is valid near resonance: i.e., the case of ω ≈ ω0, where ω0 ≡ (ǫ+− ǫ−)/ h−.
In order to get the solution we have averaged over times long enough to eliminate some of the high frequency behavior. To do this one assumes that |ω1| = |µE0/ h−| << |ω| ≈ |ω0|.
HINT: The problem is pretty much isomorphic to the spin magnetic resonance problem. e) Given that the initial state is |+i, what are the probabilities that the system is in |+i and
|−i at any later time? What are corresponding probabilities if the initial state is |−i? Do any of these probabilities have high frequency behavior: i.e., time variation with frequency of order ω ≈ ω0 or greater?
f) The factor e−iωtσz/2 in the solution
|Ψi = e−iωtσz/2e−iΩˆσt/2|Ψ(0)i
is actually physically insufficient to give the high frequency behavior—although it is right in itself—since we dropped some high frequency behavior in deriving the solution. Thus any high frequency behavior predicted by the solution can’t be physically accurate. You should have found in part (e) that the high frequency behavior from e−iωtσz/2canceled out
of the probability expressions. Is there any reason for keeping the factor e−iωtσz/2 in the