Download 1 1. Determine if the following vector operators are Her

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1. Determine if the following vector operators are Hermitian. Separate those that are not into their Her⃗ × P⃗ , (ii)
mitian and anti-Hermitan Parts: (i) R
⃗ × P⃗ , (iii) R
⃗ × L.
⃗
L
⃗ and W
⃗ be two vector operators of a certain
2. Let V
quantum mechanical system, i.e., they satisfy the
commutation relations
∑
∑
[Ji , Vj ] = i
εijk Vk
[Ji , Wj ] = i
εijk Wk
k
⃗v =
1 ( ⃗ q ⃗)
P− A .
m
c
(b) Show that
⃗v × ⃗v =
k
with the components of the total angular momentum J⃗ of the system.
⃗ ·W
⃗ is a scalar with respect to
(a) Show that V
[
]
⃗ V
⃗ ·W
⃗ .
rotations, i.e., evaluate J,
⃗ =V
⃗ ×W
⃗ is a vector under rota(b) Show that U
tions.
⃗ be a vector operator whose Cartesian compo3. Let V
nents may or may not commute with one another.
⃗×
(a) Show that the ith Cartesian component of V
⃗
⃗
⃗
⃗
⃗
(V × V ) and (V × V ) × V satisify the relations
[
]
∑
⃗ × (V
⃗ ×V
⃗) =
V
Vj [Vi , Vj ]
i
j
[
]
∑
⃗ ×V
⃗)×V
⃗ =
[Vi , Vj ] Vj
(V
i
4. Determine the form taken by the components of
⃗ ×K
⃗ in the
the angular momentum operator ⃗ℓ = R
(Cartesian) wavevector representation. What form
do they take in spherical wavevector coordinates
(k, θ, ϕ)? What form do the common eigenfunctions ψlm (k, θ, ϕ) of ℓ2 and ℓz take in this representation? [Note: This spherical wavevector coordinates are related to their Cartesian counterparts in
the usual way, i.e.,
ky = k sin θ sin ϕ,
i~q ⃗
B.
m2 c
(c) Prove the relation
m
q
d
⃗ −B
⃗ × ⃗v ⟩ + q⟨E⟩,
⃗
⟨⃗v ⟩ = ⟨⃗v × B
dt
2c
⃗ = −∇ϕ−
where the electric field operator is E
⃗
(1/c)∂ A/∂t.
6. Consider an irreducible invariant subspace corresponding to j = 1.
(a) Show that the matrix representing any component Ju = J⃗ · û of the angular momentum within this subspace obeys the relation
Ju3 = Ju . (Feel free to use any appropriate
symmetry arguments to simplify your calculation.)
(b) Show using this result that the rotation matrix
Ru (α) within this subspace can be written in
the form
j
(b) Use this to deduce the general relation
[
]
⃗ ,V
⃗ ·V
⃗ =V
⃗ × (V
⃗ ×V
⃗ ) − (V
⃗ ×V
⃗)×V
⃗.
V
kx = k sin θ cos ϕ,
(a) Show, by considering the evolution of the
⃗
mean position ⟨R(t)⟩
of such a particle, that
the velocity operator takes the form
kz = k cos θ.
⃗ =
5. In the presence of an electromagnetic field B
⃗
⃗
∇ × A, the classical Hamiltonian for a particle of
charge q takes the form
]2
q⃗
1 [
p⃗ − A(⃗
r, t) + qϕ(⃗r).
H=
2m
c
Assume that this holds for a quantum particle with
⃗ and p⃗ → P⃗ .
⃗r → R
Ru (α) = 1 − iJu sin α − Ju2 (1 − cos α).
(c) Explicitly construct the matrix Ry (α) corresponding to rotations about the y axis.
7. Consider an angular momentum subspace with j =
1
⃗ (Take ~ =
σ denote the vector operator 2J.
2 . Let ⃗
1.)
(a) Show that the matrix representing any component σu = ⃗σ · û of the operator ⃗σ within
this subspace obeys the relation [σu ]2 = 1,
and use this relation to derive an expression for the matrix representing the operator
ρu (α) = exp[−iασu ] similar to that given in
the last problem. Use this result to obtain a
similar expression for the matrix representing
the rotation operator Ru (α).
(b) Find the eigenvectors |j, mx ⟩ of J 2 and Jx as
an expansion (or column vector) in the basis
of eigenstates |j, mz ⟩ of J 2 and Jz .
(c) In the |j, mz ⟩ basis construct the unitary matrix representing Rz (π/2).
2
(d) Use the matrix of the last part to transform
the states |j, mx ⟩ that you found. Of what
component of J⃗ should these transformed
states be eigenstates?
(e) Find the matrix representing the operator Jx
after a rotation by Rz (π/2) and interpret your
result.
{Tqm |m
8. A set of 2q +1 operators
= q, · · · , −q} form
the spherical components of an irreducible tensor T
of rank q if they transform under rotations according to the relation
RTqm R+ =
q
∑
′
(q)
Tqm Rm′ m .
m′ =−q
(a) Show by considering infinitesimal rotations
Ru (ϵ) this reduces the transformation law to a
set of characteristic commutation laws associated with the components of angular momentum.
(b) Use the results of the last part, by considering rotations along the three cartesian axes
and using the known form of the matrices representing the cartesian components of J⃗ in a
standard representation to derive the following form of the relations derived in the last
part:
[Jz , Tqm ] = mTqm
[J+ , Tqm ] =
√
q(q + 1) − m(m + 1)Tqm+1
[J− , Tqm ] =
√
q(q + 1) − m(m − 1)Tqm−1 .
(c) Let {|k, J, M ⟩} denote the basis vectors of a
standard representation of eigenstates of J 2
and Jz . Use the commutation relations above
to show that Tqm |k, J, M ⟩ is an eigenstate of Jz
with eigenvalue m + M , and hence deduce the
“selection rule” that ⟨k ′ , J ′ , M ′ |Tqm |k, J, M ⟩ is
zero unless ∆M = M ′ − M = m.
9. Let Vu denote the component of a vector opera⃗ along a given direction, Ju the component
tor V
of angular momentum along the same direction,
and let |a⟩ and |b⟩ denote two arbitrary vectors lying within the same irreducible invariant subspace
S(k, j). Show that
⃗⟩
⟨J⃗ · V
⟨a|Vu |b⟩ = ⟨a|Ju |b⟩
j(j + 1)
⃗ ⟩ denotes the expectation value of this
where ⟨J⃗ · V
scalar operator taken with respect to any state in
this subspace.
10. The total angular momentum of a particle with
⃗ +S
⃗ of its orspin s = 12 is the sum J⃗ = L
bital and spin angular momentum. Let the states
|n, l, s, ml , ms ⟩ = |n, l, ml ⟩ ⊗ |s, ms ⟩ form an ONB
of (direct product) eigenstates of L2 , Lz , S 2 , and
Sz .
(a) Classify the irreducible invariant subspaces
S(n, l, s, j) of the combined state space for
such a particle. (That is, determine for given l
the values of j associated with the irreducible
invariant subspaces that actually exist, and
the number of subspaces associated with each
value of j.)
(b) Describe how to construct, for each subspace
S(n, l, s, j), the state vectors |n, l, s, j, mj = j⟩
having the maximum component of angular
momentum along the z axis.
(c) Explicitly construct the basis vectors associated with the subspaces of the form
S(n, 1, s, j), having l = 1. Determine the
Clebsch-Gordon coefficients which result from
this construction.
11. Consider a particle of spin s = 12 moving in a central potential V (r). The relativistic theory of the
electron reveals that there is a (small) interaction
between the magnetic moment associated with the
spin of the electron and the central potential in
which it moves. To lowest order, this interaction is
of the form
⃗ · S,
⃗
V = αL
where α is generally a scalar function of r which
we will, for simplicity, take to be a constant. The
total Hamiltonian can then be written
⃗ ·S
⃗
H = H0 + αL
where H0 = P 2 /2m+V (R). Assume that the states
|n, l, s, ml , ms ⟩ referred to in the last problem are
eigenstates of H0 with energy En .
(a) Show explicitly that the basis states
{|n, l, s, j, mj ⟩} whose construction was
addressed in the last problem are eigenstates
of this Hamiltonian, and determine the corresponding energies of the “full” Hamiltonian
H.
(b) Discuss how one would obtain the eigenstates
of the system if α = α(r) were a spherically
symmetric function of r.
12. Consider N particles of spin s = 12 . The total space
of this system can be viewed as the tensor product
⃗ α }) and the spin space
of the spatial space S({R
3
⃗α }). Each of these factor spaces can separately
S({S
be decomposed into irreducible invariant subspaces.
In this problem we focus only on the spin part of
the space. Let
∑
⃗=
⃗α
S
S
α
be the total spin angular momentum of the system.
Find the values of s associated with the eigenvalues
of S 2 , and the number of irreducible invariant subspaces associated with each value represented for a
system containing N particle where
(a) N = 3
(b) N = 4
(c) N = 5
(d) Discuss briefly any general information that
you can deduce about the structure of the irreducible invariant subspaces of the spin space
of N particles for arbitrary N.
13. A particle of spin s = 1 moves in a state of orbital
angular momentum l = 7.
(a) What are the irreducible invariant subspaces
(i.e., eigenspaces of J 2 ) associated with this
system?
(b) If the system is in the state |j, m⟩ = |7, 7⟩ and
a measurement of Lz is made, what values can
be obtained and what are the probabilities of
obtaining those values? Repeat the question
assuming that the operator being measured is
Sz .
(c) If the system is in the state |ml , ms ⟩ = |6, 1⟩,
and a measurement of J 2 is made, what values
can be obtained and what are the probabilities
of obtaining those values?
14. A single spinless particle of mass µ is constrained
to move on the surface of a sphere of radius a. The
Hamiltonian for the system is just the rotational
kinetic energy
H=
~2 ℓ̂2
2µa2
The particle is in a state characterized by the wave
function
[
]
2
ψ(θ, ϕ) = A 2 + (sin θ cos ϕ + i sin θ sin ϕ)
where A is a normalization constant.
(a) If the energy is measured on this state, what
values can be obtained and with what probability will each be found?
(b) What is the state of the system immediately
after a measurement which obtains each of the
possible values that can be obtained?
⃗1 , and S
⃗2 ,
15. Consider two particles of spin 12 , and let S
denote the corresponding spin angular momentum
operators. Let {|m1 , m2 ⟩} be an ONB of direct
products states for the combined
{
}spin space of this
system, with m1 , m2 ∈ + 12 , − 12 = {↑, ↓}. These
states are eigenstates of S1z and S2z . Suppose the
particles interact through a Hamiltonian
⃗1 · S
⃗2
H = −ε0 S
where ε0 is a positive constant. Find the energy
eigenvalues and their degeneracies for this system.
Construct an orthonormal basis of eigenstates for
this Hamiltonian as linear combination of direct
product states.
16. Consider two particles of spin s = 1.
Let
{|m1 , m2 ⟩} be an ONB of direct products states
for the combined spin space of this system, with
m1 , m2 ∈ {−1, 0, 1} . These states are eigenstates
of S1z and S2z .
(a) According to the angular momentum addtion
theorem, what irreducible invariant subspaces
⃗ =
associated with the total spin operator S
⃗
⃗
S1 + S2 occur for this system?
(b) Explicitly construct the eigenstates {|s, m⟩} of
S 2 and Sz for this spin-space, as linear combinations of the direct product states.
(c) From the states that you have produced, construct a table of the Clebsch-Gordon coefficients required to combine two angular momenta with j1 = j2 = 1.
17. An atomic nucleus of spin s1 = 1/2 is in an orbital
angular momentum state with ℓ = ℓ1 = 0. A single
electron (spin s2 = 1/2) is bound to this nucleus in
an orbital angular momentum state with ℓ = ℓ2 =
2.
(a) What values are possible for the quantum
numbers j1 and j2 associated with the to⃗1 + S
⃗1 and
tal angular momentum J⃗1 = L
⃗
⃗
⃗
J2 = L2 + S2 of the nucleus and the electron,
respectively.
(b) What values are possible for the quantum
numbers ℓ and s associated with the total or⃗ =L
⃗ 1 +L2
bital and spin angular momentum L
⃗=S
⃗1 + S
⃗2 , respectively.
and S
(c) What values are possible and how many irreducible subspaces exist for each value of the
quantum number j associated with the total
⃗ +S
⃗ = J1 + J2 of
angular momentum J⃗ = L
this system?
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18. Two distinguishable spinless particle of mass M
move freely on the surface of a sphere of radius
r = a . The two particle state is described by the
angular position space wave function
ψ (θ1 , ϕ1 , θ2 , ϕ2 ) = Y22 (θ1 , ϕ1 ) Y10 (θ2 , ϕ2 ) .
⃗ = ⃗ℓ1 + ⃗ℓ2 be the total angular momentum
Let L
of the system. If L2 and Lz are measured on this
system, what values can be obtained and with what
probability will they occur? [Hint: Feel free to use
an online table of CG coefficients, which you can
find, e.g., on Wikipedia. On a test such a table
would provided.
19. Consider a particle of mass M moving on the surface of a sphere of radius a, in the presence of a
weak electric field, with total Hamiltonian
H = H0 + H (1) =
~2 ℓ̂2
− eE0 Z
2M a2
where ℓ̂2 = ⃗ℓ · ⃗ℓ and where on the surface of the
sphere, in the position representation Z → z =
a cos θ.
(a) What are the energies and degeneracies of
the system in the absence of the applied
field? What are the energy eigenfunctions in
the angular position representation (in which
χ (θ, ϕ) = ⟨θ, ϕ|χ⟩, for example.
(b) Find the energy of the ground state of the
unperturbed Hamiltonian to 2nd order in the
field. [Hint: use the Wigner-Eckart theorem
to determine which matrix elements of H (1)
are not zero.
20. For a particle moving on the surface of a sphere,
let |ℓ, m⟩ denote a standard representation of eigen⃗ be a vector operator for
states of ℓ̂2 and ℓ̂z . Let V
this system which has the property that the matrtix element
⟨1, 0|Vx |1, 1⟩ = 3v0
where v0 is a positive constant. Construct the 3 × 3
submatrices [Vx ] , [Vy ] , [Vz ], representing the Cartesian components of this vector operator within the
irreducible invariant subspace S (1) , spanned by
the states {|1, m⟩|m = −1, 0, 1}.