Physics 312: Problem Set 3
1. Given that you can re-express a rotation about an arbitrary axis in terms of three rotations using the x̂, ŷ, ẑ
axis basis, show that
i
0
~
i
e− h̄ γ ẑ ·J e− h̄ β ŷ
00
·J~ − h̄i αẑ·J~
e
i
i
i
= e− h̄ αJz e− h̄ βJy e− h̄ γJz ,
where ŷ 00 is rotated an amount α from ŷ about the ẑ axis, and ẑ is rotated an amount β from ẑ about the ŷ 00
axis (refer to figure from class or Fig. 17.1 of Merzbacher). This also illustrates that there are many different
paths to obtaining the desired rotation.
2. Sakurai Ch. 3 Problem 8 (Sakurai is on reserve in the library if you want to see the problems in their original
form. If that is inconvenient, they are reproduced here):
Consider a sequence of Euler rotations represented by
D1/2 (α, β, γ) = exp
−iσ3 α −iσ2 β −iσ3 γ exp
exp
.
2
2
2
[The (1/2) superscript on D means that we are in a spin 1/2 system, so the σ matrices are the Pauli matrices.]
Because of the group properties of rotations, we expect that this sequence of operations is equivalent to a single
rotation about some axis by an angle θ. Find θ.
3. Sakurai Ch. 3 Problem 17:
Suppose a half–integer l value, say 21 , were allowed for orbital angular momentum. From
L+ Y1/2,1/2 (θ, φ) = 0 .
we may deduce, as usual,
√
Y1/2,1/2 (θ, φ) ∼ eiφ/2 sinθ .
Now try to construct Y1/2,−1/2 (θ, φ); by (a) applying L− to Y1/2,1/2 (θ, φ); and (b) using L− Y1/2,−1/2 (θ, φ) = 0.
Show that the two procedures lead to contradictory results. (This gives an argument against half–integer l
values for orbital angular momentum.)
4. Find the probability (to order 2 ) that a state |j, m = mmax = ji is rotated by an infinitesimal angle about
the y–axis and is found unchanged.
5. Use Clebsch-Gordan coefficients to block diagonalize (decompose) the 4-dimensional 2 ⊗ 2 individual-basis operators J~2 , Jx , Jy , and Jz to their 4-dimensional block diagonal composite-basis form. Find the matrix which
performs this block-diagonalization. What does this do to the group elements generated by these operators?
Hints:
• You may have to reorder your states in order to see the block-diagonal form.
• J+ = S1+ + S2+ , etc., where S are the spin-1/2 operators.
~ 2 +S
~ 2 +2S1z S2z +S1+ S2− +S1− S2+ .
• Find the operators by taking matrix elements, using for example, J~2 = S
1
2
P
P
0
0
0 0
0 0
0
0
• hm1 m2 |Ji |m1 m2 i = lm l0 m0 hm1 m2 |lmihlm|Ji |l m ihl m |m1 m2 i.
• You are to find the matrix S such that O(individual basis) = SO(combined
operators above, and the same S works for all of them.
basis) S
†
, where O is any of the four
6. “Adding” l = 1 with s = 1/2: Use raising and lowering operators as well as orthogonality to express the
combined basis states in terms of the individual basis states |l = 1, ml , s = 1/2, ms i.
7. (from Merzbacher Exercise 17.20) Clebsch-Gordan coefficients take us from the individual product basis to the
block-diagonal irreducible basis for the rotation matrices D(j) :
)
)
Dµ(j11m
(R)Dµ(j22m
(R) =
1
2
jX
1 +j2
XX
(J)
hj1 j2 ; m1 m2 |JM ihj1 j2 ; µ1 µ2 |JµiDµM (R) .
J=|j1 −j2 | M
µ
Using this, the unitarity of the D matrices, and the orthogonality of the C-G, show that
X X (J)
X
)
)∗
(R) =
DM 0 M (R)hj1 j2 ; m1 m2 |JM iDµ(j11m
(R) .
hj1 j2 ; µ1 µ2 |JM 0 iDµ(j22m
2
1
µ2
m1 M
8. Which of the following are groups?
(a) All real numbers, with group multiplication being ordinary multiplication.
(b) All real numbers, with group multiplication being addition
(c) All complex numbers except zero with group multiplication being ordinary multiplication
(d) All positive rational numbers with group multiplication between two group elements a and b being a/b
(e) The three Pauli matrices σx , σy , σz , along with the unit matrix I.
9. Show that the group SU (n) has n2 − 1 independent real parameters. Show that the group of special real
orthogonal matrices SO(n, R) has n(n − 1)/2 independent real parameters. (SO(n) matrices have their inverses
equal to their transposes and determinant equal to one.)
10. Show, using a plane wave as your original eigenstate, that the momentum operator “generates” a translation in
space.
11. We argued that a single-indexed object ui , where i=1,2, generates the 2x2 representation of SU(2), which we
wrote in its explicit most general form in class in terms of the α, β, and γ parameters. Call this most general
group element Σ. ui transforms via Σ while ui transforms as its conjugate, Σ∗ . In what follows, the summation
convention is implied.
(a) Review (we did this in class) by explicit calculation using the Σ transformation matrices that the Levi-Civita
tensor remains invariant:
Σac Σbd εab = εcd .
(b) Show by explicit calculation using the Σ matrices that the naive Kronecker-delta does not remain invariant
under SU(2) tranformations:
Σac Σbd δab 6= δcd .
(c) Now show that the Kronecker delta δji is invariant:
Σac (Σ∗ )bd δab = δcd .