University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics B (Physics 130B) Fall 2014
Worksheet 3 –
Solutions
Announcements
• The 130B web site is:
http://physics.ucsd.edu/students/courses/fall2014/physics130b/ .
Please check it regularly! It contains relevant course information!
• Greetings everyone! This week we’re going to learn about spin, rotations, representations, and all that jazz.
Problems
Suppose we are studying a system with a rotational symmetry. So we need understand how
to represent this symmetry on our Hilbert space of states. This involves creating matrices
which do all the things we expect.
1. Do a Barrel Roll
Recall that in 3-dimensional space1 we can derive the following rotation matrices from
geometry:






1
0
0
cos θ 0 sin θ
cos θ − sin θ 0
1
0  Rz =  sin θ cos θ 0
Rx = 0 cos θ − sin θ Ry =  0
0 sin θ cos θ
− sin θ 0 cos θ
0
0
1
(1)
where Ri is a rotation about the i-th axis by an angle θ. Consider a rotation with an
infinitesimal θ = δθ.
(a) Express each rotation in 1 as Ri (θ = δθ) = 1 − i(δθ)Xi for some matrices Xi .
These are the generators of rotations as we’ll see in a moment.2
Use small angle approximation cos θ ≈ 1 and sin θ ≈ θ
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2
Euclidean. Over R. Don’t get cheeky.
Note that the factor of i is conventional.
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Spoilers. The form of Xi is simply:






0 0 0
0 0 i
0 −i 0
X1 = 0 0 −i X2 =  0 0 0 X3 =  i 0 0
0 i 0
−i 0 0
0 0 0
(2)
(b) Show explicitly that each Xi is Hermitian: X † = X
Obvious
(c) I claim that the Xi of 2 satisfy the following algebra3
[Xi , Xj ] = iijk Xk
(3)
Convince yourself

0 0
X1 X2 = −1 0
0 0
of this by checking a few examples.



0
0 −1 0
0 and X2 X1 = 0 0 0
0
0 0 0


0 1 0

This implies [X1 , X2 ] = −1 0 0 = iX3 and so on.
0 0 0
Given a hermitian matrix X one can construct a unitary matrix U = e−iXa which
’evolves’ a state by an amount a. For example the Hamiltonian Ĥ is hermitian
and leads to the ’time-evolution’ operator U = e−iĤt .
In this way Ĥ generates time evolution. Can you guess where this is going?
(d) Consider the unitary matrices given by Ui = e−iXi θ for each Xi in 2. Show, using
Taylor’s theorem, that Ui = Ri ; they are the rotation matrices of 1.
Let’s do this for X1 , the rest follow very similarly.


0 0 0
U1 = e−iX1 θ = eθA for A = 0 0 −1. Now let’s Taylor expand:
0 1 0
P
P
n
n
∞ θn An
θ A
eθA = ∞
n=0 n! = 1 +
n=1 n! where now we need to note the following facts:


0 0
0
2

A = 0 −1 0  ≡ −B
A3 = −A
A4 = B
A5 = A and so on.
0 0 −1
This allows us to split the infinite sum into evens and odds and then pull out our
A and B matrices.
n−1
n
P
P
P
P
n
n n
n n
2 n
2
eθA = 1 + n,even θ n!A + n,odd θ n!A = 1 + n,even (−1)n! θ B + n,odd (−1) n! θ A
= 1 + (cos θ − 1)B + sin θA = Rx
2. What is Spin?
The fact there are spin- 21 particles is one of the most deeply quantum features of nature.
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This is known as a Lie algebra
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We can think of the spin of an electron as an additional degree of freedom. This is
represented quantum mechanically is a two dimensional Hilbert space H2 spanned by
two vectors {| ↑i, | ↓i}
Now, how can we represent rotations on this space?
Consider the following matrices:
1 0 −i
1 0 1
S2 =
S1 =
2 1 0
2 i 0
1
S3 =
2
1 0
0 −1
(4)
These, up to that factor of 21 , are known as the Pauli matrices.
(a) Show that Si are hermitian. Show explicitly that the following algebra is satisfied:
[Si , Sj ] = iijk Sk
(5)
This is the same algebra as 3, between the generators of rotations!4 Together
these imply we are constructing something like angular momentum.
Just do it.
Now let’s construct the analog of rotation matrices for these objects.
(b) Define Ui = e−iθSi and write a simple matrix expression for it.
Hint: Use the fact σi2 = 1 where σi is a Pauli matrix.
θ
e−iθSi = e−i 2 σi = 1 cos 2θ − iσi sin 2θ
(c) Now consider Ui (θ = 2π), what has happened?
Ui (θ = 2π) = 1 cos π − iσi sin π = −1
We have gone around a complete rotation and picked up a minus sign!
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Fancy math point, this is the statement SO(3) and SU (2) have the same Lie algebra.
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