MATHEMATICAL PHYSICS
PHYSICS 227, Instructor: Marcel den Nijs
SOLUTIONS HOMEWORK SET 6: [10 points total]
PROBLEM 1 [2 points]
Consider the Pauli spin matrices
ˆ
S
x
=
0
1
1
0
;
ˆ
S
z
=
0

i
i
0
;
ˆ
S
z
=
1
0
0

1
;
ˆ
1 =
1
0
0
1
a. Give the general definition for unitary matrices, and explain why the Pauli spin matricesare unitary operators.Unitary operators are matrices where the adjoint is the same as the inverse,ˆA1=ˆA†such thatˆA†ˆA= 1.Recall that Hermitian matrices have the property that they are self adjointˆA†=ˆA, which means that the square of unitary hermitian matrices is equalto one; see part (b).b. Derive that are their own inverse,c. Show that they anticommute, as ind. Consider the linear combination
(
ˆ
S
x
)
2
= (
ˆ
S
y
)
2
= (
ˆ
S
z
)
2
= 1
(
ˆ
S
x
)
2
=
0
1
1
0
0
1
1
0
=
1
0
0
1
and the same for the others
ˆ
S
x
ˆ
S
y
+
ˆ
S
y
ˆ
S
x
= 0
ˆ
S
x
ˆ
S
y
+
ˆ
S
y
ˆ
S
x
=
0
1
1
0
0

i
i
0
+
0

i
i
0
0
1
1
0
=
i
0
0

i
+

i
0
0
i
= 0
and the same for the other combinations. Note that the commutators are
nonzero
ˆ
S
x
ˆ
S
y

ˆ
S
y
ˆ
S
x
=
i
0
0

i


i
0
0
i
= 2
i
ˆ
S
z
ˆ
H
=

B
x
ˆ
S
x

B
z
ˆ
S
z
1
with
~
B
= (
B
x
,
0
, B
z
) the magnetic field (a vector with real numbers). Find the eigenvalues
and eigenvectors of
ˆ
H
and the rotation matrix that diagonalizes
ˆ
H.
The eigen values of the matrix
B
z
B
x
B
x

B
z
(
B
z

λ
)(

B
z

λ
)

B
2
x
= 0
→
λ
=
±
p
B
2
z
+
B
2
x
are equal to
±
1 times the magnitude of the magnetic field.
(
B
z

λ
)
x
=
B
x
y
→
~a
±
=
x
y
=
B
x
B
z
∓
p
B
2
z
+
B
2
x
The operator is Hermitian (symmetric in this case) such that the right and
left eigenvectors coincide and the eigenvectors are perpendicular
~a
+
·
~a

=
B
x
, B
z

p
B
2
z
+
B
2
x
·
B
x
B
z
+
p
B
2
z
+
B
2
x
= 0
The rotation matrix that diagonalizes
ˆ
H
is formed by the eigen vetors as
rows:
ˆ
R
=
~a
+
~a

It has the same effect as rotating the 3D coordinate system in such a way
that the magnetic field now points in the
z
direction,
ˆ
H
=

~
B

S
z
(group
theory needed to show how in detail; see Quantum Mechanics classes.)
PROBLEM 2 [2 points]
a. Study example 3 on page 165 in Boas, about three coupled springs strung between two
walls, Fig 12.1. Describe in your own words the motion corresponding to the eigenmode
with frequency
ω
1
=
p
K/m
and the one with frequency
ω
2
=
p
3
K/m
.
The eigenmode with
ω
1
=
p
K/m
represents the oscillatory motion where
the two masses move in tandem swinging both to the left and the right at
constant distance from each other.
The eigenmode with
ω
1
=
p
3
k/m
represents the oscillatory motion where
the two masses pulsate in opposite direction while their center of mass does
not move.
We did this as a demo in class on Friday last week using the air track.
You've reached the end of your free preview.
Want to read all 8 pages?