Physics 115A Third Problem Set
Harry Nelson
Office Hours (This Week) Th 1:303:00pm
TA: Antonio Boveia
Office Hours M 910am, Fr 13pm PLC
Grader: Victor Soto
Office Hours Th 11:0012:30pm PLC
due Monday, January 27, 2003
1. The three kets

1
)
,

2
)
, and

3
)
are represented, in a particular basis as:

1
)
.
=
1
/
2
√
3
/
2
√
2
√
3
/
2
√
2
,

2
)
.
=

√
3
/
2
1
/
2
√
2
1
/
2
√
2
,

3
)
.
=
0
1
/
√
2

1
/
√
2
.
(a) Find the matrices that represent

1
)(
1

,

2
)(
2

,

1
)(
1

+

2
)(
2

, and

1
)(
1

+

2
)(
2

+

3
)(
3

.
(b) Use the matrix representation to evaluate the action of

1
)(
1

+

2
)(
2

on

3
)
, and
interpret the result.
(c) Evaluate and interpret the trace of

1
)(
1

+

2
)(
2

.
2. Use an insertion of the ‘decomposition of unity,’ which is the relationship
∑
n
k
=1

k
)(
k

=
I
,
to derive the ‘matrix multiplication’ relationship between the matrix elements of a sequence
of two linear operators,
ΛΩ
, with matrix elements in the basis of
(
i

ΛΩ

j
)
, and the matrix
elements of the individual operators
(
i

Λ

k
)
and
(
k

Ω

j
)
.
3. In class, and on page 22 of your text, the representation of the rotation operator
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 Winter '03
 Nelson
 Physics, Linear Algebra, mechanics, Hilbert space, matrix elements, Victor Soto, 13pm PLC Grader

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