Quantum Mechanics 215A Homework Solutions #1
Sam Pinansky
October 8, 2003
People did ﬁne on this assignment. The only mistakes were on the second problem. The average
was 18
.
1
/
20.
1. (5 Points) This can be proved simply by writing out both sides:

AC
{
D,B
}
+
A
{
C,B
}
D

C
{
D,A
}
B
+
{
C,A
}
DB
(1)
=

ACDB

ACBD
+
ACBD
+
ABCD

CDAB

CADB
+
CADB
+
ACDB
(2)
=
ABCD

CDAB
(3)
= [
AB,CD
]
(4)
Done.
2. (10 Points) The problem asks you to use braket algebra to prove these, but some of you
converted them into matrix problems. This was not incorrect, but was not exactly in the spirit
of the problem.
(a) By the deﬁnition of the trace:
tr(
XY
)
≡
X
a
h
a

XY

a
i
(5)
where

a
i
is some orthonormal basis. Now we insert a complete set of states:
X
a
h
a

XY

a
i
=
X
a
X
a
0
h
a

X

a
0
ih
a
0

Y

a
i
(6)
=
X
a
0
X
a
h
a
0

Y

a
ih
a

X

a
0
i
(7)
=
X
a
0
h
a
0

Y X

a
0
i
(8)
= tr(
Y X
)
(9)
where we have used the fact that
h
a
0

Y

a
i
and
h
a

X

a
0
i
are just numbers so commute, and
the completeness of the
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 Fall '04
 Pinansky
 Linear Algebra, mechanics, Work, Hilbert space, A, dual correspondence, Y a

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