Physics 115A Midterm
Harry Nelson
Monday, Feb. 10, 2003
Closed Book; no calculators. For full credit, show your work and make your reasoning clear
to the graders.
The ‘boldface’ notation below is used for operators; thus,
Ω
is an abstract operator.
In
class we put a ‘twiddle’ under the Ω to denote that it was an operator. The symbol
.
= means
‘is represented by’.
The quadratic formula for the roots to the equation
ax
2
+
bx
+
c
= 0 is:
x
=

b
±
√
b
2

4
ac
2
a
.
1. (25 pts) Two kets have unit length:

V
1
)
, and

V
2
)
, so
(
V
1

V
1
)
=
(
V
2

V
2
)
= 1; these two
kets are never equal, that is,

V
1
) negationslash
=

V
2
)
. The two projection operators are
P
1
=

V
1
)(
V
1

and
P
2
=

V
2
)(
V
2

.
(a) Suppose

V
1
)
and
V
2
)
are represented in an orthonormal basis by:

V
1
)
.
=
bracketleftBigg
1
0
bracketrightBigg
,

V
2
)
.
=

radicalBig
1
3
radicalBig
2
3
.
i. Find the matrices that represent
P
1
and
P
2
.
ii. Use the matrix representations to find the matrix that represents the commu
tator [
P
1
,
P
2
].
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 Winter '03
 Nelson
 Linear Algebra, mechanics, Work, Hilbert space, Ω, Orthonormal basis, nonzero real number, abstract vectors

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