Problem Set 04
Note:
This problem set is due Oct 01 before midnight. Please leave it in my mailbox located
at the first floor of Rutherford building.
1.
Recall the projection operator that we discussed in the class. Our basis is given by the
kets

a
i
i
with
i
= 1
, ..., n
. In this basis we define an operator
b
P
ij
=

a
i
ih
a
j

(1)
which would become a projection operator only when
i
=
j
.
Thus all
b
P
ii
would be
projection operators. Now answer the following questions:
(a) Imagine we define another operator of the form
b
P
j
=
n
X
i
=1
α
i
b
P
ij
(2)
where
α
i
are complex numbers. Determine the matrix representation of
b
P
j
.
(b) If we define another operator of the form
b
Q
=
b
P
12
+
b
P
23
+
b
P
34
+
b
P
45
+
....
(3)
where the series terminates at
n
. Determine the matrix representation of
b
Q
.
(c) A function
F
(
x
) has the following generic binomial expansion
F
(
x
) =
∞
X
m
=0
f
m
x
m
(4)
where
f
m
are real coefficients. Find
F
(
b
P
ii
). Can you also determine the matrix represen
tation of this operator? What happens when
F
(
x
) =
e
x
?
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 Fall '05
 KeshavDasgupta
 Linear Algebra, mechanics, pj, Hilbert space, Matrix representation

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