Problem Set 07
Note:
This problem set is due Nov 01 before midnight.
Please leave it in my mailbox
located at the first floor of Rutherford building.
1.
A Hilbert space is parametrised by the basis vectors

λ
+
i
i
which are eigenstates of an
operator
A
with eigenvalues
λ
+
i
. The Hilbert space is finite dimensional with
i
= 0
, ..., n
,
and allows a Hamiltonian operator
H
which satisfies the following commutation relation
with
A
:
[
H
,
A
] =
α
H
(1)
where
α
is an integer with 0
≤
α
≤
n
. Clearly this means that the basis vectors are not
eigenstates of the Hamiltonian. Now at time
t
= 0 let us construct a state

ψ
i
=
n
X
i
=0
c
i

λ
+
i
i
(2)
with
c
i
time independent constants. Determine the evolution of this state at a time
t
=
t
0
>
0 using the variables specified in the problem. Under what condition is the system
completely solvable?
2.
If I provide you with a
n
dimensional Hilbert space in which an operator
A
is given
by an upper triangular matrix with elements 1.
An upper triangular matrix is a
n
×
n
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 Fall '05
 KeshavDasgupta
 mechanics, Hilbert space, Upper Triangular Matrix

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