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Chapter 5
APPROXIMATION METHODS FOR STATIONARY STATES
As we have seen, the task of prediciting the evolution of an isolated quantum mechan
ical can be reduced to the solution of an appropriate eigenvalue equation involving the
Hamiltonian of the system. Unfortunately, only a small number of quantum mechanical
systems are amenable to an exact solution. Moreover, even when an exact solution to
the eigenvalue problem is available, it is often useful to understand the behavior of the
system in the presence of weak external
f
elds that my be imposed in order to probe the
structure of its stationary states. In these situations an approximate method is required
for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that
renders an exact solution untenable. There are two general approaches commonly taken
to solve problems of this sort. The
f
rst, referred to as the variational method, is most
useful in obtaining information about the ground state of the system, while the second,
generally referred to as timeindependent perturbation theory, is applicable to any set of
discrete levels and is not necessarily restricted to the solution of the energy eigenvalue
problem, but can be applied to any observable with a discrete spectrum.
5.1
The Variational Method
Let
H
be a timeindependent observable (e.g., the Hamiltonian) for a physical system
having (for convenience) a discrete spectrum. The normalized eignestates
{
φ
n
i}
of
H
each satisfy the eigenvalue equation
H

φ
n
i
=
E
n

φ
n
i
(5.1)
where for convenience in what follows we assume that the eigenvalues and corresponding
eigenstates have been ordered, so that
E
0
≤
E
1
≤
E
2
···
.
(5.2)
Under these circumstances, if

ψ
i
is an arbitrary normalized state of the system it is
straightforward to prove the following simple form of the
variational theorem
:t
h
e
mean value of
H
with respect to an arbitrary normalized state

ψ
i
is necessarily greater
than the actual ground state energy (i.e., lowest eigenvalue) of
H
, i.e.,
h
H
i
ψ
=
h
ψ

H

ψ
i
≥
E
0
.
(5.3)
The proof follows almost trivially upon using the expansion
H
=
X
n

φ
n
i
E
n
h
φ
n

(5.4)
of
H
in its own eigenstates to express the mean value of interest in the form
h
H
i
ψ
=
X
n
h
ψ

φ
n
i
E
n
h
φ
n

ψ
i
=
X
n

ψ
n

2
E
n
,
(5.5)
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Approximation Methods for Stationary States
and then noting that each term in the sum is itself bounded, i.e.,

ψ
n

2
E
n
≥

ψ
n

2
E
0
,
so
that
X
n

ψ
n

2
E
n
≥
X
n

ψ
n

2
E
0
=
E
0
(5.6)
where we have used the assumed normalization
h
ψ

ψ
i
=
P
n

ψ
n

2
=1
of the otherwise
arbitrary state

ψ
i
.
Note that the equality holds only if

ψ
i
is actually proportional to the
ground state of
H
.
Thus, the variational theorem proved above states that the ground state minimizes
the mean value of
H
taken with respect to the normalized states of the space. This has
interesting implications. It means, for example, that one could simply choose random
vectors in the state space of the system and evaluate the mean value of
H
with respect
to each. The smallest value obtained then gives an upper bound for the ground state
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 Fall '10
 PaulE.
 mechanics

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