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Introduction to Quantum and Statistical Mechanics
Handout 8
Approximation Methods in Quantum Mechanics
Assigned Reading:
Chaps. 13 – 16, pp. 247 – 304.
8.1
Overview
We have learned the basic principles in quantum mechanics and how to analytically solve
the Schrödinger equation in simple potentials.
The general steps include finding the
eigenfunctions and eigenvalues of the timeindependent Schrödinger equation according to the
boundary condition (boundary value problem), express the initial condition as the expansion of
the eigenfunctions, and then the solution of the wave functions can be obtained by adding
(
29
/
exp
t
iE
n

to the eigenfunction expansion (initial value problem).
The expectation value of
all measurable quantities can then be calculated from the wavefunction.
However, the
Schrödinger equation can become unsolvable analytically either due to the increase of the
number of variables (you have seen the multipleparticle effect in phonons or you can imagine a
pack of electrons and ions forming plasma in 3D) or the complexity of the potential.
In fact,
there are only a handful of potentials that we have reasonably tractable solutions.
For most
practical problems, a good and accurate “guess” or “approximation” is not often necessary, but
also gives the best intuition when we try to design a quantum system.
Complex potential can often be treated with the perturbation method.
If the system
Hamiltonian can be written as:
1
0
ˆ
ˆ
ˆ
H
H
H
+
=
(8.1)
where
0
ˆ
H
has known eigenfunctions and eigenvalues:
)
(
)
(
ˆ
0
x
E
x
H
n
n
φ
=
(8.2)
1
ˆ
H
is called the perturbation Hamiltonian. For the wave function
ψ
(x)
that satisfies the system
Hamiltonian, i.e.,
)
(
)
(
x
E
x
H
=
, since
n
(x)
will form a complete set,
(x)
can be expressed
as the linear combination of
n
(x)
as long as
1
ˆ
H
does not change the continuity at boundaries
and behavior at infinity:
∑
=
=
N
n
n
n
x
c
x
1
)
(
)
(
(8.3)
However, since we do not know the form of
(x),
we cannot use the
braket
<
k

method to find
c
n
.
Substituting this expansion into the Schrödinger equation of
H
(x)=E
(x)
:
1
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∑
∑
=
+
n
n
n
n
n
n
c
E
c
H
H
φ
1
0
ˆ
ˆ
(
29
∑
∑
=

n
n
n
n
n
n
n
H
c
c
E
E
1
ˆ
(8.4)
Now we can use the
bracket
<
k

method to simplify the summation, i.e., we will
multiply both sides with
k
*
and integrate over the entire
x
space,
(
29
∑
=

n
n
k
n
k
k
H
c
c
E
E

ˆ

1
(8.5)
Notice that we know the form of
k
and
n
and hence the braket can be evaluated. For
each index
k
we will obtain an equation containing the
N+1
unknowns of
c
n
and
E
. What we
obtain here will be
N
linear equations plus the normalization to find
c
n
and
E
.
This is why we
call
n
k
H

ˆ

1
as the matrix element.
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