Page 255 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

246 Approximation methods

9.4 Perturbed harmonic oscillator
As illustrations of the application of perturbation theory we consider two
examples of a perturbed harmonic oscillator. In the ®rst example, we suppose
that the potential energy V of the oscillator is

2
2 2
1
1
4
V ˆ kx ‡ cx ˆ mù x ‡ cx 4
2 2
where c is a small quantity. The units of V are those of "ù (energy), while the
units of x are shown in equation (4.14) to be those of ("=mù) 1=2 . Accordingly,
3
2
the units of c are those of m ù =" and we may express c as
m ù 3
2
c ˆ ë
"
where ë is dimensionless. The potential energy then takes the form

3 4
2
m ù x
2 2
1
V ˆ mù x ‡ ë (9:42)
2 "
^ (0)
The Hamiltonian operator H for the unperturbed harmonic oscillator is
given by equation (4.12) and its eigenvalues E (0) and eigenfunctions ø (0) are
n
n
^
shown in equations (4.30) and (4.41). The perturbation H9 is
2
3 4
^
^
H9 ˆ H (1) ˆ ë m ù x (9:43)
"
^ (2) ^ (3)
Higher-order terms H , H , ... in the perturbed Hamiltonian operator do
not appear in this example.
To ®nd the perturbation corrections to the eigenvalues and eigenfunctions,
4
we require the matrix elements hn9jx jni for the unperturbed harmonic
oscillator. These matrix elements are given by equations (4.51). The ®rst-order
correction E (1) to the eigenvalue E n is evaluated using equations (9.24), (9.43),
n
and (4.51c)
ëm ù 3
2
^
1
2
3
4
E (1) ˆ H (1) ˆ hnjx jniˆ (n ‡ n ‡ )ë"ù (9:44)
n nn 2 2
"
The second-order correction E (2) is obtained from equations (9.34), (9.43), and
n
(4.51) as follows