Physical Chemistry , 1st ed.

a.Using perturbation theory, estimate the average energy of a particle having

mass mand whose motion is described by the lowest-energy wavefunction

(n1).

b.The integral in part a can be solved exactly. Explain why this calculated

value is not the exact value for the energy of a particle in this system.

Solution

a.According to perturbation theory, the energy of the particle is

EE(0)E(1)

If one assumes that Hˆ° is the Hamiltonian for the particle-in-a-box, then the

perturbation part Hˆof the complete Hamiltonian is kx. According to equa-

tion 12.17, the energy is

E

8

n

m

(^2) h
a
2
 2 E

(1)

To evaluate E(1), we must evaluate the integral

E(1)

2

a



a

0

sin 

a

x

*kxsin 

a

x

dx

where the normalization constant has been brought outside of the integral

sign, and dand the integration limits are for the 1-D particle-in-a-box. This

integral simplifies to

E(1)

2

a

k



a

0

xsin^2 

a

x

dx

This integral has a known solution (see the integral table in Appendix 1).

Evaluation of this integral specifically is left as an exercise. Substituting for

the evaluated integral, this expression becomes

E(1)

a

4

2



2

a

k



k

2

a



Therefore, the energy of the n1 level is

E

8

n

m

(^2) h
a
2
 2 
k
2
a

b.This is not an exact energy for such a system because the wavefunctions
used to determine the energies were the particle-in-a-box wavefunctions, not
wavefunctions for a box having a sloped bottom. So although the integral for
the perturbation energy is solvable analytically, it does not correct the energy
to the exactvalue of the true energy because we are not using the eigenfunc-
tions of the defined system. (Nor are we using the complete Hamiltonian
operator for the defined system.) Higher-order perturbation theory, not dis-
cussed in this text, may have a better chance of approaching the exact wave-
function and energy eigenvalues for this system.
As the above example shows, although we have defined a first-order energy
correction, we are still using the ideal forms of the wavefunctions. What we
also need is a correction to the wavefunctions.It is assumed that, as for the en-
ergy correction, the first-order correction to the wavefunction is some correc-
tion added to the ideal wavefunction to approximate the real wavefunction:
390 CHAPTER 12 Atoms and Molecules