Physical Chemistry , 1st ed.

Solution
a.The system looks like this:

where the sloped line indicates the true bottom of the box.
b.In order to determine a 2 , we need to evaluate the expression

a 2 

where PIAB stands for particle-in-a-box. The wavefunctions and energies for
the particle-in-a-box system are known, so all we need do is substitute for the
wavefunctions and the energies.

a 2 

Since all the functions in the integral are being multiplied together, they can
be rearranged (and the constants removed from the integral sign and the
denominator simplified) to yield

a 2 

In order to integrate this, we need to substitute the trigonometric identity
sin axsin bx^12 [cos (ab)xcos(ab)x] and then use the integral
table in Appendix 1. We get:

a 2 



Evaluating this at the limits and simplifying, one finds that

a 2 ^1

2

2

7

8

km

(^2) h
a
2
3

and so the approximate wavefunction is
1,real 1,PIAB

1

2

2

7

8

km

(^2) h
a
2
3
2,PIAB

a
k
 
a^2
 2 cos 
a
x

a
x
sin 
a
x

9
a
2
 2 cos 

3

a

x



3

a

x

sin 

3

a

x

 a 0





8

3

m

h

a

2

 2



2

a

k

 

1

2



a

0 

xcos 

a

x

xcos 

3

a

x

dx





8

3

m

h

a

2

 2



2

a

k



a

0

xsin 

2

a

x

sin 

1

a

x

dx





8

3

m

h

a

2

 2



a

0 



2

a

sin 

2

a

x

kx

2

a

sin 

1

a

x

dx





8

1

m

(^2) h
a
2
 2  8

2

m

(^2) h
a
2
 2

a
0
*2,PIABkx1,PIABdx

E1,PIABE2,PIAB
x
12.6 Perturbation Theory 393