Page 58 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.5 Particle in a one-dimensional box 49
V(x)

x
0 a
Figure 2.1 The potential energy V(x) for a particle in a one-dimensional box of length a.

or
d ø 4ð 2
2
ˆÿ ø (2:34)
dx 2 ë 2
where ë is the de Broglie wavelength,
2ð" h
ë ˆ p  ˆ (2:35)
2mE p
We have implicitly assumed here that E is not negative. If E were negative,
then the wave function ø and its second derivative would have the same sign.
2
2
As jxj increases, the wave function ø(x) and its curvature d ø=dx would
become larger and larger in magnitude and ø(x) would approach ( ) in®nity
as x !1.
The solutions to equation (2.34) are functions that are proportional to their
second derivatives, namely sin(2ðx=ë) and cos(2ðx=ë). The functions
exp[2ðix=ë] and exp[ÿ2ðix=ë], which as equation (A.31) shows are equivalent
to the trigonometric functions, are also solutions, but are more dif®cult to use
for this system. Thus, the general solution to equation (2.34) is
2ðx 2ðx
ø(x) ˆ A sin ‡ B cos (2:36)
ë ë
where A and B are arbitrary constants of integration.
The constants A and B are determined by the boundary conditions placed on
the solution ø(x). Since ø(x) must be continuous, the boundary conditions
require that ø(x) vanish at each end of the box so as to match the value of ø(x)
outside the box, i.e., ø(0) ˆ ø(a) ˆ 0. At x ˆ 0, the function ø(0) from (2.36)
is
ø(0) ˆ A sin 0 ‡ B cos 0 ˆ B
so that B ˆ 0 and ø(x)isnow
2ðx
ø(x) ˆ A sin (2:37)
ë
At x ˆ a, ø(a)is