bei48482_FM

of  (^) n^2 over all space is finite, as we can see by integrating  (^) n^2 dxfrom x0 to
xL (since the particle is confined within these limits). With the help of the
trigonometric identity sin^2 ^12 (1 cos 2) we find that



 (^) n^2 dx
L
0
 (^) n^2 dxA^2 
L
0
sin^2  dx
 
L
0
dx
L
0
cos dx
 x  sin 
L
0
A^2  (5.43)
To normalize we must assign a value to Asuch that  (^) n^2 dxis equalto the prob-
ability P dxof finding the particle between xand xdx, rather than merely propor-
tional to P dx.If  (^) n^2 dxis to equal P dx, then it must be true that



 (^) n^2 dx 1 (5.44)
Comparing Eqs. (5.43) and (5.44), we see that the wave functions of a particle in a
box are normalized if
A (5.45)
The normalized wave functions of the particle are therefore
Particle in a box (^) nsin n1, 2, 3,... (5.46)
The normalized wave functions 1 , 2 , and 3 together with the probability densities
 1 ^2 ,  2 ^2 , and  3 ^2 are plotted in Fig. 5.5. Although (^) nmay be negative as well as
positive,  (^) n^2 is never negative and, since (^) nis normalized, its value at a given xis
equal to the probability density of finding the particle there. In every case  (^) n^2 0 at
x0 and xL, the boundaries of the box.
At a particular place in the box the probability of the particle being present may be
very different for different quantum numbers. For instance,  1 ^2 has its maximum
value of 2Lin the middle of the box, while  2 ^2 0 there. A particle in the lowest
energy level of n1 is most likely to be in the middle of the box, while a particle in
the next higher state of n2 is neverthere! Classical physics, of course, suggests the
same probability for the particle being anywhere in the box.
The wave functions shown in Fig. 5.5 resemble the possible vibrations of a string
fixed at both ends, such as those of the stretched string of Fig. 5.2. This follows from
the fact that waves in a stretched string and the wave representing a moving particle
are described by equations of the same form, so that when identical restrictions are
placed upon each kind of wave, the formal results are identical.
nx
L
2
L
2
L
L
2
2 nx
L
L
2 n
A^2
2
2 nx
L
A^2
2
nx
L
Quantum Mechanics 179
Figure 5.5Wave functions and
probability densities of a particle
confined to a box with rigid walls.
x = 0 x = L
(^1)
(^2)
(^3)
x = 0 x = L
| 3 |^2
| 2 |^2
| 1 |^2
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