bei48482_FM

Example 5.4

Find the probability that a particle trapped in a box Lwide can be found between 0.45Land

0.55Lfor the ground and first excited states.

Solution

This part of the box is one-tenth of the box’s width and is centered on the middle of the box

(Fig. 5.6). Classically we would expect the particle to be in this region 10 percent of the time.

Quantum mechanics gives quite different predictions that depend on the quantum number of

the particle’s state. From Eqs. (5.2) and (5.46) the probability of finding the particle between x 1

and x 2 when it is in the nth state is

Px 1 ,x 2 

x 2

x 1

n^2 dx 

x 2

x 1

sin^2 dx

  sin 

x 2

x 1

Here x 1 0.45Land x 2 0.55L. For the ground state, which corresponds to n1, we have

Px 1 ,x 2 0.19819.8 percent

This is about twice the classical probability. For the first excited state, which corresponds to

n2, we have

Px 1 ,x 2 0.00650.65 percent

This low figure is consistent with the probability density of n^2 0 at x0.5L.

2 nx



L

1



2 n

x



L

nx



L

2



L

180 Chapter Five

x = 0 x = L

| 2 |^2

| 1 |^2

x 1 x 2

Figure 5.6The probability Px 1 ,x 2 of finding a particle in the box of Fig. 5.5 between x 1 0.45Land

x 2 0.55Lis equal to the area under the ^2 curves between these limits.

bei48482_ch05.qxd 2/6/02 6:51 PM Page 180