bei48482_FM

Quantum Mechanics 181

Example 5.5

Find the expectation value x of the position of a particle trapped in a box Lwide.

Solution

From Eqs. (5.19) and (5.46) we have

x 





x ^2 dx 

L

0

x sin^2 dx

 

L

0

Since sinn0, cos 2n1, and cos 0 1, for all the values of nthe expectation value of

xis

x   

This result means that the average position of the particle is the middle of the box in all quan-

tum states. There is no conflict with the fact that  ^2 0 at L2 in the n2, 4, 6,... states

because x is anaverage, not a probability, and it reflects the symmetryof  ^2 about the middle

of the box.

Momentum

Finding the momentum of a particle trapped in a one-dimensional box is not as straight-

forward as finding x. Here

 (^) n sin
 cos
and so, from Eq. (5.30),
p 


pˆ dx


* dx
 
L
0
sin cos dx
We note that
sinax cosax dx sin^2 ax
1
2 a
nx
L
nx
L
n
L
2
L
i
d
dx
i
nx
L
n
L
2
L
d
dx
nx
L
2
L
L
2
L^2
4
2
L
cos (2nxL)
8(nL)^2
xsin (2nxL)
4 nL
x^2
4
2
L
nx
L
2
L
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