bei48482_FM

so we can say that the various Enare the eigenvalues of the Hamiltonian operator Hˆ.

This kind of association between eigenvalues and quantum-mechanical operators is quite

general. Table 5.1 lists the operators that correspond to various observable quantities.

5.8 PARTICLE IN A BOX

How boundary conditions and normalization determine wave functions

To solve Schrödinger’s equation, even in its simpler steady-state form, usually requires

elaborate mathematical techniques. For this reason the study of quantum mechanics

has traditionally been reserved for advanced students who have the required profi-

ciency in mathematics. However, since quantum mechanics is the theoretical structure

whose results are closest to experimental reality, we must explore its methods and ap-

plications to understand modern physics. As we shall see, even a modest mathemati-

cal background is enough for us to follow the trains of thought that have led quantum

mechanics to its greatest achievements.

The simplest quantum-mechanical problem is that of a particle trapped in a box

with infinitely hard walls. In Sec. 3.6 we saw how a quite simple argument yields the

energy levels of the system. Let us now tackle the same problem in a more formal way,

which will give us the wave function (^) nthat corresponds to each energy level.
We may specify the particle’s motion by saying that it is restricted to traveling along
the xaxis between x0 and xLby infintely hard walls. A particle does not lose
energy when it collides with such walls, so that its total energy stays constant. From a
formal point of view the potential energy Uof the particle is infinite on both sides of
the box, while Uis a constant—say 0 for convenience—on the inside (Fig. 5.4). Because
the particle cannot have an infinite amount of energy, it cannot exist outside the box,
and so its wave function is 0 for x0 and x L. Our task is to find what is
within the box, namely, between x0 and xL.
Within the box Schrödinger’s equation becomes
 E  0 (5.37)
2 m
2
d^2
dx^2
Quantum Mechanics 177
Table 5.1 Operators Associated with Various
Observable Quantities
Quantity Operator
Position, xx
Linear momentum, p
Potential energy, U(x) U(x)
Kinetic energy, KE
Total energy, Ei
Total energy (Hamiltonian form), H U(x)
^2
x^2
2
2 m

t
^2
x^2
2
2 m
p^2
2 m

x
i
x
0 L
∞
U
Figure 5.4A square potential well
with infinitely high barriers at
each end corresponds to a box
with infinitely hard walls.
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