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equal P, then it must be true that

Normalization 





^2 dV 1 (5.1)

since if the particle exists somewhere at all times,







P dV 1

A wave function that obeys Eq. (5.1) is said to be normalized.Every acceptable

wave function can be normalized by multiplying it by an appropriate constant; we shall

shortly see how this is done.

Well-Behaved Wave Functions

Besides being normalizable, must be single-valued, since Pcan have only one value at

a particular place and time, and continuous. Momentum considerations (see Sec. 5.6)

require that the partial derivatives x, y, zbe finite, continuous, and single-

valued. Only wave functions with all these properties can yield physically meaningful

results when used in calculations, so only such “well-behaved” wave functions are ad-

missible as mathematical representations of real bodies. To summarize:

1 must be continuous and single-valued everywhere.

2 x, y, zmust be continuous and single-valued everywhere.

3 must be normalizable, which means that must go to 0 as x→   , y→   ,

z→  in order that ^2 dVover all space be a finite constant.

These rules are not always obeyed by the wave functions of particles in model

situations that only approximate actual ones. For instance, the wave functions of a par-

ticle in a box with infinitely hard walls do not have continuous derivatives at the walls,

since 0 outside the box (see Fig. 5.4). But in the real world, where walls are never

infinitely hard, there is no sharp change in at the walls (see Fig. 5.7) and the de-

rivatives are continuous. Exercise 7 gives another example of a wave function that is

not well-behaved.

Given a normalized and otherwise acceptable wave function , the probability that

the particle it describes will be found in a certain region is simply the integral of the

probability density ^2 over that region. Thus for a particle restricted to motion in the

xdirection, the probability of finding it between x 1 and x 2 is given by

Probability Px 1 x 2  

x 2

x 1

^2 dx (5.2)

We will see examples of such calculations later in this chapter and in Chap. 6.

5.2 THE WAVE EQUATION

It can have a variety of solutions, including complex ones

Schrödinger’s equation,which is the fundamental equation of quantum mechanics in

the same sense that the second law of motion is the fundamental equation of New-

tonian mechanics, is a wave equation in the variable .

Quantum Mechanics 163

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