16.4 Postulate 4 and Expectation Values 701
Ifuis a variable that can take on any real value in a certain range, we require a formula
that is analogous to Eq. (16.4-13). We define the probability thatulies betweenu′and
u′+duby(Probability thatulies betweenu′andu′+du)f(u′)du (16.4-14)The functionf(u) is called aprobability densityor aprobability distribution.Itis
analogous to the probabilitypifor the discrete case, and is a probability per unit length
on theuaxis. The mean value ofuis given by an integral that is analogous to the sum
in Eq. (16.4-13):〈u〉∫
uf(u)du (16.4-15)where the integral is over all values of the variableu. Comparison of Eq. (16.4-8) with
Eq. (16.4-15) shows that the probability of finding the particle betweenx′andx′+dx
is equal to(Probability thatx′<x<x′+dx)|Ψ(x′,t)|^2 dx (16.4-16)wherex′is some value ofx. This corresponds to(Probability density)|Ψ(x,t)|^2 (16.4-17)The probability density in this case is a probability per unit length on thexaxis. This
is an important result, which we generalize to three dimensions and to more than one
particle.The square of the magnitude of the wave function is the probability density for
finding the particle or particles. For the motion of a particle in three dimensions, the
probability that the particle lies betweenx′andx′+dxin thexdirection, betweeny′
andy′+dyin theydirection, and betweenz′andz′+dzin thezdirection is analogous
to that in Eq. (16.4-16)(Probability)|Ψ(x′,y′,z′,t)|^2 dxdydz (16.4-18)so that(Probability density)|Ψ(x,y,z,t)|^2 (16.4-19)This probability density is a probability per unit volume in three dimensions. For a sys-
tem of more than one particle moving in three dimensions, the probability density is a
probability per unit volume in a 6-dimensional space, a 9-dimensional space, and so forth.
For a stationary state the probability density is time-independent and is equal to the
square of the magnitude of the coordinate wave function. For motion in one dimension,|Ψ(x,t)|^2 ψ(x)∗eiEt/ ̄hψ(x)e−iEt/ ̄hψ∗(x)ψ(x)|ψ(x)|^2 (16.4-20)A version of Eq. (16.4-20) can be written for a wave function that depends on more(^0) 0.0 than one coordinate.
10
20
Energy or wave
function squared (arbitrary units)
0.2 0.4
x/a
0.6 0.8 1.0
Figure 16.2 The Probability Density
for Positions of a Particle in a One-
Dimensional Box.
Figure 16.2 shows the probability density for the first four energy eigenfunctions
of a particle in a box. Each graph is placed at a height proportional to the energy
eigenvalue corresponding to that wave function. These probability densities are very
different from the predictions of classical mechanics. If the state of a classical particle
in a box is known, the probability at a given time would be nonzero at only one point,
as in Figure 16.3a. Since the classical particle moves back and forth with constant