Physical Chemistry Third Edition

706 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

In the predictable case all outcomes of repeated measurements will be equal to each

other and to the mean value. The standard deviation will equal zero. In the statistical

case the outcomes will vary and the standard deviation will be nonzero.

EXAMPLE16.16

Findσx, the standard deviation of the position of a particle in a one-dimensional box of length

afor then1 state.

Solution

〈x^2 〉

2

a

∫a

0

sin

(

πx

a

)

x^2 sin

(

πx

a

)

dx



2

a

(

a

π

) 3 ∫π

0

y^2 sin(y)dya^2

[

1

3

−

1

2 π^2

]

 0. 282673 a^2

From Example 16.12 we have〈x〉a/2, so that

σx

[

0. 282673 a^2 −(a/2)^2

] 1 / 2

 0. 180756 a

The fact that the standard deviation is nonzero shows that the statistical case applies. Inspec-

tion of the probability distribution also reveals this fact, since it is nonzero at more than

one point.

EXAMPLE16.17

Calculate the probability that a particle in a one-dimensional box of lengthawill be found

within one standard deviation of its mean position if the wave function is then1 energy

eigenfunction.

Solution

The probability is given by integrating the probability density from〈x〉−σxto〈x〉+σx:

(Probability)

2

a

∫ 0. 6807566 a

0. 319244 a

sin^2

(

πx

a

)

dx

2

π

∫ 2. 13866

1. 00293

sin^2 (y)dy



2

π

[

y

2

−

sin(2y)

4

]∣∣

∣∣

2. 13866

1. 00293

 0. 65017

For most probability distributions, approximately two-thirds of the probability lies within

one standard deviation of the mean.

The probability distribution for the harmonic oscillator in thev0 state is an

example of aGaussian distribution, which is defined by

f(u)

1

√

2 πσ

e−(u−μ)

(^2) / 2 σ 2
(Gaussian distribution) (16.4-28)
whereμis the mean value and whereσis the standard deviation. The Gaussian distri-
bution is also called thenormal distributionand if its standard deviation equals unity,
it is called thestandard normal distribution.
The Gaussian distribution is name for
Carl Friedrich Gauss, 1777–1855,
a great German mathematician who
made many contributions to
mathematics.