9.3 The Velocity Probability Distribution
397
Equation (9.3-14) is a differential equation. Solving it means finding a function f that
satisfies the equation. It can be solved by a different version of separation of variables,
in which we separate a dependent variable from an independent variable. We multiply
Eq. (9.3-14) by dv x and divide it by f (v x ):
1
f
df
dv x
dv x Cv x dv x
(9.3-15)
We recognize (df/dv x )dv x as df and write
1
f
df Cv x f (v x )
(9.3-16)
An indefinite integration of both sides of Eq. (9.3-16) gives
ln(f )
Cv 2
x
2
+ A
(9.3-17)
where A is a constant of integration. We take the exponential (antilogarithm) of each
side of this equation to obtain
f (v x ) e
A e
Cv 2
x /2
(9.3-18)
We require that f be normalized:
∞
−∞
f(v x )dv x
∞
−∞
e
A e
Cv 2
x /2 dv x 1
(9.3-19)
Because we are using nonrelativistic classical mechanics, speeds greater than the speed
of light are not excluded and we must take infinite limits for the integrations. The
constant C must be negative, because otherwise the integrand in Eq. (9.3-19) would
grow without bound for large magnitudes of v x and the integral would diverge (become
infinite). We let b −C so that b is positive. We eliminate the parameter A by the
normalization condition
1 e
A
∞
−∞
e
−bv 2
x /2 dv x e
A
2π
b
1/2
(9.3-20)
where we have looked up the definite integral in Appendix C. Equation (9.3-18) now
becomes
f (v x )
b
2π
1/2
e
−bv 2
x /2
(9.3-21)
The probability distribution for all three components becomes
g(v) f (v x ) f (v y ) f (v z )
b
2π
3/2
e
−bv 2
x /2 e
−bv 2
y /2 e
−bv 2
z /2
b
2π
3/2
e
−b(v 2
x + v 2
y + v 2
z )/2
b
2π
3/2
e
−bv 2 /2
(9.3-22)
Because v 2
x + v 2
y + v 2
z v 2 , the square of the speed, g depends only on the speed,
as we anticipated. It can also be written as a function of the kinetic energy of the
particle, ε:
g(v)
b
2π
3/2
e
−bε/m
(9.3-23)
397
Equation (9.3-14) is a differential equation. Solving it means finding a function f that
satisfies the equation. It can be solved by a different version of separation of variables,
in which we separate a dependent variable from an independent variable. We multiply
Eq. (9.3-14) by dv x and divide it by f (v x ):
1
f
df
dv x
dv x Cv x dv x
(9.3-15)
We recognize (df/dv x )dv x as df and write
1
f
df Cv x f (v x )
(9.3-16)
An indefinite integration of both sides of Eq. (9.3-16) gives
ln(f )
Cv 2
x
2
+ A
(9.3-17)
where A is a constant of integration. We take the exponential (antilogarithm) of each
side of this equation to obtain
f (v x ) e
A e
Cv 2
x /2
(9.3-18)
We require that f be normalized:
∞
−∞
f(v x )dv x
∞
−∞
e
A e
Cv 2
x /2 dv x 1
(9.3-19)
Because we are using nonrelativistic classical mechanics, speeds greater than the speed
of light are not excluded and we must take infinite limits for the integrations. The
constant C must be negative, because otherwise the integrand in Eq. (9.3-19) would
grow without bound for large magnitudes of v x and the integral would diverge (become
infinite). We let b −C so that b is positive. We eliminate the parameter A by the
normalization condition
1 e
A
∞
−∞
e
−bv 2
x /2 dv x e
A
2π
b
1/2
(9.3-20)
where we have looked up the definite integral in Appendix C. Equation (9.3-18) now
becomes
f (v x )
b
2π
1/2
e
−bv 2
x /2
(9.3-21)
The probability distribution for all three components becomes
g(v) f (v x ) f (v y ) f (v z )
b
2π
3/2
e
−bv 2
x /2 e
−bv 2
y /2 e
−bv 2
z /2
b
2π
3/2
e
−b(v 2
x + v 2
y + v 2
z )/2
b
2π
3/2
e
−bv 2 /2
(9.3-22)
Because v 2
x + v 2
y + v 2
z v 2 , the square of the speed, g depends only on the speed,
as we anticipated. It can also be written as a function of the kinetic energy of the
particle, ε:
g(v)
b
2π
3/2
e
−bε/m
(9.3-23)