Physical Chemistry , 1st ed.

11.5 The Reduced Mass

Many harmonic oscillators are not simply a single mass moving back and

forth, like a pendulum or an atom attached to a massive, unmoving wall. Many

are like diatomic molecules, with two atoms each moving back and forth to-

gether as in Figure 11.6. But to describe such a system as a harmonic oscilla-

tor, the mass of the oscillator isn’t the sum of the two masses of the atoms.

Such a system needs to be defined a little differently.

We will assume that the two masses m 1 and m 2 in Figure 11.6 have positions

labeled as x 1 and x 2 but are moving back and forth as a harmonic oscillator.

We will ignore any other motion of these two masses (like translation or rota-

tion) and focus solely on the oscillation. In a purely harmonic oscillation (also

called a vibration), the center of mass* does not change, so that

m 1 d

d

x

t

^1 m

2 

d

d

x

t

^2

The negative sign indicates that the masses are moving in the opposite direc-

tions. By adding the mixed term m 2 (dx 1 /dt) to both sides, we get

m 1 d

d

x

t

^1 m

2 

d

d

x

t

^1 m

2 

d

d

x

t

^2 m

2 

d

d

x

t

^1

(m 1 m 2 )

d

d

x

t

^1 m

(^2) 
d
d
x
t
^1 d
d
x
t
^2

(where on the right side we have switched the order of the derivatives). This is
rearranged to

d
d
x
t
^1 
m 1
m

2
m 2

d
d
x
t
^1 d
d
x
t
^2
 (11.20)
It is very convenient in many cases to define relativecoordinates instead of ab-
solute coordinates. For example, specifying certain values of Cartesian coordi-
nates is a way of using absolute coordinates. However,differencesin Cartesian
coordinates are relative, because the difference doesn’t depend on the starting
and ending values (for example, the difference between 5 and 10 is the same as
the difference between 125 and 130). If we define the relative coordinate qas
q x 1 x 2
and thus

d
d
q
t

 

d

d

x

t

^1 d

d

x

t

^2

Now we can substitute into equation 11.20 to get

x 1 

d

d

x

t

^1 

m 1

m



2

m 2



d

d

q

t

 (11.21)

where we use x 1 to indicate the time derivative ofx. By performing a parallel

addition ofm 1 dx 2 /dtto the original center-of-mass expression, we can also get

x 2 d

d

x

t

^2 

m 1

m



1

m 2

d

d

q

t

 (11.22)

as a second expression.

330 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom

*Recall that the center of mass (xcm,ycm,zcm) of a multiparticle system is defined as

xcm(mi xi)/(mi), where each sum is over the iparticles in the system,miis the parti-

cle’s mass, and xiis the particle’s xcoordinate; and similar expressions apply for ycmand zcm.

(CoM)

m 1 m 2

x 1 x 2
Figure 11.6 Two masses,m 1 and m 2 ,are mov-
ing back and forth with respect to each other with
the center of mass (CoM) unmoving. This cir-
cumstance is used to define the reduced mass .