Physical Chemistry , 1st ed.

In considering the total energy of this harmonic oscillation, the potential
energy is the same as for any other harmonic oscillator but the kinetic energy
is the sum of the kinetic energies of the two particles. That is,

K^12 m 1 x^21 ^12 m 2 x^22

Using equations 11.21 and 11.22, it is easy to substitute and show that the ki-
netic energy has a simple form in terms of the time derivative of the relative
coordinate q:

K

1

2



m

m

1 

1 m

m

2

2

q^2 (11.23)

The reduced massis defined as

 

m

m

1 

1 m

m

2

2

 (11.24)

so that the total kinetic energy is simply

K ^12 q^2 (11.25)

which is a simpler expression for the kinetic energy. The reduced mass can also
be determined using the expression





1



m

1

1



m

1

2

 (11.26)

What this means is that the kinetic energy of the oscillator can be represented
by the kinetic energy ofa single mass moving back and forth,if that single mass
has the reduced mass of the two masses in the original system. This allows us
to treat the two-particle harmonic oscillator as a one-particle harmonic oscil-
lator and use the same equations and expressions that we derived for a simple
harmonic oscillator. So all of the equations of the previous sections apply, as-
suming one uses the reducedmass of the system. For example, equation 11.3
becomes



1



2

1







k

 (11.27)

The Schrödinger equation, in terms of the reduced mass, is

 2





2



d

d

x

2

 2 Vˆ(x)E (11.28)

Fortunately, our derivations need not be repeated because we can simply sub-
stitute for min any affected expression. The unit of reduced mass is mass,
as is easily shown.

Example 11.8

Show that reduced mass has units of mass.

Solution

Substituting just units into equation 11.24, we get



kg

kg



kg

kg



k

k

g

g

2

kg

which confirms that the reduced mass has units of mass.

11.5 The Reduced Mass 331