The purpose of this webpage is to show how the quantization of angular momentum in two-body rotating system leads generally to the quantization of rotational kinetic energy. In the case of the linear momentum p and kinetic energy E of a body it is simple to express their relationship as

E = p²/m

where m is the mass of the body. In the case of angular momentum and rotational kinetic energy the analysis can be carried one step further because angular momentum is quantized.

Consider two bodies of masses m and M with their centers separated by a distance s. The system rotates about the center of mass and the distances from that center of mass are given by:

mr_m = Mr_M
and hence
r_m/r_M = M/m

The separation distance s is given by

s = r_m + r_M = r_M[r_m/r_M + 1]
or, equivalently
s = r_M[M/m + 1] = r_MM[1/m + 1/M]

The expression [1/m+ 1/M] is the reciprocal of the reduced mass μ of the two bodies. Thus

s = r_MM/μ
or, equivalently
μs = Mr_M

Thus, very neatly

r_MM = r_mm = sμ

Angular Momentum

If the system is rotating at a rate ω then its angular momentum L is given by

L = mωr_m² + Mωr_M² = mr_mωr_m + Mr_Mωr_M
and, since Mr_M=mr_m
L = mωr_m[r_m+r_M]
and hence
L = mωr_ms

But mr_m is equal to μs so

L = ωμs²

The angular momentum is quantized; i.e.,

L = nh
and thus
ωμs² = nh

where n is a positive integer (known as the principal quantum number) and h is Planck's constant divided by 2π.

This means that

ω = nh/(μs²)

Kinetic Energy

The rotational kinetic energies of the two bodies are

K_m = ½mω²r_m²
and
K_M = ½Mω²r_M²

These can be expressed as

K_m = ½(mωr_m)²/m
and
K_M = ½(Mωr_M)²/M

Since both mωr_m and Mωr_M are equal to ωμs the total kinetic energy K is given by

K = ½(ωμs)²/m + ½ (ωμs)²/M
= ½(ωμs)²(1/m + 1/M)
= ½(ωμs)²/μ
and hence
K = ½ω²μs²

Since the angular momentum L is equal to μωs² and it is quantized as nh

K = ½Lω = ½nhω

But it was previously found that ω is equal to nh/(μs²) so

K = (nh)²/(2μs²)

This formula can be examined for particular cases. Consider first the case of the deuteron. Twice the reduced mass for the neutron and proton in a deuteron is 1.67374921×10^-27 kilograms. The separation distance of the centers of the nucleons in a deuteron is 2.252×10^-15 meters. Planck's constant divided by 2π in the MKS system is 1.054571×10^-34 and squared is 1.112122×10^-68. Thus for n=1

K = 1.112122×10^-68/(1.67374921×10^-27*(5.071504×10^-30))
= 1.31016295×10^-12 joules
= 8.177390 million electron volts (MeV).

When a deuteron is formed there is an emission of a gamma ray of energy 2.224573 MeV. This means that when a deuteron is formed there is a loss of 10.401963 MeV, 8.17739 MeV of which goes into its rotational kinetic energy and 2.224573 MeV of which goes into the emission of a gamma photon.

The hydrogen atom is another interesting case. The mass of the proton is about 1836 times greater than the mass of the electron. Therefore the reduced mass of the system is essentially the same as the mass of the electron, which is 9.109383×10^-31 kg. The radius of the first Bohr orbit is 5.3×10^-11 meters. Thus

K = 1.112122×10^-68/(2*9.109383×10^-31*28.09×10^-22)
K = 2.1731×10^-18 joules
K = 13.56 electron volts (eV)

This is essentially the Rydberg constant (13.59 eV) and verifies the validity of the formula

K = (nh)²/(2μs²)

(To be continued.)