Aage Bohr (Aage is pronounce approximately like the English word owe with a little eh at the end), the son of Niels Bohr, worked with Ben R. Mottelson, a Danish-American physicist, to relate collective properties of nuclei to the motion of their constituent nucleons. James Rainwater also worked on this topic and the three; Bohr, Mottelson and Rainwater; received the Nobel Prize in Physics in 1975.

At the time the three started working there was a fixation with deriving everything from the Schrödinger Equation. Instead Bohr and Mottelson wanted their analysis tied to empirical fact. They asserted

Progress in this direction has been achieved by a combination of approaches based partly on clues provided by experimental data, partly on the theoretical study of model systems, and partly on the exploration of general relations following from considerations of symmetry.

They planned for a three volume presentation but that devolved to two volumes; Volume I covering single-particle motion and and Volume II nuclear deformations.

The authors state in Volume I that the nuclei are relatively weakly bound systems.

The energy required to remove a nucleon from a nucleus is about 5-10 MeV, and the average kinetic energy of the nucleons in a nucleus is of the order of 25 MeV. These energies are small compared with the rest energies of the nucleons themselves (Mc² ≅ 1000 MeV), but also of the lightest of the hadrons, the π mesons ((m_πc² ≅ 137 MeV). In the analysis of nuclear bound states and reactions at not too high energies, it therefore a good approximation to regard the nucleus as composed of a definite number of nucleons with properties similar to those of free nucleons, and moving with nonrelativistic velocities (v²/c²≤0.1).

Aage Bohr, in a lecture given at the Danish Society for the Dissemination of Natural Science, gave the following sketch of the Collective Model of Nuclear Structure:

Though the investigation of different properties of nuclei, one thus is led to two widely different pictures of nuclear structure corresponding to the liquid-drop model and to the shell model. It is, nevertheless, not difficult to combine the most essential features of both models.

[…]

In the shell model one regards the orbital motion of the individual nucleons in the average nuclear field as the primary form of motion; in the liquid-drop model one looks at the simple collective motions of the nucleons. Both of these forms of motion are observed experimentally in nuclei, so one is led to seek a more general description of nuclear structure by considering the total state of motion as a superposition of the two previously mentioned basic components.

Such a description may be regarded as a generalization of the shell model in which the nuclear field, previously considered constant, is now to be considered to be a dynamic variable; the variation of the collective nuclear field is linked with the vibrating shape of the nucleus.

He goes on in that lecture to describe the best method to elucidate nuclear structure is the scattering of charged particles, protons or alpha partices, off of target nuclei. He explains how this works in terms of the collective model by way of an analogy.

As an analogy we can imagine a comet coming very close to our earth. […] A principal effect of the passage of the comet would be that the rotation of the earth would be slightly changed, partly by the oblate shape of the earth and partly because of the strong oceanic tidal wave that would be set in motion.

In his lecture Aage Bohr gave only one equation. It was for the discrete level of rotational energy of a nucleus. It is

E_rot = (|p|²/(2J)) = h²I(I+1)/(2J))

where p is the angular momentum, J is the moment of inertia and h is Planck's constant divided by 2π. The symbol I stands for the quantum number of the angular momentum. For a nucleus with an even number of nucleons it is an integer and for a nucleus with an odd number of nucleons it is an integer plus a half unit.

The derivation of this equation is of some interest. For an object with a moment of inertia of J spinning at an angular rate of ω the angular momentum p is Jω and the rotational energy E_rot is ½Jω². Thus E_rot in terms of the angular momentum p is p²/(2J). Therefore if there are no outside forces or fields the Hamiltonian of the system is

H = p²/(2J)

But before going further here is what the Old Quantum Physics of Niels Bohr would have to say about the system. According to that theory the angular momentum is quantized as

Jω = Ih

where I is an integer. It is important to note that ω is in units of radians per second and the constant is h-bar, h rather than h, per se. Therefore the rotational energy is given by

E_rot = p²/(2J) = (Ih)²/(2J) = h²I²/(2J)

This essentially the same as the equation given in the Bohr-Mottelson paper except that the rotational energy is proportional to I² rather than I(I+1).

The quantum mechanical analysis proceeds from the Hamiltonian. The Hamiltonian function is converted into the Hamiltonian operator H by replacing p by −ih∇. Thus

H = −h(1/2J)∇²

The time-independent Schrödinger equation for the spinning object is then

−(h/2J)∇²ψ = E_rotψ

where ψ is the wave function for the system.

Using the technique of separation of variables two differential equations are derived that have solutions only if the quantum number is I(I+1) where I is an integer.

(To be continued.)