Figure 42.8 showed the potential energy of two interacting nucleons. To solve Schrödinger’s equation for the nucleus, we would need to know the total potential energy of all interacting nucleon pairs within the nucleus, including both the strong force and the electrostatic force. This is far too complex to be a tractable problem.

We faced a similar situation with multielectron atoms. Calculating an atom’s exact potential energy is exceedingly complicated. To simplify the problem, we made a model of the atom in which each electron moves independently with an average potential energy due to the nucleus and all other electrons. That model, although not perfect, correctly predicted electron shells and explained the periodic table of the elements.

The shell model of the nucleus, using multielectron atoms as an analogy, was proposed in 1949 by Maria Goeppert-Mayer. The shell model considers each nucleon to move independently with an average potential energy due to the strong force of all the other nucleons. For the protons, we also have to include the electrostatic potential energy due to the other protons.

FIGURE 42.9 shows the average potential energy of a neutron and a proton. Here $r$ is the distance from the center of the nucleus, not the nucleon–nucleon distance as it was in Figure 42.8. On average, a nucleon’s interactions with neighboring nucleons are independent of the nucleon’s position inside the nucleus; hence the constant potential energy inside the nucleus. You can see that, to a good approximation, a nucleon appears to be a particle in a finite potential well, a quantum-mechanics problem you studied in Chapter 40.

FIGURE 42.9 The average potential energy of a neutron and a proton.

Three observations are worthwhile:

1. The depth of the neutron’s potential-energy well is $\approx 50\text{MeV}$ for all nuclei. The radius of the potential-energy well is the nuclear radius $R=r_0{A}^{1\text{/}3}$.

2. For protons, the positive electrostatic potential energy “lifts” the potential-energy well. The lift varies from essentially none for very light elements to a significant fraction of the well depth for very heavy elements. The potential energy shown in the figure would be appropriate for a nucleus with $Z\approx 30$.

3. Outside the nucleus, where the strong force has vanished, a proton’s potential energy is $U=(Z-1)e^2\text{/}4\pi {ϵ}_{0}r$ due to its electrostatic interaction with the $(Z-1)$ other protons within the nucleus. This positive potential energy decreases slowly with increasing distance.

The task of quantum mechanics is to solve for the energy levels and wave functions of the nucleons in these potential-energy wells. Once the energy levels are found, we build up the nuclear state, just as we did with atoms, by placing all the nucleons in the lowest energy levels consistent with the Pauli principle. The Pauli principle affects nucleons, just as it did electrons, because they are $\text{spin}\text{-}\frac{1}{2}$ particles. Each energy level can hold only a certain number of spin-up particles and spin-down particles, depending on the quantum numbers. Additional nucleons have to go into higher energy levels.