Before the discovery of the neutron, one theory of the nucleus proposed that the nucleus contains protons and electrons. For example, the helium-4 nucleus would contain 4 protons and 2 electrons instead of—as we now know to be true—2 protons and 2 neutrons. 

(a) Assuming that the electron moves at nonrelativistic speeds, find the ground-state energy in mega-electron-volts of an electron confined to a one-dimensional box of length 5.0 fm (the approximate diameter of the 4He nucleus). (The electron actually does move at relativistic speeds. See Problem 77.) 

(b) What can you conclude about the electron-proton model of the nucleus? The binding energy of the 4He nucleus—the energy that would have to be supplied to break the nucleus into its constituent particles—is about 28 MeV. 

(c) Repeat (a) for a neutron confined to the nucleus (instead of an electron). Compare your result with (a) and comment on the viability of the proton-neutron theory relative to the electron-proton theory.