An Intuitive Argument For the localized Linear momentum state
There is no simple equation to estimate the linear momentum state energy for close-coupled nuclei. An intuitive way to look at the problem is to compare a proton (+1) potential with a +10 potential as in figure 2-19.
Figure 2-19
The orbital of the electrons slide down the potential well until their wavelength fits exactly around the nuclei. The +10 nuclei will have an orbital radius one tenth of the proton. If a pair of protons is placed on the +10 potential as in figure 2-20, the potential of the protons mimics the potential of the +10 potential in one dimension. Since these are linear momentum states, one dimension is sufficient. Figure 2-20 also shows how the width of the electron probability will decrease, localizing the electron using linear momentum states.
Figure 2-20
Using a simple but more accurate numerical method much, like in the solid state section, we can calculate the n=1, energy level as a function of proton separation. The electron energy and the repulsive energy as a function of proton separation are shown in figure 2-21. The sum of the electron energy and the repulsive proton energy is also shown, with an energy minimum of ~8,200eV at about 0.007-angstrom separation. This would form the pseudo potential to calculate the proton probability function using the Schrodinger equation. A few of the probability functions from the proton-proton model are shown in figure 2-22. We will do more accurate calculations with the Helium-deuterium system in chapter 3. First, we must visit the 3-dimensional hydrogen wave function, setting us up to solve the 3-dimensional wave function for the linear momentum state electron, plus the deuteron and alpha particle wave functions needed to calculate fusion rates.
Figure 2-21
Figure 2-22
