Normalizing Ψ and Radial Probabilities
It is instructive to look at the interpretation of the probability functions of Ψ=AB and the observational consequences. The normalization of Ψ is automatic with the normalization of “A” and “B”. The normalization of “A” is standard, as shown in equation 3-11:
The two angular wave functions are normalized so that all 3 spatial integrals will be equal to one. The radial probability graph, 3-7, would be the result of an experiment to measure the probability of finding the particles at a given separation, ro, regardless of orientation.
The normalization of “B” is just the one-dimensional integral over all x, which is aligned with ro, as in equation 3-12. This wave function has no inherent volume component to it. This is the same as the radial wave function for hydrogen, which is one dimensional. In fact, all the wave functions have orthogonal components that are in some sense independent. The wave function, “B”, becomes space filling only in combination with “A”. In a sense “/A/2” plays the role of “ci”, the coefficient we used in chapter 2, when adding states together to create another localized state. To be more precise, ci would contain a volume element giving ci=/A/2dV. In this case, adding all the ci's is simply the same normalization integral as equation 3-11. This is the reason the localized linear momentum state leads to such a high energy bond. If you do an experiment to discover the electron probability as a function of x, regardless of orientation (same as the nuclei in figure 3-7), it will give the same distribution as shown in figures 3-5, with the maximum value centered around the helium nucleus. Remember that the electron ground state wave function does not change much with ro, and is normalized for each value of ro. The electron, helium and deuterium nuclei probabilities, as a function of r, are shown in figure 3-9. This extreme localization of the electron supplies the energy to overcome the coulomb potential.
Figure 3-ll shows the local probability of each of the three particles. This would be the probability found with an experiment to measure the distribution of the respective particles. The difference in how the probabilities, in Figures 3-9 and 3-11, are displayed is shown in the cartoon of the linear momentum states in figure 3-10. Basically all that is done is the values in Figure 3-9 are divided by the radius squared, r2. Figure 3-9 depicts the relative probability of finding the particles at a given radius anywhere around the center of mass. Figure 3-11 shows the local concentration of a given particle that would be counted in an experiment to locate that particle. Figure 3-11 shows how extreme the local probability concentration is at the center of mass. This happens because all of the Linear Momentum States cross at the center of mass. It is almost as if the electron has been placed between the two nuclei!
