The localized electron and the linear momentum state
Suppose that, the helium kinetic energy has brought the helium and deuterium nuclei close together, say 0.2 angstroms. For a moment (neglecting the 2 bound angular momentum state (1s) electrons on the helium) there will be a potential well formed by the positive charges that looks like figure 3-3. This is very similar to the potential seen in the interstitial atom model (chapter 2), with the two atoms without core electrons.
The dotted lines suggest a one-dimensional potential well. Basically, a two atom metal crystal. This is perfect for a linear momentum state electron to enter, except that it violates the Heisenberg uncertainty principle. The uncertainty principle is violated because the electron state is one-dimensional. To be a viable state, it must have values for all space and be consistent with the uncertainty principle.
Now remember that we are trying to find the wave function Ψ, not just “A” or “B”. The wave function Ψ is composed of the multiplication of the two wave functions “A” and ”B” [AB= A(ro,θ,φ )B(x)=R(ro) Θ(θ) Φ(φ)B(x))= R Θ Φ B]. We now make the assertion that “B” is correlated with “A”. This just means that “B” is always aligned through ro, the line (vector) drawn through the two nuclei, with orientation represented by θ and φ. Any other localized linear momentum state will not have a stable harmonic, and thus will not exist. This is not an unprecedented assumption. When calculations are made with Density Function Theory, it is tacitly assumed that the electrons are correlated to the position of the nuclei. This is the essence of the Born-Oppenheimer approximation. All nuclei will have a 3-dimensional distribution of positions, just as in this theory. To find the actual electron distribution using Density Function Theory, the electron distributions of all the infinite nuclear positions would be summed; each adjusted for the probability of the particular nuclear position. That is no different from what is done here. The only difference is that the linear momentum state has availabe (existence) only a one-dimensional state for each position of the nuclei.
The BCS theory of superconductivity assumes the correlation between 2 electrons and a phonon, and there is some correlation between the two electrons in helium. The correlations occur because it lowers the energy of the system of particles. In the case of superconductivity, the energy is lowered by a fraction of an electron volt, as a pair electrons spend more time a little deeper in the nuclear potential well, caused by a phonon. In helium, there is a tendency for the electrons to repel each other, decreasing the energy slightly. Each of these actions decreases entropy, but the overall energy of the system of particles is lowered more than the loss in entropy. Entropy decreases because some of the particles spatial combinations are eliminated.
In the case of superconductivity, the correlation lowers the energy by a fraction of an electron volt. The localized linear momentum state, as we will see later, reduces the energy of the system by tens of thousands of electron volts. The kinetic energy thus gained, is spread between the 3 particles (electron, alpha & deuteron) and the repulsion of the nuclei. The work done giving kinetic energy to the alpha and deuteron particles and overcoming the repulsive force, plays the same role as the emitting of a photon in the hydrogen atom, allowing the electron to fall to lower states. The electron can only fall as far into the potential well as it gives up the same amount of energy as work or photons.
We will see that the ground state wave function of “A” is analogous to the electron wave function in atomic hydrogen. The average radius, compared to hydrogen is orders of magnitude smaller, but the spherically symmetrical ground state probability distribution is similar. The vector ro, can take on the values of zero to infinity and point in all directions (φ=0-2pi; θ=0-pi). Since “B” has a presence everywhere that the vector ro points, “B” is also a spherically symmetrical distribution. Thus, there is no uncertainty principle contradiction. The electron wave function is the sum of an infinite number of wave functions; one for each of the infinite number of spatial orientations and magnitudes of ro.
In the case of a hydrogen atom, the electron had an infinite number of orbits around the proton; some circular, even more ellipses. In an intuitive sense, they all add up to give the hydrogen probability function. In the case of the localized linear momentum state, there is only one stable harmonic, straight through the nuclei. There are, however, an infinite number of nuclei locations in the He-d molecule. The electron spatial wave function is properly represented by the total wave function Ψ=AB. Fortunately, we only have to calculate “B” as a function of ro, since “B” is correlated with “A”. The solutions to “B(ro)” will also give us E*(ro), needed to calculate “A” and E.