Nuclei Wave Functions
figure 4-2
Now we must consider the situation from a quantum mechanic point of view. The deuterium is held in the tetrahedral site (figure 4-2), and thus has some wave function describing the nuclear position. This has been studied in the literature using Density Function Theory and the zero point (or ~n=1) energy estimated to be 0.18 eV. If we assume a Hookien force constant (= [constant]x[displacement from equilibrium position] see Figure 4-3a), the force constant can be calculated using the given” zero point energy” for hydrogen. The force constant can be calculated from the Schrodinger equation by using the zero point energy as a constant and the force constant a variable. As a reality check, Figure 4-3b compares the “back calculated” force constant with what might be intuitively expected for hydrogen. Using the hydrogen force constant and double the mass (for deuterium), the wave function for the deuterium nuclei can be calculated.using the same wave equation. Figure 4-3 Shows the radial probability distribution expressed as the local probability (Ψ2) and as a function of radius ([rΨ]2, as we discussed in the last chapter).
figure 4-3c
figure 4-3a
When the ~200-500 eV helium atom becomes channeled, it is a “free particle” in the direction of travel and will have a wave length consistent with its De Broglie wave length (determined by its momentum). Transverse (orthogonal) to the direction of travel, the helium will have a wave function determined by the lateral forces depicted in Figure 4-3c, much like the deuterium above, except in only two radial dimensions. The Schrodinger equation can be used to calculate the wave function orthogonal to the direction of travel. As a potential, we will use twice the force constant of hydrogen, as the helium has a plus two charge. As seen in figure 4-3b, this is a reasonable estimate
in the absence of a more rigorous Density Function calculation. The form of the energy operator in the Schrodinger equation (eq 4-1) must be changed a little as in equation 4-2 (cylindrical coordinates) to accommodate confinement in just two dimensions. We will again, ignore the angular momentum component of the Schrodinger equation.
Figure 4-4 shows the probability distributions for the orthogonal dimensions relative to the channeled helium velocity vector (the side to side part of the wave function). The “zero point” energy (n=1) for the helium nucleus (transverse
figure 4-3
figure 4-4direction relative to travel), is calculated to be 0.084eV. This is the critical piece of information needed to calculate the allowed states of the “He-d” molecule. After discussing the electron transition to the localized linear momentum state, we will address the significance of this “zero point” energy.