Free Electrons and Linear Momentum States
Before moving on, the appropriate form of the energy operator, (Del)2, must be chosen for equations 3-6 & 3-7 (from the previous section). This is a critical factor to understanding Fleischmann Pons fusion, and so will be dwelled on to some extent. For equation 3-7, the Schrodinger equation is a familiar central force problem, much like the hydrogen atom. Thus, the polar coordinate representation will be used to solve for A(ro) and E (just as with the hydrogen atom). The quantum solutions for the two angular coordinates, φ and θ, will be identical to the hydrogen atom solutions, and not reproduced here. We will let A(ro,θ,φ)=R(ro) Θ(θ) Φ(φ), as assumed in the case of hydrogen.
Before choosing a form for the energy operator, (del)2 in equation 3-6, we should digress into a discussion of free electrons in metals. Understanding the concept of free electron linear momentum states is essential to developing the high energy bonding mechanism. A detailed discussion of the origins of the free electrons can be found in any introductory solid-state physics texts.
For locally bound electrons in angular momentum states, there are 4 quantum numbers describing the state;
1) n =1, 2, 3,… principle,
2) l =1,2,…n-1 orbital angular momentum,
3) m =l, -l+1…0…l-1, l magnetic
4) spin. +1/2 or -1/2
The first quantum number arises naturally from the solution to the radial wave function. The second two quantum numbers come from the solutions to wave functions for the two angular coordinates, θ and φ (i.e. angular momentum). The 4th quantum number is the electron spin.
Free electrons, lacking the component of angular momentum, have the 4 quantum numbers of the 3 dimensions, plus the electron spin;
1) nx =1,2,3…. X-coordinate
2) ny =1,2,3…. Y-coordinate
3) nz =1,2,3…. Z-coordinate
4) spin. +1/2 or -1/2
In other words, the local forces from each nucleus, on the free electrons, are not strong enough to pull the electrons into an orbit matching the wavelength of the electrons. The only restricting boundary on the electron is the outer edge of the crystal lattice (not the potential well formed by a spherically symetrical nuclear charge). Hence, the boundary conditions for solving the free electron Schrodinger equation, are the three dimensional boundaries of the crystal. This is the same as the wave functions that spanned the little 20 angstrom “crystal" in chapter 2. In practice, most physisits use a boundary condition accounting for the repeating crystal structure, since calculations accomodating the huge number of atoms in a real crystal is impossible.
Although it is dangerous to use intuition with quantum mechanics, it is instructive to think of a wave function as representing all the various paths that a particle may take in a confining potential, at a given total energy, and summing the results of those paths as concentrations. (This of course is not what is happening, as it belies the wave nature of quantum mechanics) If we picture the analogy of the moon, we see that an isolated earth and moon results in an orbiting moon as in figure 3-1, analogous to a hydrogen atom.
If we form a 2 dimensional array (a crystal) of earths, with the distance between the earths less than 2 * r, we get the situation in figure 3-2.There can be no orbit of an earth in this array, as an adjacent earth pulls the moon away, before it can be pulled into an orbit, and hence there can be no angular momentum. This same situation arises when the partially filled orbitals of a metal combine to form a continuous path for an electron. The important thing to note is that there can be no states (orbits) around the atoms, only straight paths through, or by them. The forces involved, just will not sustain orbits. In quantum mechanic terms, all of the free electron orbiting states will interfere and cancel themselves. The straight path of the free electrons are the only stable harmonics. These paths are called “linear momentum states”, as apposed to the angular momentum states corresponding to orbiting electrons. The fact that there are no orbiting states available, for the free electrons in the metal crystal, is critical to the high energy bonding electron