Electron Wave Functions
In chapter 2 we explored the distribution of the “free electron” probability throughout a crystal. There is another quantum mechanical analysis of this problem that allows for a more intuitive approach for our use here. We saw in chapter 2 that the electron probability for all the free electrons was spread throughout the entire crystal. This is tantamount to admitting that we don’t know where the electron is in the electron’s “universe” of the crystal. In terms of the uncertainty principle (p*r=h), we don’t have any idea what r is, which means that p can take on specific values (i.e. almost infinite uncertainty of position allows almost infinite precision in momentum knowledge). This is a long-winded way to say that we can treat the electron as a “free particle”, with a discrete energy wave function.
[CAUTION: We are entering a mathematics zone; you can accept the precepts to follow, and skip the math or hang in there. We will try to keep it as intuitive as possible]
With the assumption of a free particle state for the electron, we need another boundary condition to define our problem. Remember that in chapter 2 the boundary condition on the electron was the containment in the low potential volume of the nuclear environment (the crystal). This worked well, but leaves us with an unsatisfactory approach to the localized linear momentum state electron. This seems unsatisfactory because all of the atoms in the interior of a crystal would be expected to behave pretty much the same, and have the same concentration of the various electron energies. Physicists have figured a way around this quandary in intuition, by using free electrons and imposing a boundary condition that just says: the wave function in each of the 3 (orthogonal) dimensions must be periodic with the sample size (~arbitrarily chosen). In mathematical terms ψ(x)= ψ(x+L) (same for y & z); where L is the sample size. Wikipedia (Fermi energy) gives a reasonable derivation of this concept. The bottom line is that the velocities of linear momentum state electrons in a metal can be pictured as a sphere of velocity vectors (Fermi Sphere: the length of the vector representing the velocity/momentum, the direction it points the actual physical direction in space, for the wave function), with each position in the sphere representing a velocity, or momentum. The probability of each electron velocity (momentum) at a given point in space is not specifically addressed with this model.
Constructing a Localized Electron Wave function From the Free Electron Wave Functions
This means that at every point in the crystal, every electron velocity in the “Fermi sphere” is represented in a wave function. The “free electron” wave functions radiate from this single point (again ~arbitrarily chosen). This is exactly the same thing we said in chapter 2, namely; each electron is represented everywhere in the crystal (i.e. each point is crossed by every wave function). In chapter 2 we represented the wave function in cartesian coordinates, here the functions are represented in radial coordinates. The difference is an intuitive one. With the Fermi sphere represented at every physical point in the crystal, we can now intuitively see how an electron transitions from the Fermi sea of electrons, to a localized electron in linear momentum states. If we focus on the point at the center of mass of the colliding helium-deuterium pair, we can start to picture the transition. First we express all of the free electron wave functions originating from a point as as:
Ψk(r,θ,φ) ;
For each radial direction (set of θ & φ) there are linear momentum quantum numbers kx, ky, kz (=0; +2 π/L ; +4 π/L ;……) and k2 = kx2+ ky2 +kz2 (directly analogous to the way we found the velocity vector in chapter 2).
These are analogous to the “n” quantum numbers for the linear momentum states in chapter 2. For each radial direction of Ψk(r,θ,φ) (set of θ & φ), there is an infinite set of wave functions with the proper combinations of kx, ky, kz =>k. Now, if we want to construct a localized electron wave function, we can use some of the Ψk(r,θ,φ) wave functions to construct any wave form that we want using Fourier transformation.