Solid State Electrons- “free electrons”
A Small One Dimensional Universe
We start with the simplest quantum model to show how discreet energy levels arise from the solutions to the Schrodinger equation. These energy levels are the quanta in “Quantum Mechanics”.
Our universe, studied here, is 20 angstroms long, with only one dimension. To start with there will be one particle, an electron. The solution to the Schrodinger equation will give two pieces of information;
1) the allowable energy levels of the system
2) the wave function
The square of the wave function gives the probability of finding a particle at a given position.
The behavior of the particles in the 20 angstrom universe (the real universe too) is described by their wave function, commonly referred to as ψ (psy). The wave function of a particle has a value everywhere (although it may be vanishingly small in most places). The square of the wave function, ψ2, represents the probability of finding a particle at a given position. If we add up the probability of finding a particle in all space, it will be one. This just means, if it exists, the particle will be found somewhere. In the 20 angstrom universe we should be able to calculate the wave function of a particle (say an electron or proton) for all position of x, from 0 to 20 angstroms. We can then square the wave function, ψ2, to get the probability of finding the particle at any region in the 20 angstrom world.
We will see that the position probability of a particle has strange behaviors at this small scale. First, we must introduce the one dimensional Schrodinger equation.
One Dimensional Schrodinger Equation (time independent)
The first term can be thought of as the kinetic energy, the second term the potential energy. The right hand side of the equation is the sum of the kinetic and potential energy, the total energy, E, a constant. The particle energy, E and wave function ψ, are found by solving the Schrodinger equation.
We will start by putting one electron in our 20 angstrom world. There are an infinite set of wave lengths that can be contained in this small world, but to keep things manageable, we will concern ourselves with only the first 10. A graph of the wave functions for n=1 to n=10 is shown in figure 2-4.
Figure 2-4
The “n” is known as the linear momentum state quantum number. There is a quantum number for each of the three dimension, nx, ny, nz. Each wave function is depicted at the appropriate energy level in figure 2-4. Figure 2-5 shows the probability functions,ψ2, for the first 3 energy levels. This simple example is a crude representation of the free electrons in metals. Each of these states, represented by the wave functions, can hold 2 electrons of opposite spins. It is instructive to add the probability densities of all the wave functions together. The graphical result of adding all 10 of the probability distributions together is shown in Figure 2-6. Note that the graph is trending toward constant electron concentration throughout the one dimensional space, as would be expected in a real metal crystal.
Figure 2-5
Figur 2-6
One of the principles of quantum mechanics is that a particle may be in more than one state, or energy level, at any given time. In the case of the particle being in more than one state, the wave functions are added together, before being squared, to calculate the probability distribution. It is the ability of the wave functions with negative values to cancel positive values of another state, that lead to interference patterns and other strange behavior. Each state has its own probability of being found, represented by the coefficient “ci”. A particle in our little 20 angstrom world could then have a wave function represented by a summation of the square root of the coefficient, times the wave functions ci0.5 Yi(i=1 to 10). This works since Yi is already normalized, then the sum of the coeficients will be 1.
Figure 2-7
If each wave function, or state of the particle, is allowed to be 10% (ci=0.1 ;i=1 to 10) of the total wave function, we get the probability function shown in figure2- 7. Note that the electron is now localized, with the width of probability held to about a 2 angstrom region. This is the origin of the Heisenberg uncertainty principle. By giving the particle a range of energies, or momentum, we have localized it. With only one value of the momentum, the particle was spread over the entire 20 angstrom space, regardless of the energy level. By increasing the uncertainty of the energy (n=1 to 10), the electron was localized. It is important to note that the electron can be in more than one state at a time.
We have seen that, given a restricted space, we can define the electron probability distribution and energy. We have also seen that by filling all the energy levels with electrons we can give a constant concentration of electrons, as in a metal. With the fractional addition of states, we saw how an electron is localized by giving it a range of energy states.
Justification for studying one-dimensional world
There needs to be some justification for using a one-dimensional model of the universe. To start with lets introduce a 2-dimensional world. The results are just as applicable to 3-dimensions. For this exercise our 2-dimensional universe will be 20 angstroms square, just to keep things easy. So the wave function will have an x and a y component, say ψ = ψx * ψy . The wave functions that will satisfy the Schrodinger equation will be the same as shown for the 1-dimensional system, but there will be a wave component in both the “x” and “y” directions. Thus, there will be 2 “quantum numbers”, nx, and ny . Both quantum numbers can take on integer values from 1, 2 , 3…..infinity. Remember the quantum number refers to how many humps the curves have, or in other words, (n-1) = number of times the wave function crosses zero. At first glance it may seem like the electrons can only travel in the x or y direction, which intuitively doesn't’t seem right. But remember that there are now 2 directional components to the wave function. Each solution to the wave function represented by the nx and ny , also has an energy, E(nx) = nx2 *A (where A=h2/mL). Each energy, E(nx) and E(ny), will have a momentum associated to the energy and hence a velocity, and the velocities will add as vector quantities. This is not strictly true, since we cannot really talk about velocities in a wave function, but intuitively, it is easier to grasp. Remembering that;
E = ½mV2 = n2A
Then,
V = n(A/½m) ½
And adding Vx and Vy as vectors,
Vxy = (Vx2 + Vy2 ) ½ = {(nx(A/½m) ½)2 + (ny(A/½m) ½)2 } ½
= {(nx2(A/½m) + (ny2(A/½m} ½
Exy = ½mVxy = ½m[{(nx2(A/½m) + (ny2(A/½m} ½]2
Exy = A(nx2 + ny2)
Thus if nx and ny are large (as in a metal) the velocity vector can point in any direction. These states are called “linear momentum states”. This is commonly referred to as “Fermi’s sphere” of states, in momentum space. The surface of the sphere represents the highest ground state energy. The linear momentum state electrons move in straight lines unless interrupted by phonons or flaws in a metal crystal.