Hydrogen wave function
The simplest atomic structure is a proton plus an electron; atomic hydrogen. The electron in atomic hydrogen can be analyzed as orbiting the proton, much like the moon orbiting the earth. The electrostatic force is analogous to the gravitational force. Both forces having a 1/r2 dependence. The centripetal force in both cases keeps the electron/moon in orbit. The difference arises in what orbits are allowed in each system. The moon could orbit at any distance from the earth by adding energy to raise it to a higher orbit, or taking away energy to lower it. The electron, whoever, can only orbit at a distance that allows its wave-length to fit in the orbit. This is a crude explanation, but is similar to the way Neils Bohr pictured the electron in hydrogen when he first proposed the electron orbiting the nucleus in 1913. It was not until over 10 years later when in 1924 DeBroglie proposed that particles had wavelengths, was the path clear over the next few years for the development of quantum mechanics .
Above we explored the 1-dimensional wave function. Here we will quickly solve for the 3-dimensional hydrogen wave function. We will not delve into this problem in any detail, as there are many good derivations available on-line or in any beginning quantum mechanics text. The 3-dimensional Schrodinger equation is commonly written as;
Where 2 (del squared) is the second degree volume differential operator. This means that we take the derivative twice, with respect to the 3 spatial dimensions. This can be done, as we alluded to in the linear momentum case, with;
d2/dx2 + d2/dy2 + d2/dz2
Since, as the name implies, this system is linear, the cartesian coordinates of X, Y, Z, was the best choice. In the case of a central force, like the positive proton acting on the negative electron, it is convenient to use polar coordinates. This means that the distance from the proton to the electron, is given by a radius, r. The rest of space is filled by turning the radius vector through two angles, θ and φ, as seen in figure 2-23.
Figure 2-23
For now we will neglect the angular dependence of ψ, hence neglecting θ and φ. This is the same as assuming that the wave function has a symmetrical solution around the proton. This is a reasonable assumption for the ground state of the electron in hydrogen.
The full Schrodinger equation in polar coordinates with the e2/r electro-static potential is shown in equation 2-1.
This is difficult to solve mathematically, but fortunately, this is a good place to introduce the numerical method that has been, and will be used here often. Again, it is not necessary to follow this technique, in order to follow the rest of the argument. It may be interesting, to the uninitiated, how simple it can be to solve such equations to relatively good accuracy.
We will actually be solving this equation for rψ, not ψ. This is not a real handicap as we can just divide rψ by r to obtain ψ, which is trivial in a spreadsheet. We will dwell briefly, on the finite difference technique, then show the results of solving for the 2 lowest energy levels of hydrogen. We will then compare these results with the exact analytical calculations.
First rearrange equation 2-1 with the assumptions above, letting U= rψ we get equation 2-2. To solve this we simply divide the r axis into appropriately small spaces, ∆r, and instead of using calculus, we will use algebra to solve the equation. Now we set up the equation in what is called finite difference form, equation 2-3.
The Uj just means the jth number in the column in the spreadsheet. If we rearrange this equation to get Uj+1, we get equation 2-4.
This means if we make a column with the value of r, incremented by ∆r, and another column with U; all we need is to guess at U1 and U2, and we can calculate the rest of the U’s in the column. The guessing of U1 and U2 is the same as putting in a starting slope for the curve as (U2-U1)/∆r is just the slope of U at r=0. As it turns out, the starting slope doesn’t matter in the eventual calculation with a given energy. The slope only affects the normalization. The wave function is easily normalized simply by adding up all the values in the EXCELtm column of the wave function squared, and dividing each entry by this sum. This is ridiculously easy to set up as columns in a spreadsheet. It would be more efficient to write a program, but the ease of the spreadsheet is too compelling, and as we will see, we must guess at the value of “E” in the equation, as it is also an unknown. Each guess requires a judgment as to how good it is, by looking at the curve. This too could be programmed, but the behavior of the wave function with various energies would be lost. So let us look at the graphs of a couple of guessed values of E, high and low of the actual value and a “good” solution to the equation. Figure 2-24 shows the graphical trials of the high, low and correct values for the n=1 energy level. Figue 2-25 gives the graphical solutions to the n=1 and n=2 energy levels and shows that the energy calculation is good to 3 significant figures.
Figure 2-24
Figure 2-25
