Quantum Mechanics without the maths or philosophy

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Point 5: Potentials

All the examples above begin with "A single electron is floating around in free space".
You can add a term to the equation to represent the fact that the electron is attracted to a proton. What's happening here is as follows:

• The experimenter puts an electron in a region of space near a proton
• The electron has a very uncertain momentum
• The electron does one of two things. There is a chance that it escapes (resulting in what looks like an initial explosion).
• There is a chance that the electron will be bound to the proton.
• Although bound to the proton, the electron's wavefunction moves around.

Summary
You can easily add something to the Schrödinger equation that makes the electron interact with a proton. When we try this, we get a simulation in which the electron seems to stick around. In previous examples, the electron tended to move freely round its enclosure. Now we've got an electron whose wavefunction stays high at the centre of the box - which is where the nucleus is.
To express that differently, no matter how long you wait, the electron will always have a probability of being near the nucleus. An electron plus a nucleus, when bound like this, equals an atom.

Point 6: Spectra

In the previous experiment, the electron clearly moves about. The electron is negatively charged. We know that charged particles that accelerate generate electromagnetic radiation.
We also know that real atoms also emit electromagnetic radiation when they're excited.
We can use a spectrometer to see what frequencies of radiation are given off by real atoms.
We can simply plot the average movement of the electron in our new quantum theory - shown below: To the right of the animation of the atom, we have a graph. The y-axis is the average vertical velocity of the electron. The x-axis is time.
If you compare the frequencies coming from real life hydrogen to the frequencies coming from our crude simulation, they match almost perfectly (at least if you do the calculation in 3d).
Summary
We can show experimentally that this simple Schrödinger equation gives accurate predictions of the frequencies of radiation coming off real hydrogen atoms.
This is evidence that not only is the theory vaguely correct in that it is a wave and stuff, it's actually quite accurate.
This is a skippable point, but it is interesting if you wish to know why people believe that quantum mechanics works.

Point 7: Two electrons

We started off with a simple wave equation. Then we added a term to it that is the attraction between the electron and a nucleus. The result let us accurately predict the emission spectrum for atomic hydrogen.
Now let's consider a 2-electron example.
In order to show 2 electrons in a 2d animation, we're going to have to constrain each electron to be just in one dimension. You, the reader, have to stare at the following picture until you understand the following:

• There are two electrons.
• Each electron is in a 1-dimensional enclosure.
• Electron 1 is most likely to be at the left.
• Electron 2 is most likely to be in the middle.

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