Re: Schrodinger's Equation and Basic Quantum Mechanics Refresher
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Reply #1 –

This thread is a spin-off of the discussion here:

http://theskepticalforum.org/index.php?topic=347.0As alluded to in that thread the Bohr model of the electron in a hydrogen atom describes the AVERAGE behavior of an ensemble of hypothetical electrons based on the energy state of the electron. The change in energy associated with "orbits" of the electron is described by

E = h * nu

But if we have average behavior, what the is the equation that describes the probability distribution from which we get averages from. For example, here is one distribution with an average, the normal distribution. Graphically depicted the distribution looks like:

In terms of an equation, it is described exactly as:

Unfortunately, the distribution of positions of a quantum "particle" like an electron isn't so easy to depict as it is dependent on context.

But let's look at a specific example, the electron "orbiting" the hydrogen atom. The average position accords with the bohr model. If we plotted a graphical distribution of the distance of the electron from the center of the nucleus of the hydrogen atom, the probability distribution for the various "orbits" would look like:

That is to say, if we had a machine that recorded positions of the electron in a partical energy state (like 1s) over time, and plotted how many times the electron appeared at various distances, it would look like the plot above.

To mathematically capture this, Shrodinger had to make an equation to predict that behavior and hence the famous Shrodinger equation: