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The Schrödinger equation was arguably the strangest equation invented/discovered in its time (1925/6). Its purpose was to describe the behaviour of an electron in a hydrogen atom as well as possible.
It has proved to be incredibly successful at describing the hydrogen (and all other atoms) providing the shape, orientation and energy levels of electrons as they orbit the nucleus of the atom. The equation gives a curve called the wave function and an energy level of each electron. The hydrogen energy levels calculated from the equation matched the observed energy levels of real hydrogen very well. This matching of results with observation (the colours of light from hydrogen spectra) started to establish the Schrödinger equation as a very successful description of atoms.
However, the wave function curve described by the equation was and is very difficult to understand. The wave function itself is a complex number - meaning it contains real numbers and the imaginary number, i, which is the square root of -1. The number i is difficult enough to "think" about even as an abstract concept - but it becomes even stranger when we try to apply i to physical reality.
Currently the best way for using i in a description of physical reality is to take the size or amplitude of the wave function and consider that value to be only the probability of finding the electron at a certain point or small region of physical space. This complex problem has remained ever since the first success of the equation. Basically it is logical and "easy" to get numbers out of the equation but impossible to picture, even understand, what those numbers represent in a direct description of the objects they are describing. The discussion and wrestling with this has remained with us as well as its success and beauty.
Here is the Schrödinger equation:
The amplitude of wave function, |ψ|², produces a curve in 1 dimension or surface in 2 dimensions or volume density in 3 dimensions.
Recently (2008) with the development of attosecond lasers (very high frequency/short wavelength light) observations of electron can now be made. One of the first videos is here:
Because the Schrödinger equation can be solved "quite" easily numerically we can produce diagrams that model the real electron behaviour. The video below shows a 2 dimensional numerical model of an electron starting in a region of space, being "released" and then "falling" towards a nucleus.
The code to solve a simple 1 or 2 dimension Schrödinger wave equation model is available here: How to draw ... (example codes) .
The negatively charged electron is attracted to the static positively charged nucleus. As the equation is numerically solved the electron wave function spreads towards a more evenly distributed, but blurred region around the nucleus. The stable state relates to a discrete quantum energy level of the electron. The visual result is very beautiful and also is not too dissimilar from the patterns observed in the 2008 video. The music is a Yamaha keyboard and soprano saxophone. Mathematics, physical phenomena and beauty often fit. Paul Dirac emphasised the importance of mathematical beauty directly in relation to Schrödinger's development of his equation:
"Schrödinger got this equation by pure thought, looking for some beautiful generalisation of deBroglies ideas, and not by keeping to the experimental development of the subject in the way Heisenberg did. ... I think that there is a moral to this story, namely that it is more important to have beauty in one's equations than to have them fit experiment." See Quotes: Paul Dirac.
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