Waves and the Wave equation

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For those who are interested, we'll attempt to give a closer glimpse at wave behavior here and to try to show where the Schr–dinger Equation comes from and what it is.
We've already looked at a diagram representing how the oscillating electric and magnetic fields of a light wave is pictured (right, top). We can simplify this by looking at just the electric field, for example, giving a one-dimensional sine wave for the variation of the field amplitude with distance along the x-axis (right, bottom). This equation, represented by the red curve, is

The maximum amplitude is A and the wavelength is l. The wave equation for light embodies both wave behavior and particle behavior. At "short" wavelengths, geometric shadowing is exlained by this equation. At "long" wavelengths, interference and diffraction effects emerge.

At any position along such a wave pattern (the x-coordinate in one dimension), the slope can be determined. That is,we see how rapidly the function (electric field) is changing as we step along the x-axis. In calculus, this slope is the first derivative and is represented by

The slopes at various locations are shown here by the solid blue lines.  
A negative slope is indicated A negative slope is indicated A zero slope is indicated
The curvature of the function is a measure of how rapidly the function is curving. (If you think about it, the curvature then is an indication of how fast the slope is changing at a given position.) Since a slope in calculus is a rate of change, the curvature is the rate of change of the first derivative, which is what's then called the second derivative and is represented by . In the illustrations below, the black lines represent the slope of some function (the function is not shown but could be the wave functions illustrated above). The colored curves represent curvature, green being positive curvature and red being negative curvature.  

Positive slope Zero slope (as in the rightmost illustration of the wave previously indicated by the blue line) Negative slope (as in the two leftmost illustrations of the wave)

If the slope isn't changing, the curvature is zero and the function keeps heading in the direction indicated. If the curvature is positive at a particular point, the slope turns upward. The larger the magnitude of the curvature, the more rapidly the curve breaks. A negative curvature represents the function breaking downward from its current slope. What you see in the sinusoidal behavior of the light wave is that where the amplitude is positive, the curve breaks towards the axis. The amplitude and its second derivative are both positive. When the function crosses the x-axis and thereby becomes negative, the curvature switches to negative as well. Where the amplitude is negative, the curve breaks up towards the axis. The amplitude and its second derivative are both negative. How frequently these switch back and forth is, of course, determined by the wavelength. The overall relationship between the curvature and the amplitude is related to the wavelength in a very simple manner.

The above expression is just an alternative way of expressing the wave behavior. For a fixed wavelength, as shown in all the figures above, the ratio of the curvature of the amplitude (numerator on left side of above equation) to the amplitude (denominator on left) is constant (right side of above equation) and generates the sine function shown with whatever wavelength is on the right in the above equation.

Now on to a wave equation for matter.

Schr–dinger just takes this equation and uses deBroglie's expression for the wavelength of a particle.

If we substitute deBroglie's expression into the wave equation, we get an equation (below) in which the relationship between the wave's curvature and its amplitude is determined by the particle's momentum, p. The momentum, p, is in turn related to the kinetic energy. The kinetic energy is the total energy, E, minus the potential energy, which we symbolize by V. The potential energy is related to any forces that might be acting on the particle. What we should recognize here is that the wavelength is no longer constant, but varies throughout space as determined by the manner in which the forces acting on the particle differ from place to place.

This is the Schr–dinger Equation (for a one-dimensional system) and is usually algebraically rearranged (for mathematical reasons) and presented as

In the case of the hydrogen atom, the above presentation is extended to three dimensions in a straightforward manner giving the three-dimensional Schr–dinger Equation
in which the potential energy is the electrostatic potential of attraction between the electron of charge "e" and the nucleus of general charge "Ze". Obviously, this implies that the wavelength varies in a complicated fashion throughout the three dimensions.

Solving the math implies finding functions, y(x,y,z), that "work". The solution also gives values of E, the allowed total energy of a particle/wave being subjected to the forces embodied in whatever V is relevant. For a nucleus attracting an electron to it through the Coulomb force, the total energy solutions are restricted to only certain, discrete values. They turn out to be identical to the ones given by Bohr's equation.

For each total energy, the results for
y(x,y,z) are the various shapes and sizes that we have been displaying.