8.5 Schr¨odinger’s Equation

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Since this is not on the syllabus I shall just mention that Quantum Physics

leads to the study of Schr¨odinger’s Equation:

@2

@x2 +

@2

@y2 +

@2

@z2

− V (x, y, z, t) = k

@

@t

where V is a potential field. This equation gives a description of the state of

a single particle. How it was set up remains rather mysterious, but having

got it, courtesy of Schr¨odinger, we can check to see if it works.

Calculating the possible solutions to this equation for a given potential function

gives results generally analogous to the distinct wave solutions to the

176 CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS

Figure 8.5: Soap Film on a rectangular wire

vibrating string. They form a discrete set which look a little bit like the

terms in a Fourier series.

This has a good deal to do with the fact that the energy levels of electrons

in an atom take discrete values which in turn has a good deal to do with

the periodic table of elements and also with the spectral lines which are seen

when looking at diffracted light.

Remark 8.5.1. This should give the (correct) impression that partial differential

equations are rather important when trying to understand how the

universe works.

Remark 8.5.2. Having said something about the setting up of the classical

PDEs, (and having mentioned one modern one) we turn now to the issue of

solving them.

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