# Schroedinger’s Equation...an historical, heuristic approach.

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Schroedinger’s Equation...an historical, heuristic approach

Radical view of light Planck and Einstein re-introduced the particle notion for light a wave “becomes” a particle and still a wave … hmmm

de Broglie... If light (ie “a wave”) can be a particle then maybe a particle (electron?) can “be a wave” - what are the implications of this “leap of reason”?

Compare and contrast: Waves & Particles Waves are extended Waves are continuous Waves conform to wave equations Waves diffract and interfere Waves have amplitude, frequency and velocity Particles are points Particles are discontinuous Particles obey equations of mechanics Particles “bounce” Particles have mass, size(?) and velocity

ParticleWaves, Wavicles or just Weirdness … If de Broglie is correct then we can ascribe a wavelength to a particle:

Schroedinger... Once at the end of a colloquium I heard Debye saying something like: “Schroedinger, you are not working right now on very important problems… why don’t you tell us some time about that thesis of de Broglie’s… in one of the next colloquia, Schroedinger gave a beautifully clear account of how de Broglie associated a wave with a particle, and how he could obtain the quantization rules… When he had finished, Debye casually remarked that he thought this way of talking was rather childish… To deal properly with waves, one had to have a wave equation. Felix Bloch, Address to the American Physical Society, 1976

Schroedinger’s key assumptions concerning a quantum-mechanical wave equation... 1It must incorporate the relations: 2Since normal waves “add” linearly (principle of superposition), so too must the solutions to the qm- wave equation. This means the solutions must be linear.

from the first assumption... Kinetic + Potential = Total Energy

from the second assumption... Introduce “psi” as the solution to a wave equation. It could include terms like: where “psi” is the wavefuntion

now, assume that psi describes a travelling wave... Psi should have the form: The derivatives should produce the expressions listed in the 1st assumption, so...

from assumption 1... We get our “lambda-term” from a 2nd spatial derivative, ie since we need a We get the “nu-term” from the 1st time derivative, ie

Also, since contains a potential term V(x,t), our equation should contain a V(x,t) factor… hence we are led to “guess” an equation of the form: we now must solve for  and .

wrinkles (there are ALWAYS wrinkles!) It will be left as an exercise for you to show why will not work, instead we will try this… With a little bit of work we find:

so, without further delay...