A simple argument for Schrödinger's equation is actually remarkably simple, though Quantum Mechanics takes the equation itself as an axiom. I thought I'd write a journal entry on it. I would like to see if this were Schrödinger's own argument; but I haven't been able to find an English translation of his paper Quantization as an Eigenvalue Problem yet.

People had discovered three major facts about the world at the very small scales:

(1) General waviness. When you force electrons and light to go through a very narrow slit, you get interference patterns that suggest something "wavelike" occurring behind the scenes. Schrödinger was convinced that there was some sort of waviness underlying the universe; he was merely at a loss to getting the details.

(2) Photoelectric effect. This was the celebrated equation E = h f. It was first proposed by Max Planck, in an attempt to explain why the radiation from blackbodies followed the form it did. Einstein appropriated his equation in his paper explaining the photoelectric effect, where it could be more adequately tested. E is the energy of the particle, h is a special constant calculated first by Planck, and f is the frequency of the photon.

(3) The De Broglie wavelength. Two years before Schrödinger's equation showed up, Louis de Broglie (pronounced loo-ee duh broy; it's a mix of French and Italian) hypothesized that electrons could be scattered the same way as photons could; with a wavelength λ = ^h/_p, or p = ^h/_λ. Here h is, again, Planck's constant; and p is the momentum of the particle (p = mv, where m is mass and v is velocity).

This uses Planck's original constant; but in waves, usually we use Planck's reduced constant to make things easier to read. See, in waves, it's customary to define a wave number k = ^2π/_λ and an angular frequency ω = 2 π f.

It turns out you can make both of these 2π terms disappear by defining a reduced constant, called ħ, as ħ = ^h/_2π. Then these two observed equations become:

E = ħ ω
p = ħ k

Okay, we've been off in la-la land. There is really only one more thing needed for this derivation, and that is to suggest that the total energy E can be written as kinetic plus potential energy:

E = K + P

We don't know the potential energy yet; let's just write it as V(x). But we *do* have some forms for the kinetic energy:

K = ¹/₂ m v² = ^p2/_2m

From here, we can write the above equation as just:

E = ^p2/_2m + V(x)

Now, if you want to find a wave equation, you might want to try to turn this into a differential equation. Suppose that, according to fact (1), I decided to try a wavey function f, and I wanted to get this result out of the wave equation:

E f = ^p2/_2m f + V f

The prototypical wave, well-known in all physics, is written in complex notation as f(x,t) = e^{i(kx - ωt)}. I want a differential equation that gives me back the above expression of E = K + P when I solve this equation.

Well, it's not quite obvious. But if I want an E, I should take a derivative with respect to time of f. That derivative will pull out an ω, and with that ω, I can use E = ħ ω to get an E.

So, take a derivative with respect to time:

^∂f/_∂t = - i ω f

ω f = i ^∂f/_∂t

Now use E = ħ ω to say that:

E f = ħ ω f = i ħ ^∂f/_∂t

And that gives us the E f that we desired.

Similarly, to get a p² out of our function f, we should try to get a k², and to do this, take two derivatives with respect to space:

^∂2f/_∂x² = - k² f

k² f = - ^∂2f/_∂x²

And complete the argument the same way that you did the previous argument:

^p2/_2m f = ^ħ2k2/_2m f = - ^ħ2/_2m ^∂2f/_∂x²

And now, plug these all into the earlier expression E = K + P, to get:

i ħ ^∂f/_∂t = - ^ħ2/_2m ^∂2f/_∂x² + V f

And that is actually Schrödinger's equation, derived with simple calculus and simple physics.