Science Friday: Schrödinger's Wave Equation (Birth of Quantum Mechanics)

Now to finish chapter four of George Gamow's classic, Thirty Years That Shook Physics. The last part of the chapter presents Schrödinger's earth shattering wave equation.

The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. De Broglie's Wavy Particles

4b. After the personal investment Gamow showed in the material thus far, I was a little struck by how short and impersonal his discussion of Schrödinger's wave equation is. While Heisenberg gets an entire chapter, including a fair amount of formulaic discussion, Schrödinger gets short shrift--he doesn't even get a chapter for himself when his wave equation is the foundation of quantum mechanics!

Did the two never meet? Schrödinger was in Austria, Gamow in Copenhagen and Cambridge, later America. I have a feeling (please correct me someone), that Gamow did not particularly think too much of Schrödinger.

So de Broglie was "too slow" to develop a mathematical theory for his matter waves. Although I haven't read chapter 5 yet, I get the impression that Gamow was partial to another version of Schrödinger's equation, a more complicated, less intuitive approach created by Heisenberg the year before Schrödinger. In short, Gamow was part of what would become the Copenhagen interpretation of quantum physics, and Schrödinger really wasn't.

I have two other books at my side as I work through Gamow. One is The Amazing Story of Quantum Mechanics, an attempt to present the development of quantum mechanics something like a comic book. (explanations are excellent, but the combination of comic books and science fiction seems a little awkward to me sometimes). The other is Quantum Generations.

Quantum Generations quotes Schrödinger as saying that he was "repelled" by Heisenberg's matrix approach (164). Even when it came to his own equation, Schrödinger thought Max Born's interpretation of it was nuts (Amazing Story, 69). Like Planck, Schrödinger sounds like he didn't actually like what he had discovered. He considered Bohr's theory of light emission "monstrous" and "really inconceivable" (Generations, 165).

The origins of his equation came after a lecture he was presenting on de Broglie's particle-wave duality of electrons. A senior colleague in the audience, Peter Debye, challenged him--if it is is a wave, then it should have a wave equation. The author of Amazing Stories then tells us that Schrödinger then went on Christmas break in the Swiss Alps with a woman who wasn't his wife and developed the equation (65).

Here is the equation, which I have wanted to understand since high school:

Amazing Stories actually does a pretty good job of unpacking it, although I'm not there yet. Interestingly, although the equation eventually worked to predict line spectra, Schrödinger wasn't sure what it meant physically. Since it took of the form of something per volume, he thought it might be a way to calculate charge per volume.

But Max Born--who seems to be one of the smartest cookies in the room as far as I can tell, just not the one who made the key discoveries themselves--suggested that this is a "probability distribution," something that was not to Schrödinger's taste at all. Born was suggesting that the electron cannot be pinned down to one location at any one time. It is not a particle in an orbit. There are rather probabilities that the electron will be in particular places.

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Here is how a person might go from algebra to understanding this equation (something like what Collier has done with relativity):

1. Waves - basic relationships between wavelength, frequency, and velocity (define k and w)
2. Trigonometric functions
3. Radian measure
4. Imaginary numbers
5. Momentum
6. Planck's constant and equation
7. exponential functions
8. basic Ψ wave function - Ψ = e to the i(kx-wt). See DrPhysicsA here.
8. kinetic energy and potential energy
9. differentiation - first and second derivatives
10. simple time independent Schrodinger equation in one dimension in a simple form (See here)
11. partial derivatives
12. The nablus operator
13. Hamiltonians, eigenvalues, Laplace operators
13. Time dependent Schrodinger equation
14. The next steps might look at the predicted hydrogen spectra and quickly lead to Dirac's relativistic version.

There are no doubt other steps I'm missing, but am not far enough along to see.