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# Re: Schrodinger equation origins

• From: John Denker <jsd@AV8N.COM>
• Date: Sat, 21 Feb 2004 00:51:22 -0800

Quoting Larry Smith <larry.smith@SNOW.EDU>:

One of my intro Modern Physics texts says regarding the genesis (origin) of
the Schrodinger wave equation "Like the classical wave equation, the
Schrodinger equation relates the time and space derivatives of the wave
function. Schrodinger's reasoning is somewhat difficult to follow and is
not important for our purposes.

I agree with that. See item (1) below.

In any case, the Schrodinger equation
cannot be derived, just as Newton's laws of motion cannot be derived."

That's baloney. The F=ma law can very usefully be derived from
classical mechanics via the Euler-Lagrange equation. The Schrödinger
equation can be derived from the Heisenberg equations of motion
(and/or vice versa). Also the SE can be seen as the non-relativistic
leading term in an expansion approximating the Klein-Gordon equation
or the Dirac equation.

Perhaps the author meant something like "... cannot be derived in
terms of stuff you learned in high school."

Can the list give me a sense of Schrodinger's line of reasoning in
developing his wave equation?

1) That's an interesting question, but be warned: like most
history questions, the answer has only very limited pedagogical
utility. As the saying goes: learning proceeds from the known
to the unknown. Before he started this line of reasoning,
physics, including particle mechanics (Lagrangians, Hamiltonians,
Euler-Lagrange equations, et cetera) as well as physical optics
(classical wave mechanics) and the ray-optics approximation.
Also statistics and partial differential equations

Nowadays nobody teaches things in that order; i.e. students in
the "Intro Modern Physics" course are not expected to have already
mastered physical optics, statistics, classical mechanics, and
partial differential equations. Therefore to push students along
Schrödinger's line of reasoning would be to explain one thing
they don't yet understand (Schrödinger's equation) in terms of
N other things they don't yet understand. Not recommended!

2) But it's even worse than that: Remember that folks like Planck
and Einstein and Feynman were amazed by parts of what Schrödinger
did. They called it a stroke of genius. And they ought to know.

Now the point is that you can't explain a stroke of genuis.
Almost by definition it involves "reasoning" that cannot be
followed.

3) I'm not a historian, I just play one on TV. I don't know
the reasoning Schrödinger used the first time. But I can tell
a plausible story about how it *could* have happened.

a1) Physical optics (waves).
a2) The wave equation thereof.
a3) The methods of stationary phase ...
a4) which leads to Fermat's principle of least time,
a5) which leads to geometric optics (rays).

True fact: Schrödinger read deBroglie's thesis and was mightily
impressed.

Conjecture: Schrödinger could have written down a table. The
first column is given above, (a1) through (a5). The second column
might at an early stage have looked like:
b1) deBroglie waves
b2) ???? some wave equation, yet to be determined ????
b3) ????
b4) Hamilton's principle of least action,
b5) which leads to classical particle motion.

Then, working backwards from (b4), you could fill in (b3) by
guessing that action plays some role analogous to phase.

That's the easy part. Anybody can get that far (given a
sufficiently strong physics background). The next step
is to try every wave equation you can think of, to see if
it can be plugged into the remaining slot (b2) in the table.

The hard part is to think of something even roughly resembling
Schrödinger's equation ... and then if you think of it, to not
immediately reject it. I imagine I would have not given it a
second thought, because it is first-order in time (unlike other
wave equations, which are second-order in time). It *looks* like
a diffusion equation, not a wave equation. True fact: It cannot
possibly withstand scrutiny of its behavior in the short-time limit.

But Schrödinger did stick with it. He didn't just write down
the equation, he *solved* it in some interesting cases, showing
that it (at least sometimes) made the correct predictions. He
also showed that in some limit it was formally equivalent to the
already-existing Heisenberg (matrix-mechanics) approach to QM.

References:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html
http://www.nobel.se/physics/laureates/1933/schrodinger-lecture.html
I recommend the physics in
Feynman Volume III chapter 16
... but I believe some of the historical remarks on page 16-13
are incorrect: Schrödinger's equation was not the first QM