Chronology |
Current Month |
Current Thread |
Current Date |

[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |

*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,

Schrödinger already knew a tremennnndous amount about classical

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.

True fact: Schrödinger knew about

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

equation ever known. Heisenberg and deBroglie had already done

a lot. Not to mention Einstein's photoelectric result. And,

arguably, the Planck radiation law.

- Prev by Date:
**Re: Back to Basics vs. Hands-On Instruction** - Next by Date:
**Re: equipotentials** - Previous by thread:
**Re: Schrodinger equation origins** - Next by thread:
**Re: Schrodinger equation origins** - Index(es):