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Feynman’s Derivation of the Schrödinger Equation

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Re: Feynman’s Derivation of the Schrödinger Equation

#21

Post by krastanov »

cyphar wrote: post #16

The Schrödinger equation was not derived , a more accurate word would be guessed . It's a fundamental law, like F = ma (although that is actually more of a definition than a law). There is no fundamental explanation why electrons obey the Schrödinger equation, just like there's no fundamental explanation why macroscopic objects obey F = ma.

A lot of things can be derived from more basic or at least more beautiful principles. It makes perfect sense to say that Schroedinger's equation can be derived from Lagrangian Mechanics, even if this is not historically how it happened.

An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial time". Schroedinger did not know about NP-completeness, but this notion does give us clues why the equation is what it is.

It is always instructive to see how a "fundamental" law can be derived or at least constrained from other laws. Thermodynamics and the notion of entropy would not have been invented if engineers did not "rederive" what was considered fundamental at the time.

Re: Feynman’s Derivation of the Schrödinger Equation

#22

Post by spdustin »

Not many people know that Schrödinger wrote a a paper involving a dog, and to show solidarity with a fellow scientist, opted to model the dog after a subject of Pavlov's experiments. I called the UIUC library to ask if they had a copy of the paper just a few months ago, but I couldn't remember the title of the paper. The researcher that answered told me, "That sure rings a bell; I'm not sure if it's here or not, though."

Re: Feynman’s Derivation of the Schrödinger Equation

#23

Post by santaclaus »

kurthr wrote: post #6

I find that often reading a scientist's dissertation will often tell you something about how they approach the rest of their research. Somewhere I actually have a paper copy of Feynman's Thesis under Wheeler, which covers exactly this material. He was clearly enamored of Lagrangian formulations quite early (I didn't know that it was due to a High School teacher). You can find this actual thesis on-line at CERN: http:…

> I didn't know that it was due to a High School teacher

That's one badass high school physics teacher! Ours grew pale at the mention of a derivative, let alone calculus of variations.

Re: Feynman’s Derivation of the Schrödinger Equation

#24

Post by nilkn »

krastanov wrote: post #21
cyphar wrote: post #16

The Schrödinger equation was not derived , a more accurate word would be guessed . It's a fundamental law, like F = ma (although that is actually more of a definition than a law). There is no fundamental explanation why electrons obey the Schrödinger equation, just like there's no fundamental explanation why macroscopic objects obey F = ma.

A lot of things can be derived from more basic or at least more beautiful principles. It makes perfect sense to say that Schroedinger's equation can be derived from Lagrangian Mechanics, even if this is not historically how it happened. An especially beautiful constraint is "Schroedinger's equation has to be linear because otherwise you can build a (quantum) computer that can solve NP-complete problems in polynomial…

I think you're both right.

The grandparent post is right in the sense that reductionism always reduces, ultimately, to a guess (in this case, Lagrangian mechanics is the guess).

You're right, though, in that a basic framework can be established, using minimal assumptions and axioms, in which most of classical physics and even much of quantum physics can be derived via (mostly) rigorous mathematical arguments. But that framework itself is, at the end of the day, a guess. It's simply the best guess we have at the given moment in time that fits the evidence (e.g., by allowing us to derive from it known rules and patterns that past generations have verified fit the evidence).

Re: Feynman’s Derivation of the Schrödinger Equation

#25

Post by ahuibers »

kurthr wrote: post #6

I find that often reading a scientist's dissertation will often tell you something about how they approach the rest of their research. Somewhere I actually have a paper copy of Feynman's Thesis under Wheeler, which covers exactly this material. He was clearly enamored of Lagrangian formulations quite early (I didn't know that it was due to a High School teacher). You can find this actual thesis on-line at CERN: http:…

The actual thesis is fun to look at! A typeset version is available: http://www.amazon.com/Feynmans-Thesis-Approach-Quantum-Theor...

Re: Feynman’s Derivation of the Schrödinger Equation

#26

Post by ahuibers »

mrcactu5 wrote: post #13

Feynman's "Quantum Mechanics and Path Integrals" is about $18 on Amazon http://www.amazon.com/Quantum-Mechanics-Path-Integrals-Emend... That is a serious graduate level book - written in his own style.

This article might be one of the best intros: Space-Time Approach to Non-Relativistic Quantum Mechanics R. P. Feynman Rev. Mod. Phys. 20, 367 – Published 1 April 1948

Caltech has this available: http://authors.library.caltech.edu/47756/1/FEYrmp48.pdf

There are some other good treatments too, although I can't find them at the moment.

Re: Feynman’s Derivation of the Schrödinger Equation

#27

Post by EGreg »

spdustin wrote: post #22

Not many people know that Schrödinger wrote a a paper involving a dog, and to show solidarity with a fellow scientist, opted to model the dog after a subject of Pavlov's experiments. I called the UIUC library to ask if they had a copy of the paper just a few months ago, but I couldn't remember the title of the paper. The researcher that answered told me, "That sure rings a bell; I'm not sure if it's here or not, thou…

"Rings a bell." Hah

Re: Feynman’s Derivation of the Schrödinger Equation

#28

Post by calhoun137 »

cyphar wrote: post #16

The Schrödinger equation was not derived , a more accurate word would be guessed . It's a fundamental law, like F = ma (although that is actually more of a definition than a law). There is no fundamental explanation why electrons obey the Schrödinger equation, just like there's no fundamental explanation why macroscopic objects obey F = ma.

It's actually not true to say that Schrodinger's equation, or F=ma, cannot be derived.

This is because in order to derive something, you need to first start from a set of assumptions. You can pick any assumptions you want, that is your privilege, as it is mine. If you want F=ma to be an assumption, you can go on and derive things from that.

But truly, F=ma is not the most general assumption in theoretical physics.

If you look in any decent book on classical mechanics, you will find that F=ma in fact is derived from a much more general principle, the Lagrangian.

That is the same Lagrangian that is the subject of the OP, which is used to derive the Schrodinger equation; and it's not just a coincidence.

Re: Feynman’s Derivation of the Schrödinger Equation

#30

Post by malanj »

EGreg wrote: post #27
spdustin wrote: post #22

Not many people know that Schrödinger wrote a a paper involving a dog, and to show solidarity with a fellow scientist, opted to model the dog after a subject of Pavlov's experiments. I called the UIUC library to ask if they had a copy of the paper just a few months ago, but I couldn't remember the title of the paper. The researcher that answered told me, "That sure rings a bell; I'm not sure if it's here or not, thou…

"Rings a bell." Hah

Actually both parts are a joke, but only if you don't look for it.
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