Deriving Schrodinger's Equation w/o Boundary Conditions

[Divisionbyzer0]

I've seen the nonrelativistic time-dependent SE "derived" from the Fourier Integral Theorem and de'Broglie's relations (I saw this in the book Bohm wrote when he still believed in Copenhagen "Interpretation"-- "Quantum Theory")

It seemed perfectly fine a derivation to me, given that we accept that wave functions are an admissable device for explanations of empirical results, that de'Broglie's relations hold under those situations when wave functions are admissable for description, and that physically admissable wave functions can be synthesized by the Fourier integral.

 

[wavemaster]

Hmm. How about this:

Following (Fourier transform) holds for any wave packet
\psi(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \phi(k) e^{i(kx-wt)} dk

Let us assume that matter is a kind of wave, so obeys this equation. Now, to relate the wave with quantities we all know and love, we assume (as an axiom) deBroglie hypotesis, and state following two:

E = \hbar w
p = \hbar k

and re-write the wave equation.

\psi(x,t) = \frac{1}{\sqrt{2 \pi} \hbar} \int_{-\infty}^{\infty} \phi(p) e^{\frac{i}{\hbar}(px-Et)} dp

By merely taking partial derivatives of both sides

\frac{\partial \psi(x,t)}{\partial x} = \frac{1}{\sqrt{2 \pi} \hbar} \int_{-\infty}^{\infty} (\frac{\partial}{\partial x}) \phi(p) e^{\frac{i}{\hbar}(px-Et)} dp

We get
\frac{\partial \psi(x,t)}{\partial x} = \frac{i}{\hbar} p \psi(x,t)
or
\frac{\hbar}{i} \frac{\partial \psi(x,t)}{\partial x} = p \psi(x,t)

Doing the same steps with partial time derivative, we can derive this

-\frac{\hbar}{i} \frac{\partial \psi(x,t)}{\partial t} = E \psi(x,t)

And this's it. This's how we can extract energy and momentum from wave function. As we all know, it's the common way to abstract these equations from wave functions, and name rest operators and eigenvalues.

\frac{\hbar}{i} \frac{\partial }{\partial x} \Rightarrow \hat{p}
-\frac{\hbar}{i} \frac{\partial }{\partial t} \Rightarrow \hat{E}

Writing Schrödinger, or Klein-Gordon equation is straightforward here. For classical mechanics, we have
\frac{p^2}{2m} + V = E
Since we have to extract momentum and energy from wave function, we'd rather write

\frac{\hat{p}^2}{2m}\psi + V\psi = \hat{E}\psi
which is actually
\frac{p^2}{2m}\psi + V\psi = E\psi
\frac{p^2}{2m} + V = E

If we preferred the relativistic case, we'd have
(E^2 - {(pc)}^2 - {(m_0 c^2)}^2) = 0
and by our usual substitution
(\hat{E}^2 - {(\hat{p}c)}^2 - {(m_0 c^2)}^2)\psi(x,t) = 0

Like it?

But I think, this doesn't justifiy we can derieve Schrödinger equation. It turns out that the equations we borrowed from classical mechanics are a consequence of quantum theory, and actualy they are derived from QM.

 

[Divisionbyzer0]

Yep that's it for a free particle.

I would ask what motivates acceptance of the deBroglie hypothesis in general?

Also can you demonstrate the other direction?-- Would this just be a statement in form to the classical laws but in terms of expectation values or what?

Oddly, I took a few courses on undergraduate QM but they didn't seem to help my understanding of it that much.

All I got was a bunch of statements of axioms and a bunch of mathematics, which while very nice and fun to do, isn't very insightful as to what's going on IMO. Alot of the books seem to lose sense of the historical context of the development of the subject, and they often don't motivate the material very strongly, and they often stay away from discussing what it means, or the statement of what it means isn't very clear..

 

[jtbell]

would ask what motivates acceptance of the deBroglie hypothesis in general?

Experimental evidence. We can produce diffraction and interference effects in beams of electrons, neutrons, atoms, buckyballs... which are in agreement with wavelengths predicted by de Broglie's equation \lambda = h / p.

 

[masudr]

Of course we can derive Schrödinger's equation. It does not provide anything new which can't be derived from more basic physics.

I don't think the derivation given works for systems (as useful as it may be for single particles). And it does not get any nearer in deriving the true Schrodinger's equation:

\hat{H} |\psi(t)\rangle = i\hbar \frac{d}{d t} |\psi(t)\rangle

where

H(x_1,p_1,x_2,p_2...) -> \hat{H}(\hat{X_1}, \hat{P_1}, \hat{X_2}, \hat{P_2}...)

i.e. \hat{H} is the same function of the position and momentum operators as the Hamiltonian for the corresponding classical system is of the position and momentum variables.

 

[pieterenator]

When we see cause-effect relationships in nature we make up rules to predict the outcomes of certain events. The point is, sometimes you can choose which of these rules are "fundamental" or "not derivable" because if you pick two rules to be "fundamental" you can derive 3 others whereas if you pick a different set of 4 rules to be "fundamental" you might be able to "derive" a rule that other people consider "non-derivable." Some people would argue that you want to derive the most things from the least number of rules, but having more rules can sometimes be conceptually neater.

Analogously, if you call one of the SE "fundamental" then you don't quite have to do as much math as if you were to call other things "fundamental" and derive the SE. What is "fundamental" (i.e. what are your axioms), is somewhat subjective. However, you can still predict the outcome of any experiment no matter what you call "fundamental," so all this is just splitting hairs (I won't go into quantum uncertainty here, just know that I'm aware that sometimes you cannot predict exactly the outcome of an experiment in quantum mechanics). The point is nothing in the classical realm can explain quantum mechanics, and even thinking of things in quantum mechanics as "waves" is suspect since we are not talking about waves in the classical sense (matter waves don't in general obey classical wave equations).

So the SE is derivable if you don't choose it to be fundamental, but you might as well choose it to be fundamental and then it is, by definition, not derivable. It's simpler to choose the SE to not be derivable, so no one is lying to you when they say the SE is not derivable.

The most important thing to keep in mind is that QM cannot be "derived" from classical mechanics.

 

[pieterenator]

Another thing to keep in mind is that it is not easy to derive the SE from other fundamental rules. So whenever you do find a derivation of the SE that looks too easy I would be very suspicious of logical and/or mathematical errors.

 

[Xeinstein]

Another thing to keep in mind is that it is not easy to derive the SE from other fundamental rules. So whenever you do find a derivation of the SE that looks too easy I would be very suspicious of logical and/or mathematical errors.

This paper illustrate a simple derivation of the Schroedinger equation
Would you be very suspicious of logical and/or mathematical errors in it?

http://arxiv.org/abs/physics/0610121

 

[Marco_84]

Schroedinger's equation cannot be derived.It is accepted as a postulate.

I think that's wrong, I've seen how it is derived in few books... and the idea was to follow the common classical rules but adding the Debroglie idea (wave particle duality) it is easy to write down fermat principles and Maupertuis ones and equating them you pass through Helmotz equation arriving at the Schrodinger one...

if somebody want i think i have it somewhere...

bye marco

 

[pieterenator]

This paper illustrate a simple derivation of the Schroedinger equation
Would you be very suspicious of logical and/or mathematical errors in it?

http://arxiv.org/abs/physics/0610121

Haha... "simple" is relative.

Also, this is not a "derivation" the way I meant it. That is to say, they're not using fundamental rules to prove new rules, they're "generalizing" and "approximating." It's safe to say that they probably wouldn't know how to generalize and approximate if they didn't know what the end result "should" be (the Schroedinger Equation). Back before quantum mechanics, if somebody used this kind of argument to justify a new "master equation," he would have been laughed at. The reason the Schroedinger equation was accepted was because of the empirical evidence for it, not because someone somehow derived it.

Not that this kind of thing doesn't have any value. And I'm not implying that they made any unacceptable logical leaps. I'm just saying that they didn't strictly prove Schroedinger's Equation using classical (or relativistic) principles... they had to extend, generalize, and approximate.

But the answer is yes. I would be very suspicious. Justifiably so.

 

[reilly]

Absolutely correct; pieterenator has it exactly right. In the cited paper, there's nothing more than a plausibility argument, and a nice one at that.

There is no derivation of the Schrodinger E., nor of the 2nd Law of Newton, nor of the Conservation of anything,...Indeed, to the uncritical, non-Humean eye, Noether's Thrm is a derivation of Conservation Laws -- but how could we even begin to examine all the assumptions that lurk behind Emily Noether, which must be done in order to give a sound evaluation of derivation vs. plausibility argument?

Who cares, if nothing can be derived? All we can do is make our best guess, and see what happens. This is as true for Newton as it is for Einstein, Heisenberg, et. all. Elegant as their work is, no one believed any of it until the work was successfully checked by experiment. Mathematicians use axioms; physicists use experiments.

Regards,
Reilly Atkinson

 

[samalkhaiat]

There is no derivation of the Schrodinger E., nor of the 2nd Law of Newton

If the chosen set of axioms does not contain Schrodinger/ Newton equations, then the formalisim should provide ways for DERIVING them from the axioms. Any failure means that your axioms contain a piece of garbage.

...Indeed, to the uncritical, non-Humean eye, Noether's Thrm is a derivation of Conservation Laws -- but how could we even begin to examine all the assumptions that lurk behind Emily Noether, which must be done in order to give a sound evaluation of derivation vs. plausibility argument?

Noether's theorem is a THEOREM! "given A, then B"; where A is the invariance under some Lie group, and B is a statement about a conserved quantity that can be DERIVED SYSTEMATICALLY from the action integral. If this is not a derivation, can you tell us what is it that we do when we play with equations and produce B from A?

Who cares, if nothing can be derived?

I do, because I lose my job if I cann't derive anything!

Mathematicians use axioms; physicists use experiments.

I am a physicist and I only care about a self-consistent set of axioms, I leave experiments for the more able peopel who work on them!

Regards

sam