You cannot derieve Schrödinger Equation .

[gulsen]

"You cannot derieve Schrödinger Equation".

Bah. We're being told this over and over again. Then the game guy invents operators to extract momentum and energy from wavefunction, then puts them in Newtwon equation! He's saying exactly this:
$$\frac{p^2}{2m} + V = E$$
Should I look amazed when this equation is consistent with $$\frac{d<p>}{dt} = <-\frac{dV}{dr}>$$. It has a name BTW, Ehrenfest's theorem. Aside from what a great discovery this is, what Dirac wrote seems just to be relativistic version of it. Put operators in $$E^2 - (pc)^2 - (mc^2)^2 = 0$$. With some tricks to make it linear.
And why are quantum teachers are proudly (I simply hate the look in their face, when pleasured by uncertainty) trying to sell us: "you can Not derieve Schrödinger equation any way, it is a fundamental law of nature!"
Then I stare blankly at them and say: "What I'm saying right now is wrong."

 

[cartuz]

Bah. We're being told this over and over again. Then the game guy invents operators to extract momentum and energy from wavefunction, then puts them in Newtwon equation! He's saying exactly this:
$$\frac{p^2}{2m} + V = E$$
Should I look amazed when this equation is consistent with $$\frac{d<p>}{dt} = <-\frac{dV}{dr}>$$. It has a name BTW, Ehrenfest's theorem. Aside from what a great discovery this is, what Dirac wrote seems just to be relativistic version of it. Put operators in $$E^2 - (pc)^2 - (mc^2)^2 = 0$$. With some tricks to make it linear.
And why are quantum teachers are proudly (I simply hate the look in their face, when pleasured by uncertainty) trying to sell us: "you can Not derieve Schrödinger equation any way, it is a fundamental law of nature!"
Then I stare blankly at them and say: "What I'm saying right now is wrong."

May be the problem here is in the fact that Quantum Axiomatic not unificated. If we read papers of different authors, we can see that thay used a different axioms. Very often Schrödinger equation is one of axioms because it is not derived but postulate.

 

[abszero]

The way that you "derive" the Schrödinger equation is that you assume that the hamiltonian is the generator of finite time translations. As for the form of the hamiltonian, it comes from making a classical correspondence and introducing operators that obey the canonical commutation relations, or by arriving at the momentum and position forms from the assumption that the momentum is the generator of infinitesimal space translations, and then out comes the form of the momentum operator in the position representation.

 

[dextercioby]

This symmetry-based deduction of SE is found in Sakurai's book and I'm not really a big fan of it. The axiomatic formulation of QM in the Dirac formulation is preferrable to any other approach to finding an evolution equation for quantum states.

Daniel.

P.S. SE is really a consequence in other formulations of QM: von Neumann's, Feynman's and Schwinger's... But in Dirac's it's an axiom.

 

[pmb_phy]

Bah. We're being told this over and over again. Then the game guy invents operators to extract momentum and energy from wavefunction, then puts them in Newtwon equation! He's saying exactly this:
$$\frac{p^2}{2m} + V = E$$
Should I look amazed when this equation is consistent with $$\frac{d<p>}{dt} = <-\frac{dV}{dr}>$$. It has a name BTW, Ehrenfest's theorem. Aside from what a great discovery this is, what Dirac wrote seems just to be relativistic version of it. Put operators in $$E^2 - (pc)^2 - (mc^2)^2 = 0$$. With some tricks to make it linear.
And why are quantum teachers are proudly (I simply hate the look in their face, when pleasured by uncertainty) trying to sell us: "you can Not derieve Schrödinger equation any way, it is a fundamental law of nature!"
Then I stare blankly at them and say: "What I'm saying right now is wrong."

What they are saying is that the Schrödinger Equation cannot be derived in a vaccum, i.e. you'd need other postulates to derive it. E.g. you'd have to postulate things like "Replace observables in classical equations with corresponding operators." But postulates must be independent, otherwise they are not postulates. I may be missing a point of logic here where, perhaps, it can be shown that you need more postulates that are postulated in the list of postulates that QM is based on. The list of postulates may not be unique either. Perhaps there are other lists of postulates in which the Schrödinger Equation does not appear. This is what you'd have to show in order to say that you can derive the Schrödinger Equation from other postulates.

Pete

 

[gulsen]

I agree that you cannot derieve operators from something else, they're just to extract necessary stuff from wavefunction representing the particle, and since this wavefunction isn't anywhere else (perhaps de Broglie, we may remember), OK, they're basic things.
But,I'm not talking about operators, whom cannot define anything mechanical on their own. I'm talking about something solidly related to the real world: Schrödinger equation. And what I'm saying is, it's a combination of classic mechanics and wavefunction + some new math. They did derieve Schrödinger equation from wavefunction operators and classic mechanics, and still I hear: "It's a fundamental nature law, that cannot be derieved from something else". But wasn't that exactly how Schrödinger did derieve it?

 

[abszero]

This symmetry-based deduction of SE is found in Sakurai's book and I'm not really a big fan of it. The axiomatic formulation of QM in the Dirac formulation is preferrable to any other approach to finding an evolution equation for quantum states.
Daniel.
P.S. SE is really a consequence in other formulations of QM: von Neumann's, Feynman's and Schwinger's... But in Dirac's it's an axiom.

The thing is, if you use the symmetry-based deduction, it sets you up quite well for thinking about Noether's Theorem, so in that sense it's quite useful.

The Schrödinger equation arises out of some postulates of quantum mechanics, particularly that observables are generators of some sort of unitary transformation of a system. In the case of the hamiltonian, it's the generator of time evolution. For momentum, it's the generator of space translation. Angular momentum gets rotation. If you think of it like this, then you "derive" the Schrödinger equation as such, but you cannot arrive at it from anything classical because it isn't classical. Classical-based arguments are not correct.

 

[jtbell]

But wasn't that exactly how Schrödinger did derieve it?

Schrödinger was inspired to his equation by making an analogy between mechanics and optics. In this analogy, quantum mechanics corresponds to classical mechanics in a similar way as wave optics corresponds to geometrical optics. I posted a more detailed description sometime last year, based on one of Schrödinger's papers. Let's see if I can find it... ah, here it is.

 

[Triss]

The beginning of Schrödinger's article "An Undulatory Theory of the Mechanics of Atoms and Molecules", E. Schrödinger Phys. Rev. 28, 1049–1070 (1926) is quite an interesting read if you care to see how Schrödinger used the optical-mechanical analogy. It can be found at http://link.aps.org/abstract/PR/v28/p1049

 

[masudr]

...the case of the hamiltonian, it's the generator of time evolution...but you cannot arrive at it from anything classical because it isn't classical...

Well technically that's not all true: the Hamiltonian is also the classical generator of time translations (i.e. via the Poisson bracket), and similar stuff can be said for momentum generating space translations etc.