Can schrodinger equation be proved

[vincentchan]

Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the Schrödinger equation from the more fundemantal principle in Physics...

 

[pmb_phy]

Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the Schrödinger equation from the more fundemantal principle in Physics...

I'm not sure, but I don't think so. I've only seen it used as a postulate in quantum mechanics. I've never seen anyone derive it from more fundamental principles.

Pete

 

[dextercioby]

There are 2 current widely-used approaches.
*In the axiomatical construction,SCHROEDINGER's EQUATION IS POSTULATED.In my classification,it is the IV-th postulate of (nonrelativistic) QM in Dirac's formulation,Schroedinger's picture...

**In the second approach,the equation is derived from classical mechanics (Hamilton formulation),plus Quantum level construction using Stone's theorem...Read the wonderful chapter #2 of J.J.Sakurai's masterpiece...A bit of Goldstein wouldn't hurt,just to warm you up.

Daniel.

 

[vincentchan]

can you give me a link to Stone's theorem, or any good textbook about it... i have absolutely no idea what the Held it is...
i am so sure the shrodinger equation can be derive from a minimal priciple, because i took a class in 3 dimensional calculus of variational, and I remember in one of my homework problem, I minimize the energy in E field and got the $\nabla ^2 V = 0$, therefore, I deeply believe the laplacian in schrodingers' equation is came from the hamilton's principle...
I just want to know am I correct...

 

[Kane O'Donnell]

It won't work out as simply as that, vincentchan. The reason is, the laplacian in the Schrödinger equation comes from quantising the momentum operator and then applying it twice. As dextercioby has mentioned, if one wants to justify the existence of the Schrödinger equation, you have to do a bit of work. For example, in Mackey, The Matehmatical Foundations of Quantum Mechanics, I believe the Schrödinger equation finally appears around page 219.

On the other hand, it would be *very* nice to have a theory which shows *why* Dirac's commutator postulate is correct.

I tried to do a simple summary of what Stone's theorem says but it got a bit technical. Basically it says that some of the unitary operators on a Hilbert space have an associated self-adjoint operator. The Hamiltonian is the self-adjoint operator associated with the time-evolution operator, and the technical details of the association give the time-dependend Schrödinger equation.

Cheerio,

Kane

 

[dextercioby]

On the other hand, it would be *very* nice to have a theory which shows *why* Dirac's commutator postulate is correct.

You mean (graded) Dirac brackets,right?

I tried to do a simple summary of what Stone's theorem says but it got a bit technical. Basically it says that some of the unitary operators on a Hilbert space have an associated self-adjoint operator. The Hamiltonian is the self-adjoint operator associated with the time-evolution operator, and the technical details of the association give the time-dependend Schrödinger equation.

I'll forgive you on that one...We both know how they're called...

Daniel.

 

[BlackBaron]

I deeply believe the laplacian in schrodingers' equation is came from the hamilton's principle...
I just want to know am I correct...

Since the quantum wavefunction $\psi(x,t)$ behaves like a classical field, it's equations of motion (Schrödinger's equation) can be derived by making extremun certain action (see for example, José, J. V. and Saletan, E. J., "Classical Dynamics: A Contemporary Approach", Cambridge University Press, 1998, chapter 9 (I really like this book )), but hey, that's just mathematics, remember that even if the wavefunction behaves like a classical field, it's not one, it's physical interpretation is not classical at all.

As dextercioby said, the usual approach, in the "canonical" formulation of quantum mechanics, is postulating Schrödinger's equation.
But then, there's also the Feynman's path integral formulation of quantum mechanics, where you have a different set of fundamental postulates and you can derive Schrödinger's equation (and the other "canonical" postulates) from them. You can look that up in Feynman's original paper: R.P. Feynman, "Space-time Approach to Non-relativistic Quantum Mechanics", Rev. Mod. Phys. 20 (1948) 367, where he also shows how in the classical limit you get Hamilton's principle. But if you prefer, ask me, and I'll give you the details here.
Since you can also derive Feynman's set of postulates from the "canonical" set, I wouldn't say any of them is more fundamental than the other, they're just different alternatives.
You should also be able to derive Schrödinger's equation from the Heisenberg's formulation, where you postulate the analogous to Hamilton-Jacobi equations as your fundamental equations of motion. Again, I wouldn't call this formulation to be more fundamental, specially since the usual approach is to derive it from the "canonical" one.

I think it's interesting to notice the following:
In classical mechanics, you have Newton's equation (along with the others Newton's laws of motion), and from it you can derive the lagrangian and hamiltonian formulations. Nevertheless, lagrangian and hamiltonian formulations turned out to be more fundamental since they both can incorporate Maxwell electrodynamics and Einstein's relativity, which goes well beyond Newton's equation.
In quantum mechanics, you have Schrödinger's equation (along with the other "canonical" postulates) and from them you can derive the path integral formulation (along with the many other formulations of quantum mechanics). As long as I know, all formulations of quantum mechanics are completely equivalent (the difference is that some things are easier to calculate in one or the other), but one wonders if someday one formulation will go beyond the others, just like what happened in classical mechanics .

 

[Kane O'Donnell]

You mean (graded) Dirac brackets,right?

I'm just referring to Dirac's postulate that the commutator of two quantum mechanical observables is the classical Poisson bracket multiplied by $$i\hbar$$.

I don't know what you're talking about with the other thing though. Stone's theorem applies to a strongly continuous one parameter group $$U:\mathbb{R}\to\mathcal{U}(\mathcal{H})$$ with elements $$U_{t}$$ being unitary operators on $$\mathcal{H}$$.

Cheerio,

Kane

 

[Loren Booda]

Reasonable assumptions concerning the Schroedinger equation include the de Broglie (momentum-displacement) and Einstein (energy-frequency) postulates, along with the conditions of linearity, constant potential for free particles and reliance on the Hamiltonian energy formulation.

From Robert Eisberg and Robert Resnick, QUANTUM PHYSICS of Atoms, Molecules, Solids, Nuclei, and Particles; 2nd ed., page 129.

 

[Kane O'Donnell]

The de Broglie/Einstein relations arise naturally from the free-particle solution to the Schrödinger equation. Often this is used as the justification for the form of the equation, but this route is really only suitable for historical reasons or undergraduate level quantum mechanics.

Linearity turns out to be an unreasonable assumption, there are, for example, non-linear forms of the Schrödinger equation and I believe the interaction term in the extension of the Dirac equation to QED is non-linear.

Kane

 

[marlon]

Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the Schrödinger equation from the more fundemantal principle in Physics...

The Scrodinger equation is a partial differential equation. Now, you know that the energy E can be written as E =p²/2m right ?

The physical system is described (in most easy form) by a plane wave like
Aexp(i(px-Et)/hbar) (why this equation for a wavefunction is valid, can be concluded from the double slit experiment, allthough you cannot localize a particle with such wavefunction...)...Now you see that if you calculate the first derivative of this wavefunction with respect to x, you'll get momentum p and if you take the second derivative you get p² (i am omitting the constants here). Then the first derivative with respect to t will yield the energy. Just calculate what i said and then via E=p²/2m you can set these two calculated derivatives equal to one and other in one equation: what you get is the Schrödinger equation. In the same way, just apply this for the more general wavefunctions that are superpositions of wavepackets...

regards
marlon

 

[dextercioby]

Well,Marlon,i guess your idea works iff you postulate the (quantum spinless nonrelativistic) free particle's wavefunction and then u'd still have to derive the equation which has the advantage to be useful in any occasions,while the plane wave,not...

Yes,that's one alternative way to do it.But why postulate something that is useful to obtain the "real thing"...?? Leave the Schroedinger's equation in the postulates and build everything from it...

Daniel.

 

[Kane O'Donnell]

The approach Marlon has taken is another one that is often used as an introductory justification to undergrad students.

After all, 2nd year undergrads won't have much exposure to axiomatics (maybe they've used the group axioms in mathematics, but that's about it) and it's a bit rough to say to a physics student, "These are our axioms" without at least some justification as to why they're reasonable.

From a mathematical perspective, of course, the whole point of axioms is to have some level at which you have to stop looking for proofs and instead try to find contradictions or counter examples. From a physical point of view at an undergrad level, a bit of justification helps, even if it's mostly Lies-To-Students, as Pratchett would put it.

Kane

 

[cepheid]

I don't know QM and so have no idea what the hell you guys are talking about. But, I have a question (being an undergrad student), about the last post. Is it really "lies to students", or just a simple confirmation that whatever the bigwigs set to be their axioms are actually compatible/in accordance with basic physics that came before? Because if those "justifications" are just BS, and we're actually being lied to, then that's disillusioning...

 

[reilly]

Let's be clear: physics is an empirical science. In fact, there are no derivations for Newton's Laws, nor for Maxwell's Eq.s, nor for the Schrödinger Eq. But there are plenty of phenomological arguments for support. Yes, there are lagrangians for E&M and QM, but they are derived from a "phenomonological point of view, and are extensively discussed in many textbooks. We keep 'em, because they work. Ultimately you pays your money and takes your chances -- the various basic equations of physics come from inspired creativity and sharp intuition.

Physicists trying to build a formal, axiomatic edifice for QM and physics have made valuable contributions -- the CPT Thrm, the idea of inequivalent vacuums, and a solid foundation for scattering theory (S Matrix) within the context of field theory. Their non-axiomatic collegues have had great success with solids, molecules, atoms and particles, with the stuff of nature.

Regards,
Reilly Atkinson

 

[Kane O'Donnell]

Is it really "lies to students", or just a simple confirmation that whatever the bigwigs set to be their axioms are actually compatible/in accordance with basic physics that came before? Because if those "justifications" are just BS, and we're actually being lied to, then that's disillusioning...

You've taken me a bit out of context there. By "Lies-To-Students", I'm making a direct reference to Terry Pratchett's book "The Science of Discworld". He and his co-authors make the point that there are lot of things we tell students/children/parents/non-scientists/etc that are not *precisely* true but are close enough so that:

(a) It doesn't take too much to 'upgrade' their knowledge to the next level, and

(b) It isn't so wrong as to totally send them down the wrong path.

The ideas behind axiomatics are very interesting. Let's just compare mathematical axiomatics and physical axiomatics for a minute. For example, one of simplest yet most powerful set of axioms is those of a group, which I will list here. A group $$(G, \times)$$ is a set G with a binary operation $$\times: G\times G\to G$$ such that:

(Group Axiom 1) $$(a\times b)\times c = a\times(b\times c)$$ for all $$a, b, c \in G$$. (Associativity)

(Group Axiom 2) There exists an element $$e \in G$$ such that $$e\times g = g\times e = g$$ for all $$g\in G$$. (Existence of an Identity)

(Group Axiom 3) For each $$g\in G$$ there exists an element $$g^{-1} \in G$$ such that $$gg^{-1} = g^{-1}g = e$$. (Existence of Inverses)

(Note that there is a slightly different set of group axioms also used that only require that the group have left inverses and a left identity or right inverses and a right identity, but in that case the double sidedness of inverses and the identity are simple proofs)

These axioms can be used to derive a large number of properties of groups. However, notice that in mathematics, one can define an object by axioms (provided they're self-consistent) and then the task is to find objects that satisfy those axioms. For example, the real numbers $$\mathbb{R}$$ are a group with the binary operation $$+$$, as are the set of symmetries of an equilateral triangle with the binary operation of composition, and so on. There are lots of things that behave like a group, so anything we can prove about groups applies to all of those things.

In Physics, however, we're restricted by the fact that we're trying to describe objects that exist whether we invent them or not, so we have to work the other way. We conduct experiments to see what objects have what properties, what properties objects share, which things are universal, etc. Then we try and write down a fundamental set of rules that the universe obeys and which give rise to all the phenomena we are able to observe experimentally.

In mathematics, it is impossible to write down a finite set of axioms that can be used to prove or disprove every possible mathematical statement (this is a very rough version of Godel's incompleteness theorem). Instead, you have to investigate each area of mathematics bit by bit. This doesn't mean we don't find connections (in fact, there are plenty of connections between all branches of mathematics) but we can't ever claim that we have "discovered" all of mathematics, simply because there are always statements we can't prove or disprove using our current 'toolbox'.

In Physics, we have much the same problem except we say that we can only write down explanations for phenomena we know about and there might always be phenomena that occur that lie outside our range of experiments. For those phenomena, we can't ethically or philosophically claim to have an explanation unless we can somehow find a way to test it.

The question of whether Physics can be axiomatised was one of Hilbert's 23 problems, I believe, and remains unanswered. We have certainly made some areas of physics axiomatic, for example quantum mechanics, electromagnetism and I would guess that gravitational theory probably has some fundamental axioms but I don't know enough about it to comment. However, we are always searching for *new* axioms that explain the old axioms. Why? Because axioms are statements taken as true without proof, and the very idea makes a scientist's skin itch a bit.

This is sort of similar to how in mathematics, to understand the group axioms you need to know about set theory, which is itself described by axioms. The set theory axioms are usually taken as the fundamental axioms of mathematics. We don't have an equivalent set of axioms for physics yet, because not many physicists believe that the ones we have at the moment are beyond explanation. If you can derive them, they aren't axioms anymore, see? So we keep looking deeper until we really can't claim to be able to go further.

So yeah, our physical axioms are 'fitted to the data' in a certain sense, but that's because they have to be. The real test of a set of axioms is whether the theory that comes out of them checks out with experiment. In the old days this didn't take much, because experiments were very limited in their scope. These days, it takes a hellofalot to fool an experimental physicist, their entire job involves crushing theorists, they are very good at it, and there isn't much room for crackpottery. It is a testimony to the outstanding accuracy of quantum electrodynamics that it has *never* been experimentally contradicted with those chaps poking at it all day long for 50 odd years...

Cheerio!

Kane O'Donnell

 

[Kane O'Donnell]

Just another point to note: When I was in second year, our QM lecturer told us we would now learn what was "really" going on in the theory. We protested and said we'd been told that the year before, and he said, "Yes, you were, and last year you *did* learn what was really going on, and so will you this year, and next year they'll tell you the whole truth again". He said this with a very cheeky grin on his face. We learned very different things in all three years.

Such is the nature of Lies-To-Students.

Kane

(who wasn't mislead, just held back until the waters had subsided enough so that he wouldn't drown on the first attempt)

 

[marlon]

Well,Marlon,i guess your idea works iff you postulate the (quantum spinless nonrelativistic) free particle's wavefunction and then u'd still have to derive the equation which has the advantage to be useful in any occasions,while the plane wave,not...

Wrong...

This approach works for any kind of wavefunction...all such functions obey the Schrödinger equantion, so what you state here cannot be true...

regards
marlon

 

[dextercioby]

What do you mean,that u can use the plane waves without getting Schroedinger's equation??

then u'd still have to derive the equation which has the advantage to be useful in any occasions,while the plane wave,not...

What is wrong there??


Daniel.

 

[marlon]

What do you mean,that u can use the plane waves without getting Schroedinger's equation??


What is wrong there??


Daniel.

I think you are missing my point here. All i am saying is that you can apply the E=mc² argument to derive (READ : prove) the actual Schrödinger equation by using the most general form for a wavefunction : ie superposition of wave-packets...that's all. The system to do so, i explained in my first post with the derivatives. I never said anything like you say on plane-waves...Ofcourse you will always need the Schrödinger-equation. It is the very basis of QM


marlon

 

[dextercioby]

So there was nothing wrong with what i had previously stated...Then why that "favorite" word of yours (and mine,too ) "wrong"...??

Daniel.

 

[marlon]

Well, the thing that disturbed me a bit is that you say "leave the Schrödinger equation" with the postulates of QM. This makes it sound like the Schrödinger equation just fell out of the sky. Indeed, it can be seen as a postulate but you got to be able to tell why this equations has the structure that is has ...you know, why a second derivative WTR positions and one derivative WTR time and why are they equal. That is all that i wanted to answer here

marlon

Besides, you talk about an alternative approach...I don't see what this justification (of the STRUCTURE of the Schrödinger equation) is the alternative of ?? How can you prove this otherwise? Enlighten me please

 

[dextercioby]

The second order derivatives come via
$$H=\frac{\vec{p}^{2}}{2m} +V(\vec{r})$$
+second postulate+and the particular choise of representation...

However the equation is much more general and describes the time evolution of any Q system,no matter whether the Hamiltonian is quadratic in momenta...

Daniel.

 

[marlon]

I still don't see how that explains the actual Schrödinger equation.

marlon

 

[dextercioby]

Hold on,what are you trying to say?The SE is indeed postulated in the general form it has (Dirac formalism).Sakurai shows how it can be derived...As a consequence of other principles...
The particularity of Sakurai's book is the absence of AXIOMATIZATION...
I simply rejected the way of deriving the SE from the wavefunctions...It's the other way around,logically correct.

Daniel.

 

[marlon]

Hold on,what are you trying to say?The SE is indeed postulated in the general form it has (Dirac formalism).Sakurai shows how it can be derived...As a consequence of other principles...
The particularity of Sakurai's book is the absence of AXIOMATIZATION...

What are these other principles. Can you state them for me, please

I simply rejected the way of deriving the SE from the wavefunctions...It's the other way around,logically correct.

Daniel.

Logically, correct. I would diagree, though i don't understand what you are saying here. What is unlogical about deriving the SE from the wavefunctions. In my opinion, it is the best and most logical way to explain why the SE has the particular structure that it has

marlon

 

[dextercioby]

1.Conservation of sum of probabilities (the sum of the square modulus of the Fourier coefficients) with time evolution (similar,the conservation of normalization in time) => unitary operator of time evolution => inifinitesimal time evolution operator=>=>(on the basis of Stone's theorem) pops up the self-adjoint generator of infinitesimal time translations=> (analogy with CM) =>Hamiltonian (the operator) as the generator of time translations.Then the ODE for the time evolution operator and then SE for state vectors...

I'm leaving the principles' identification to you...

2.How on Earth would u find the wave-functions without GETTING THE SE FROM SOMEWHERE (either by postulating it,or as Sakurai "digs" it)?

Daniel.

P.S.What do you mean "particular structure"...It's DAMN GENERAL AND ABSTRACT,at least in the Dirac formulation and no choise of representation whatsoever...

 

[marlon]

1.Conservation of sum of probabilities (the sum of the square modulus of the Fourier coefficients) with time evolution (similar,the conservation of normalization in time) => unitary operator of time evolution => inifinitesimal time evolution operator=>=>(on the basis of Stone's theorem) pops up the self-adjoint generator of infinitesimal time translations=> (analogy with CM) =>Hamiltonian (the operator) as the generator of time translations.Then the ODE for the time evolution operator and then SE for state vectors...

Sighs, this is all very true but (as i thought) you still are not getting what we are talking about (this is proven by your very last question in your post). What you say here does not PROVE the SE...

2.How on Earth would u find the wave-functions without GETTING THE SE FROM SOMEWHERE (either by postulating it,or as Sakurai "digs" it)?

Very easy,...the wave function (or at least the concept thereof) comes from the double slit experiment. The general form of a wavefunction (superposition and Fouriertransforms for goinf from coordinate space to momentumspace ...) is used to derive the SE...

You are confusing this with the actual solutions of the SE in various cases. Those wavefunctions indeed come after solving the SE, but again that is not the point. Every wavefunction is (in most general case) such a superposition so there really is no problem. I think you are again persuing a useless crusade just to prove me wrong. I strongly urge you to check out a QM-textbook and just look at how the SE is introduced. I have heard people whinning about the fact that this way is just the introductory way if introducing the SE but there ain't no other way...C'est tout...

P.S.What do you mean "particular structure"...It's DAMN GENERAL AND ABSTRACT,at least in the Dirac formulation and no choise of representation whatsoever...

This clearly proves you don't know what we are talking about. I am just referring to the actual SE that is all, i am sure you know its equation.

No need to bring such vague terms as Dirac-formulation and representation of the SE...we are ONLY talking about the SE

marlon

 

[Kalimaa23]

Reasonable assumptions concerning the Schroedinger equation include the de Broglie (momentum-displacement) and Einstein (energy-frequency) postulates, along with the conditions of linearity, constant potential for free particles and reliance on the Hamiltonian energy formulation.

Yup, this is how Schrödinger did it. It's also how I was taught it. I believe everyone should see that it came from SOMEWHERE, instead of having it appear out of thin air (like in Griffiths for example).

Remember, all that fancy operator stuff came quite a bit later, historically, as did Born's probabilistic interpretation.

 

[dextercioby]

Very easy,...the wave function (or at least the concept thereof) comes from the double slit experiment.

How do we get the mathematical expression for the wavefunction?Analogy with optics + DE BROGLIE's CONDITION...without the latter,we wouldn't make any difference between light waves & matter waves...

. I strongly urge you to check out a QM-textbook and just look at how the SE is introduced.

I did.Sakurai's example is the most interesting.I would still like more what i was taught in school.Postulating it and worrying not about its origin,but about its consequences...

I have heard people whinning about the fact that this way is just the introductory way if introducing the SE but there ain't no other way...C'est tout...

That's what YOU think...It doesn't necessary make it correct,not to mention unique...

This clearly proves you don't know what we are talking about. I am just referring to the actual SE that is all, i am sure you know its equation.

So you were talking about this:
$$i\hbar \frac{\partial \Psi(\vec{r},t)}{\partial t} =-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi(\vec{r},t) +V(\vec{r},t)\Psi(\vec{r},t)$$

In that case,i wasn't talking about that...I was stalking about the one that's postulated...Not derivable...But i know what I'm talking about...

No need to bring such vague terms as Dirac-formulation and representation of the SE...we are ONLY talking about the SE

They are not vague...Maybe to you they are...

Daniel.

P.S.The postulated equation should not be called SE,but Dirac's equation...