>Rock Brentwood schrieb: >> On Aug 8, 2:39 pm, kushal <atmabo...@gmail.com> wrote: >>> What are the characteristics of the Schrodinger's Equation? Are they >>> the single particle trajectories that we get from Newton's Force laws?
>> It's a parabolic differential equation, same form as the heat equation. >The Schoedinger equation is _not_ a parabolic equation, but >a hyperbolic equation.
This obviously includes the Schroedinger equation d^2(psi)/dx^2 = k d(psi)/dt; k = -2mi/h-bar and heat equation as examples.
>Parabolic equations cannot have plane wave solutions
... which however has nothing to do with the sense in which Schroedinger equation has wave solutions; and is therefore irrelevant.
>> Equivalently, it's a hyperbolic differential equation in one >> higher dimension with a first-order differential constraint. >parabolic and hyperbolic is never equivalent.
Likewise, this has nothing to do with what you're reply you quoted, nor anything to do with what you're rpelying to, and is therefore irrelevant.
A parabolic differential equation is always equivalent to the combination of a hyperbolic differential equation in one higher dimension plus a linear differential constraint.
In particular, dY/dt = d^2Y/dx^2 can be equivalently expressed as d^Z/dudt = d^2Z/dx^2 dZ/du = m Z with the equivalence given by Z = Y exp(imu).
>> Rock Brentwood schrieb: >>> On Aug 8, 2:39 pm, kushal <atmabo...@gmail.com> wrote: >>>> What are the characteristics of the Schrodinger's Equation? Are they >>>> the single particle trajectories that we get from Newton's Force laws? >>> It's a parabolic differential equation, same form as the heat equation.
>> The Schoedinger equation is _not_ a parabolic equation, but >> a hyperbolic equation.
> This obviously includes the Schroedinger equation > d^2(psi)/dx^2 = k d(psi)/dt; k = -2mi/h-bar > and heat equation as examples.
You don't seem to see that implicit in the Wikipedia article is the unstated assumption that all coefficients are real. The classification given in Wikipedia makes sense only if A,B,C are real. To see this, look at the bottom links to the hyperbolic/elliptic case, where one requires B^2-4AC>0/<0.
For complex coefficients, a corresponding classification must look differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>> Parabolic equations cannot have plane wave solutions
> ... which however has nothing to do with the sense in which > Schroedinger equation has wave solutions; and is therefore irrelevant.
There are no different senses how partial differential equations can have plane wave solutions.
Arnold Neumaier wrote: > Rock Brentwood wrote: ... > ... The classification > given in Wikipedia makes sense only if A,B,C are real. To see this, > look at the bottom links to the hyperbolic/elliptic case, where one > requires B^2-4AC>0/<0.
> For complex coefficients, a corresponding classification must look > differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>>> Parabolic equations cannot have plane wave solutions
>> ... which however has nothing to do with the sense in which >> Schroedinger equation has wave solutions; and is therefore irrelevant.
> There are no different senses how partial differential equations can > have plane wave solutions.
Do you mean that a plane-wave solution is only described by
exp( i k.r - i w t )
with k a real three-vector and w a real number?
If you allow an imaginary part for w then also a pure imaginary w should be allowed but then the heat equation (parabolic) actually does have a plane wave solution.
If you don't allow an imaginary part I think it conflicts with the way "plane wave" is used in solving problems in physics, and also with wikipedia (now that someone mentioned it anyway :-)
So are there different senses in which a wave can be plane?
Jos Bergervoet wrote: > Arnold Neumaier wrote: >> Rock Brentwood wrote: > ... >> ... The classification >> given in Wikipedia makes sense only if A,B,C are real. To see this, >> look at the bottom links to the hyperbolic/elliptic case, where one >> requires B^2-4AC>0/<0.
>> For complex coefficients, a corresponding classification must look >> differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>>>> Parabolic equations cannot have plane wave solutions >>> ... which however has nothing to do with the sense in which >>> Schroedinger equation has wave solutions; and is therefore irrelevant. >> There are no different senses how partial differential equations can >> have plane wave solutions.
> Do you mean that a plane-wave solution is only described by
> exp( i k.r - i w t )
> with k a real three-vector and w a real number?
Yes. Would anyone call the solution with w=i and k=0 a plane wave?
Solutions with Im w < 0 and sufficiently small make physical sense in fluid flow problems with damping, but one would not call them plane waves but damped waves.
> If you allow an imaginary part for w then also a pure imaginary w > should be allowed but then the heat equation (parabolic) actually > does have a plane wave solution.
But this is not the usual conception of a plane wave. Parabolic equations (with the established definition) such as the heat equation are well-known not to give rise to waves in the usual sense of the word.
> If you don't allow an imaginary part I think it conflicts with the > way "plane wave" is used in solving problems in physics, and also > with wikipedia (now that someone mentioned it anyway :-)
Wikipedia makes the realness assumption implicitly without finding the need to stress it. But that it is present in http://en.wikipedia.org/wiki/Plane_wave shows in the formula for the real part, which would have to be different if omega were complex.
Arnold Neumaier wrote: > Jos Bergervoet wrote: >> Arnold Neumaier wrote: >>> Rock Brentwood wrote: ... >>> ... The classification >>> given in Wikipedia makes sense only if A,B,C are real. To see this, >>> look at the bottom links to the hyperbolic/elliptic case, where one >>> requires B^2-4AC>0/<0.
>>> For complex coefficients, a corresponding classification must look >>> differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>>>>> Parabolic equations cannot have plane wave solutions >>>> ... which however has nothing to do with the sense in which >>>> Schroedinger equation has wave solutions; and is therefore irrelevant. >>> There are no different senses how partial differential equations can >>> have plane wave solutions.
>> Do you mean that a plane-wave solution is only described by
>> exp( i k.r - i w t )
>> with k a real three-vector and w a real number?
> Yes. Would anyone call the solution with w=i and k=0 a plane wave?
> Solutions with Im w < 0 and sufficiently small make physical sense > in fluid flow problems with damping, but one would not call them > plane waves but damped waves.
Where do we find the authoritative definition of these things?
>> If you allow an imaginary part for w then also a pure imaginary w >> should be allowed but then the heat equation (parabolic) actually >> does have a plane wave solution.
> But this is not the usual conception of a plane wave.
Why not? A plane wave with some arbitrary exponential time dependence seems to be extremely "usual"..
> Parabolic equations (with the established definition) such as the > heat equation are well-known not to give rise to waves in the usual > sense of the word.
>> If you don't allow an imaginary part I think it conflicts with the >> way "plane wave" is used in solving problems in physics, and also >> with wikipedia (now that someone mentioned it anyway :-)
> Wikipedia makes the realness assumption implicitly without finding the > need to stress it.
The definition there requires "surfaces of constant phase are infinite parallel planes". No real omega needed there..
> .. But that it is present in > http://en.wikipedia.org/wiki/Plane_wave > shows in the formula for the real part, which would have to be > different if omega were complex.
Many things in Wikipedia would have to be different to be correct (although not too many, I admit) so I still don't see this as a final answer.
If it really is part of the definition I would want to see it stated in the definition. Not implicitely elsewhere.
Jos Bergervoet wrote: > Arnold Neumaier wrote: >> Jos Bergervoet wrote: >>> Arnold Neumaier wrote: >>>> Rock Brentwood wrote: > ... >>>> ... The classification >>>> given in Wikipedia makes sense only if A,B,C are real. To see this, >>>> look at the bottom links to the hyperbolic/elliptic case, where one >>>> requires B^2-4AC>0/<0.
>>>> For complex coefficients, a corresponding classification must look >>>> differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>>>>>> Parabolic equations cannot have plane wave solutions >>>>> ... which however has nothing to do with the sense in which >>>>> Schroedinger equation has wave solutions; and is therefore irrelevant. >>>> There are no different senses how partial differential equations can >>>> have plane wave solutions. >>> Do you mean that a plane-wave solution is only described by
>>> exp( i k.r - i w t )
>>> with k a real three-vector and w a real number? >> Yes. Would anyone call the solution with w=i and k=0 a plane wave?
>> Solutions with Im w < 0 and sufficiently small make physical sense >> in fluid flow problems with damping, but one would not call them >> plane waves but damped waves.
> Where do we find the authoritative definition of these things?
I haven't looked for an online source. The classification of PDEs into elliptic, parabolic and hyperbolic (at a particular point of a solution - in most generality the type may deprend on the solution and the point), due to Hilbert and Courant, I think, can be found in many textbooks on PDEs. But these commonly also assume real coefficients (and often implicitly).
For example, Logan, An introduction to nonlinear PDEs, writes on p.5: ``There are three fundamental types of equations: those that govern diffusion process, those that govern wave propagation, and those that govern equilibrium phenomena. Equations of mixed types also occur.'' Then They discuss 2nd order equations without first order terms and with a linear second-order part au_xx+2bu_xt+cu_tt, and say ``Classification is made on the basis of the combination of the second order derivatives in the equation. If we define the discriminant Delta by Delta=b^2-ac then [the equation] is hyperbolic if Delta>0, parabolic if Delta=0, elliptic if Delta<0.'' If the coefficients were allowed to nonreal (an implicit assumption throughout the book, but nowhere stated explicitly), this would not be a classification!
The intent of the classification is to tell quickly by a simple criterion which kind of solutions to expect, and hence which kind of methods are needed to study the solutuions. This determines the category of the equation. In case of real coefficients, it turns out that the answer can be given in terms of the discriminant. In the complex coefficient case, one must instead analyze an associated eigenvalue problem that determines the microlocal structure of the principal part of the equation.
>>> If you allow an imaginary part for w then also a pure imaginary w >>> should be allowed but then the heat equation (parabolic) actually >>> does have a plane wave solution. >> But this is not the usual conception of a plane wave.
> Why not? A plane wave with some arbitrary exponential time > dependence seems to be extremely "usual"..
usual, yes, but not usually called a plane wave. Else the term loses its differentiating meaning.
Plane waves are commonly meant to be periodic in time, with a real period. No textbook that I know of calls a spatial phenomenon with a time law of exp(- ut) (u real) a wave.
>> Parabolic equations (with the established definition) such as the >> heat equation are well-known not to give rise to waves in the usual >> sense of the word.
>>> If you don't allow an imaginary part I think it conflicts with the >>> way "plane wave" is used in solving problems in physics, and also >>> with wikipedia (now that someone mentioned it anyway :-) >> Wikipedia makes the realness assumption implicitly without finding the >> need to stress it.
> The definition there requires "surfaces of constant phase are > infinite parallel planes". No real omega needed there..
>> .. But that it is present in >> http://en.wikipedia.org/wiki/Plane_wave >> shows in the formula for the real part, which would have to be >> different if omega were complex.
> Many things in Wikipedia would have to be different to be correct > (although not too many, I admit) so I still don't see this as a > final answer.
Wikipedia is not the arbiter but just gives a view of the matter. There is consensus among PDE researchers about this question, and this determines what is correct.
> If it really is part of the definition I would want to see it > stated in the definition. Not implicitely elsewhere.
Then you must start a discussion about that point on the wikipedia page, referring to the present discussion here at s.p.r. Probably things will be cleared up soon by those caring about formal accuracy. The writers of the page probably simply failed to imagine that someone would read the article with complex constants in mind.
Arnold Neumaier wrote: > Jos Bergervoet wrote: >> Arnold Neumaier wrote: >>> Jos Bergervoet wrote: >>>> Arnold Neumaier wrote: >>>>> Rock Brentwood wrote: ... >>>>> ... The classification >>>>> given in Wikipedia makes sense only if A,B,C are real. To see this, >>>>> look at the bottom links to the hyperbolic/elliptic case, where one >>>>> requires B^2-4AC>0/<0.
>>>>> For complex coefficients, a corresponding classification must look >>>>> differently, and B^2-4AC=0 no longer characterizes the parabolic case.
>>>>>>> Parabolic equations cannot have plane wave solutions >>>>>> ... which however has nothing to do with the sense in which >>>>>> Schroedinger equation has wave solutions; and is therefore irrelevant. >>>>> There are no different senses how partial differential equations can >>>>> have plane wave solutions. >>>> Do you mean that a plane-wave solution is only described by
>>>> exp( i k.r - i w t )
>>>> with k a real three-vector and w a real number? >>> Yes. Would anyone call the solution with w=i and k=0 a plane wave?
>>> Solutions with Im w < 0 and sufficiently small make physical sense >>> in fluid flow problems with damping, but one would not call them >>> plane waves but damped waves.
>> Where do we find the authoritative definition of these things?
> I haven't looked for an online source.
That wouldn't be necessary. It just has to be authoritative!
> ... The classification of PDEs into > elliptic, parabolic and hyperbolic (at a particular point of a solution > - in most generality the type may deprend on the solution and the > point), due to Hilbert and Courant, I think, can be found in many > textbooks on PDEs. But these commonly also assume real coefficients > (and often implicitly).
I think an authoritative definition which is not "implicitly" would have my preference..
And the one single question I was mainly interesting in here (and asking about) was the definition of a "plane wave".
> For example, Logan, An introduction to nonlinear PDEs, writes on p.5: > ``There are three fundamental types of equations: those that govern > diffusion process, those that govern wave propagation, and those that > govern equilibrium phenomena. Equations of mixed types also occur.'' > Then They discuss 2nd order equations without first order terms and with > a linear second-order part au_xx+2bu_xt+cu_tt, and say > ``Classification is made on the basis of the combination of the second > order derivatives in the equation. If we define the discriminant Delta > by Delta=b^2-ac then [the equation] is hyperbolic if Delta>0, parabolic > if Delta=0, elliptic if Delta<0.'' If the coefficients were allowed to > nonreal (an implicit assumption throughout the book, but nowhere stated > explicitly), this would not be a classification!
He's not talking about the definition of "plane wave", it seems.
> The intent of the classification is to tell quickly by a simple > criterion which kind of solutions to expect, and hence which kind of > methods are needed to study the solutuions.
The real sinusoidal time dependence is usually studied by the same methods as the damped exponential, so no distinction seems to be needed if that is (implicitely) what governs our definition.
> ... This determines the category > of the equation. In case of real coefficients, it turns out that the > answer can be given in terms of the discriminant.
But what is "the answer"? He didn't even raise the question about the definition of a plane wave.
> In the complex > coefficient case, one must instead analyze an associated eigenvalue > problem that determines the microlocal structure of the principal part > of the equation.
Thanks, Nice stuff! But in there I see no ruling against using the name "plane wave" if there is damping, as long as the constant phase planes *in space* are parallel planes.
>>>> If you allow an imaginary part for w then also a pure imaginary w >>>> should be allowed but then the heat equation (parabolic) actually >>>> does have a plane wave solution. >>> But this is not the usual conception of a plane wave.
>> Why not? A plane wave with some arbitrary exponential time >> dependence seems to be extremely "usual"..
> usual, yes, but not usually called a plane wave. > Else the term loses its differentiating meaning.
Now we are (finally) getting to the point! Having parallel planes as equal-phase surfaces in space is a well-defined differentiating meaning. How do you mean that it is lost?!
> Plane waves are commonly meant to be periodic in time, with a > real period. No textbook that I know of calls a spatial phenomenon > with a time law of exp(- ut) (u real) a wave.
I would be interested in a textbook that you *do* know of that gives the definition of plane wave and explicitely confirms your point. (Even then it wouldn't be clear whether just one textbook is authoritative..)
>>> Parabolic equations (with the established definition) such as the >>> heat equation are well-known not to give rise to waves in the usual >>> sense of the word.
>>>> If you don't allow an imaginary part I think it conflicts with the >>>> way "plane wave" is used in solving problems in physics, and also >>>> with wikipedia (now that someone mentioned it anyway :-) >>> Wikipedia makes the realness assumption implicitly without finding the >>> need to stress it.
>> The definition there requires "surfaces of constant phase are >> infinite parallel planes". No real omega needed there..
>>> .. But that it is present in >>> http://en.wikipedia.org/wiki/Plane_wave >>> shows in the formula for the real part, which would have to be >>> different if omega were complex.
>> Many things in Wikipedia would have to be different to be correct >> (although not too many, I admit) so I still don't see this as a >> final answer.
> Wikipedia is not the arbiter but just gives a view of the matter. > There is consensus among PDE researchers about this question, and > this determines what is correct.
>> If it really is part of the definition I would want to see it >> stated in the definition. Not implicitely elsewhere.
> Then you must start a discussion about that point on the wikipedia > page, referring to the present discussion here at s.p.r. > Probably things will be cleared up soon by those caring about > formal accuracy. The writers of the page probably simply failed to > imagine that someone would read the article with complex constants > in mind.
Hmm.. I would have hoped that since the days of Scipione del Ferro people had become more flexible!
Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: > > If it really is part of the definition I would want to see it > > stated in the definition. Not implicitely elsewhere. > Then you must start a discussion about that point on the wikipedia > page, referring to the present discussion here at s.p.r. > Probably things will be cleared up soon by those caring about > formal accuracy. The writers of the page probably simply failed to > imagine that someone would read the article with complex constants > in mind.
Oh, for goodness sakes: it's wikipedia, the "encyclopedia anyone can edit". Since you seem to care, and know what you are talking about, why not take a minute or two to fix it yourself?
As regarding the definition of plane waves, I agree (from a more practical point of view than others) that allowing imaginary parts to omega or k will likely get you into trouble. This is because plane wave solutions are often used as a basis on which to solve a more general problem. And (e.g.) if I then propagate a wave forward in time, an imaginary omega will mean the normalisation of my basis changes, which makes things unnecessarily messy. The saem holds for imaginary parts to k when spatially propagating.
===== Moderator's note ===============
Isn't this a discussion about semantics only? I'd call the Schroedinger equation parabolic despite the fact that it contains an imaginary coefficient. After all you can do a Wick rotation and get a real parabolic equation (type of heat-conduction equation). As long as one does the right analytic continuations it's all fine (at least from a naive phycicist's point of view).
-- ---------------------------------+--------------------------------- Dr. Paul Kinsler Blackett Laboratory (Photonics) (ph) +44-20-759-47734 (fax) 47714 Imperial College London, Dr.Paul.Kins...@physics.org SW7 2AZ, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/
> Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: > > > If it really is part of the definition I would want to see it > > > stated in the definition. Not implicitely elsewhere.
> > Then you must start a discussion about that point on the wikipedia > > page, referring to the present discussion here at s.p.r. > > Probably things will be cleared up soon by those caring about > > formal accuracy. The writers of the page probably simply failed to > > imagine that someone would read the article with complex constants > > in mind.
> Oh, for goodness sakes: it's wikipedia, the "encyclopedia anyone > can edit". Since you seem to care, and know what you are talking > about, why not take a minute or two to fix it yourself?
> As regarding the definition of plane waves, I agree (from a more > practical point of view than others) that allowing imaginary > parts to omega or k will likely get you into trouble. This is > because plane wave solutions are often used as a basis on which > to solve a more general problem. And (e.g.) if I then propagate > a wave forward in time, an imaginary omega will mean the normalisation > of my basis changes, which makes things unnecessarily messy. > The saem holds for imaginary parts to k when spatially propagating.
> ===== Moderator's note ===============
> Isn't this a discussion about semantics only?
The plane wave question was meant to be just that! It is precisely the semantics I was interested in. Whether it's "plane wave", or "Coulomb gauge" or "off mass-shell", we just need to know what things mean..
Or if there is no standardized definition we should admit it.
p.kins...@ic.ac.uk wrote: > Arnold Neumaier <Arnold.Neuma...@univie.ac.at> wrote: >>> If it really is part of the definition I would want to see it >>> stated in the definition. Not implicitely elsewhere.
>> Then you must start a discussion about that point on the wikipedia >> page, referring to the present discussion here at s.p.r. >> Probably things will be cleared up soon by those caring about >> formal accuracy. The writers of the page probably simply failed to >> imagine that someone would read the article with complex constants >> in mind.
> Oh, for goodness sakes: it's wikipedia, the "encyclopedia anyone > can edit". Since you seem to care, and know what you are talking > about, why not take a minute or two to fix it yourself?
I know the literature and can read between the lines, thus I don't need a perfect wikipedia. In this sense, I do not care.
It would be a full time job to improve all inacccuracies that Wikipedia has in my areas of expertise. And my time is quite limited.
But if Jos Bergervoet cares and wants to have a more authoritative answer to the problem than I can give, he can and should start a discussion there.
For I consulted several books and nowhere found a definition that is general, precise, and excludes complex frequencies explicitly. It is usually clear only from the context that the frequency of a plane must be real. For example, it is obvious that no book at all considers an exponentially decaying solution of the heat equation as a plane wave.
> >> Rock Brentwood schrieb: > >>> On Aug 8, 2:39 pm, kushal <atmabo...@gmail.com> wrote: > >>>> What are the characteristics of the Schrodinger's Equation? Are they > >>>> the single particle trajectories that we get from Newton's Force laws? > >>> It's a parabolic differential equation, same form as the heat equation.
> >> The Schoedinger equation is _not_ a parabolic equation, but > >> a hyperbolic equation.
> > This obviously includes the Schroedinger equation > > d^2(psi)/dx^2 = k d(psi)/dt; k = -2mi/h-bar > > and heat equation as examples.
> You don't seem to see that implicit in the Wikipedia article is the > unstated assumption that all coefficients are real. The classification > given in Wikipedia makes sense only if A,B,C are real. To see this, > look at the bottom links to the hyperbolic/elliptic case, where one > requires B^2-4AC>0/<0.
> For complex coefficients, a corresponding classification must look > differently, and B^2-4AC=0 no longer characterizes the parabolic case.
> >> Parabolic equations cannot have plane wave solutions
> > ... which however has nothing to do with the sense in which > > Schroedinger equation has wave solutions; and is therefore irrelevant.
> There are no different senses how partial differential equations can > have plane wave solutions.
> Arnold Neumaier
Neumaier is correct, and in particular he's physically correct, because the S eqn is a limit of the Dirac eqn through the Pauli eqn (2- component neutrino) and the "i" survives from there. It's not a parabolic equation because of the way initial data is posited, and that's also true physically.
It is a kind of diffusion equation with an imaginary coefficient of diffusion, which is however physically silly There's a big difference between a clock and a shock absorber.
It does however bring up an interesting subtlety of spacetime - propagation is built into geometry by the indefinite metric and that survives both limiting processes (Dirac->Pauli->Schr) so these "diffusive waves" are the leftover vestige of spacetime.
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