Exploring the Relationship Between Schroedinger and Bohm's Quantum Mechanics

[Rothiemurchus]

The schroedinger equation is an inspired postulate - it cannot be derived
from fundamental principles.The pilot wave of David Bohm's version of quantum mechanics is also postulated.Has anyone tried to derive the
pilot wave and schroedinger equation from physical laws?

 

[ZapperZ]

The schroedinger equation is an inspired postulate - it cannot be derived
from fundamental principles.

Oh really?

There is a reason why the Schrodinger equation is often called the Hamiltonian of the system. Now, unless you are also claiming that the Hamiltonian/Lagrangian principle of classical mechanics are also an "inspired postulate" (so then why pick on the Schrodinger equation only?), then maybe you should look at the connection between the two and how Schrodinger actually derived his infamous equation[1].

There are several postulates in QM. The Schrodinger equation isn't one of them.

Zz.

[1] E. Schrodinger, Phys. Rev. v.28, p.1049 (1926); or look here: http://fangio.magnet.fsu.edu/~vlad/pr100/100yrs/html/chap/fs2_14001.htm

 

[marlon]

This is a very nice reference ZapperZ.

It is always nice to see how very well established theories like QM are constructed from "scratch". I always have been interested in the original documents of people like Schrödinger, Dirac, Einstein and so on.

Isn"t there some link available where we could see the original version of the incorporation of magnetic monopoles into the Maxwell-equations...by the hand of Dirac

regards
marlon

 

[humanino]

Zz ! That is the greatest link ever ! Why did not I ask this before, I can only blame myself. Thank you so much Zz.

 

[ZapperZ]

Hey, I'm glad you two liked it. If you notice, that link has several other landmark papers in QM. I have been collecting a lot of legitimate physics links over the years, so it doesn't hurt to ask. You might just get it! :)

Zz.

Addendum: If one is questioning something, it is ALWAYS a good idea to see the origin of that "thing" and how it came into being.

 

[Rothiemurchus]

The Lagrangian, kinetic energy - potential energy - turns up in many classical situations but why, as someone else on this forum has said,
should that energy difference be such an important part of nature, which it clearly is?
Of all the possible combinations such as KE + (PE) ^2, ( KE)^3 + (PE) ^3 etc. why
KE - PE?

 

[humanino]

If you add two things, make sure they have the same units. You cannot add GeV and GeV^2, that is meaningless.

Why the difference of KE and PE ? The sum is already the total energy. It turns out that, you can perform a mathematical operation called a Legendre transformation... that is not the explanation you want, is it ?

The action is the number nature optimize during a process. Imagine falling in a gravitational potential : to minimize the action, you want to "need a minimum of KE" while "use as much PE as possible". Does that make sens ?

 

[ZapperZ]

The Lagrangian, kinetic energy - potential energy - turns up in many classical situations but why, as someone else on this forum has said,
should that energy difference be such an important part of nature, which it clearly is?
Of all the possible combinations such as KE + (PE) ^2, ( KE)^3 + (PE) ^3 etc. why
KE - PE?

Again, this is an example on why, if one did not learn from the ground up, things will seem to appear out of nowhere.

We have gone over this in other threads of the importance of calculus of variation, and in particular the principle of least action. This is the only means of understanding the origin of the Lagrangian/Hamiltonian approach to classical mechanics. I strongly suggest you look this up.

Zz.

 

[Rothiemurchus]

Yes it makes sense.
The action can be maximised or minimised can't it -Hamilton's principle?
Doesn't Feynman's sum over all histories minimise the action?
And is the action minimised in GR ?
But again why should nature favour maximum/minimums?

 

[ZapperZ]

Yes it makes sense.
The action can be maximised or minimised can't it -Hamilton's principle?
Doesn't Feynman's sum over all histories minimise the action?
And is the action minimised in GR ?
But again why should nature favour maximum/minimums?

For once, let's not stray too far from the tree. Do you still think the Schrodinger equation is an "inspired postulate"?

Zz.

 

[humanino]

I'm french : Pierre Louis Moreau de Maupertuis' principle one century before
Please follow Zz advice. Otherwise this site would grow indecently. Use the search function. You cannot search more than once in 5 minutes.

Yes the action is stationnary.
Yes.
Yes.
Because Nature is elegant. You can also choose to formulate the laws in terms of optimization.

--------
EDIT : I also want to add that in France, Hamilton is not recognized as the great mathematician he was. It always bothered me.

 

[Rothiemurchus]

Zapper Z:
For once, let's not stray too far from the tree. Do you still think the Schrodinger equation is an "inspired postulate"?

Rothie M:
When I studied chemistry it was derived for us!
But because I am not a physicist and I read a physics book that said it was "an inspired postulate" I thought maybe I had been mislead on my chemistry course.
However I am currently reading a 500 page book on the maths of QM and will soon be
on the page about Schrodinger's equation.
But Tom Mattson said in reply to one of my questions that the pilot wave of Bohm's theory was a "sourceless differential equation."
Is this why Bohm's theory has less credibility than standard QM?

 

[bd1976]

hmm...

Mr.Rothiemurchus, your original post was correct in a sense. Schrödinger’s equation was a kind of inspired postulate.

Well actually to clarify it wasn't so much of a postulate as a differential equation manufactured so that its solutions fit the observed measurements on qm systems. However Schrödinger’s equation was not and currently can not be derived from any known principles. How and why it arises are questions that currently have no answer.

In modern quantum theory the postulate is that each dynamical variable in quantum mechanics (position, momentum etc...) can be represented by quantum mechanical operators. But these once again mysteriously arise they cannot be derived from any know principle - they just are!

This is just one of the reasons why quantum mechanics is viewed so sceptically by so many people.

 

[humanino]

First of all, welcome bd1976.

Did you read Zz post (#2)
Zz is a professional physicist, taking on his precious time for a noble cause : communicating science. So please, read post #2.

 

[bd1976]

hmm...

Thanks for the welcome. Happened to be reading this thread and thought I'd offer a bit of help. First of all I don't claim to be an expert on qm. (thats the goal of a lifetime!) But I have taken four undergraduate courses on the subject and can say that no one can derive the wave equation based on underlying physical principals. And If you can, then you have revolutionised physics as we know it today and should put in for your rightful nobel prize!

As to Bohm theory. I don't know very much about it. From what little I can remember I think its a hidden variable theory proposed to explain some of the conceptual problems with quantum mechanics. However as I remember (this is pretty fuzzy) the theory introduces conceptual problems that are as bad as the ones its trying to solve. Amoung other things I think its non-local and has some problems when you try and integrate relaitivity into the theory. So basically the reason its not widely accepted is that its just as bad as quantum mechanics and its harder to work with!

 

[humanino]

I see your point of view : nobody knows where QM comes from.

The principles of QM cannot be derived. Yet the wave equation can be derived. Schrodinger equation can be obtained from the classical hamiltonian by using the operator representations for p and E. In turn, those can be derived in the Hilbert space. The only thing you must postulate can be for instance the correspondance between the classical Poisson bracket, and the commutation rules in QM.

Can classical mechanics be demonstrated as well ?

 

[humanino]

So basically the reason its not widely accepted is that its just as bad as quantum mechanics and its harder to work with!

Please ! Don't

 

[ZapperZ]

Mr.Rothiemurchus, your original post was correct in a sense. Schrödinger’s equation was a kind of inspired postulate.

Well actually to clarify it wasn't so much of a postulate as a differential equation manufactured so that its solutions fit the observed measurements on qm systems. However Schrödinger’s equation was not and currently can not be derived from any known principles. How and why it arises are questions that currently have no answer.

In modern quantum theory the postulate is that each dynamical variable in quantum mechanics (position, momentum etc...) can be represented by quantum mechanical operators. But these once again mysteriously arise they cannot be derived from any know principle - they just are!

This is just one of the reasons why quantum mechanics is viewed so sceptically by so many people.

I will address that last part first. During the 100 year anniversary of QM in 1999, there was a unanimous proclamation that QM is the most successful theory of mankind so far by the physics community.

So three questions: (1) who are these "so many people" who are viewing QM so sceptically; (2) are they skeptical about the INTERPRETATION of QM or the FORMALISM of QM; and (3) do these skeptical people realize that they are using the application of QM in their modern electronics every single second of the day?

I don't know where this "manufactured" accusation of QM came from. I would think a "manufactured" idea to fit a particular result would have difficulties in fitting other results, especially when they come from wildly different phenomena. How do you manufacture a result that agrees with the band theory of semiconductors used in your modern electronics to also agree with BE condensation in atomic gasses that also describes light diffraction from a single slit?

Let me repeat, the Hamiltonian/Schrodinger equation didn't come out of nowhere! Read that paper! QM has a set of postulates that requires the assignment of PHYSICAL MEANINGS to mathematical formalism. This is not unusual. Look at all of physics and you will see that at some point, there HAS to be a set of postulates that simply cannot be derived via First Principles. That is what made this physics and not mathematics.

I'm also puzzled by another point. If there ARE people who are "skeptical" of QM simply because they (incorrectly) think QM is simply manufactured to fit the results, why aren't you and these people also complaining about classical E&M? Coulomb's law, for example, is PURELY phenomenological. I dare you to open a classical E&M text and find me its derivation.

As I've repeated often in this section of PF, why are people using their prejudice and ignorance against QM and yet, they completely ignore the very same "problem" in other parts?

Zz.

 

[ZapperZ]

I see your point of view : nobody knows where QM comes from.

The principles of QM cannot be derived. Yet the wave equation can be derived. Schrodinger equation can be obtained from the classical hamiltonian by using the operator representations for p and E. In turn, those can be derived in the Hilbert space. The only thing you must postulate can be for instance the correspondance between the classical Poisson bracket, and the commutation rules in QM.

Can classical mechanics be demonstrated as well ?

That is incorrect. We KNOW where QM comes from. We know where a lot of physics came from. We however have no explanation for why certain things are that way. Physics are not meant to explain. Our understanding of our world is based on our ability to describe with accuracy how things behave.

Unlike people who work in other areas of our society who have no qualms in giving reasons (sometimes with alarming certainty that they are right) why certain things occur, we can only go by what we can test and verify. We dare not go beyond that by offering things that we simply have no business in saying. That is not a weakness, but the strength of physics and well-defined sciences that is unmatched in other areas of study. The fact that something WORKS seems to be sniffed and trivialized at.

When I used to teach intro physics in college many years ago, I came in the first day and wrote on the board "Conservation of Energy (or mass/energy)" and "Conservation of momentum (both linear and angular)".

I then told the students that throughout their undergraduate career, EVERYTHING they will be studying are nothing more than various manifestations of those two principles. Think about it. All the things you will be studying in undergraduate classical mechanics, E&M, QM, Thermo, etc, etc., are nothing more than various manifestation of those two principles!

Now, if you have learn anything about Noether theorem, you would know that for every conservation law, there is a corresponding symmetry principle behind it. So the conservation law is nothing more than a manifestation of that symmetry. The two conservation laws I mentioned above are directly due to the time-symmetry of our physical universe and the translation symmetry/isotropic nature of classical empty space.

These two symmetry principles are NOT DERIVABLE! They are just the way our universe is! These, in fact, are our starting postulates for almost all of our physical description of our universe! [I'll smack anyone with a large boson who brings up the CP violation in some weak decay events :)]

Again, this is why physics is physics and NOT mathematics. One should not confuse between the two and then think that it is why QM, for example, has a problem.

Zz.

 

[bd1976]

hmm...

Can classical mechanics be demonstrated as well ?

The problem with these questions is that your talking about pretty deep stuff. I'm fairly sure that I don't have any of the qualifications needed to answer this question but I'll give it a go in the hope of being corrected by someone wiser.

I think the problem is that theories in physics do not (as I think most people believe) attempt to explain the natural world. A physical theory is really just a set mathematical relationships between measurable physical quantities from which predictions can be made. To explain this considers Newtonian mechanics.
Basically here the fundamental concept is that objects interact via forces. The theory then goes on to say that force is related to the rate of change of velocity via the equation f = ma. Now the theory was wildly successful accurately predicting the motion of projectiles, rockets even the planets. However for all that, Newtonian mechanics says nothing about what a force actually is.. how it acts, or what mass actually is! It doesn't explain these things at all. Physics has to start with some principle concepts which can not be derived.
However Newtonian mechanics makes sense conceptually. I think its easy to grasp the concept of force and acceleration because we feel these things all the time in the real world. Also Newtonian mechanics produces real, testable predictions. Quantum mechanics is different because it just doesn't make the same kind of sense conceptually. First of all the fundamental equation arises by an indirect inductive argument and the theory makes no actual predictions rather it can only generate probabilities. And also there’s all the stuff about the collapse of the wave function etc... Suffice it to say Einstein didn’t like quantum mechanics and that’s a good enough reason to think it’s dodgy for me!

(p.s Thinking about that stuff seriously tested by brain and I’m sure it’s probably all very dodgy.)

 

[bd1976]

hmmm...

oops.. I think I already was corrected by someone wiser!

 

[ZapperZ]

oops.. I think I already was corrected by someone wiser!

No, bd. I think it is a plus to have someone else that also presents the same idea. It indicates that I'm not making this up as I go along. :)

However, as far as Einstein goes, his opposition was based on TASTES, not on any concrete observations. We need to clearly distinguish between an opposition to the INTERPRETATION versus FORMALISM. If you have a phenomenon, and that phenomenon violates the prediction of a theory, that is a problem with the FORMALISM. If you simply say "I don't like the way it looks", then you are challenging something not on solid evidence but on purely a matter of personal preference and taste. This is NOT a valid challenge in physics.

Furthermore, Einstein NEVER said QM was wrong. He had problems with the Copenhagen Interpretation of QM, and based on that, he stated that QM must be "incomplete", the same way classical mechanics is incomplete or an approximation of a more general relativistic theory.

Zz.

 

[bd1976]

hmmm...

Well. (Better make this quick seeing as I'm only just keeping my head above the water here) No one is going to disagree with the fact the fact that quantum mechainics is a successful theory in that it dosen't make any predictions that do not fit the observed facts. However I am not alone (or at least I very much hope I am not alone ) in thinking that quantum mechanics as it stands contains some real conceptual problems. For example the only things in the theory that have real world counter parts are measurements. However no one can really say what a quantum mechanical measurement is, or exactly how it affects the wavefunction which in turn dosen't exist physically yet... arrggg. *brain gives out*

Well anyway suffice it to say that quantum mechanics is differen't to the other physical theories in that, the principle object, "the wavefunction" has no physical significance. I think that the conceptual problems with the theory are so extensive that they point to something being incorrect or something missing from the theory itself.

p.s On a completely differn't note. One thing that I did get from my courses was a sense of depression about physics. It seems that the only real way for physics to advance is for a phenonomen to be discovered that is not currently explained by any theory. However it seems that dispite an ever increasing amount of science we are no closer to finding such a phenonomen than we were 50 years ago! What if such a thing is never found? There's a depressing thought. Hummanity could be stuck at its present level of understanding forever!

 

[metacristi]

The schroedinger equation is an inspired postulate - it cannot be derived
from fundamental principles.The pilot wave of David Bohm's version of quantum mechanics is also postulated.Has anyone tried to derive the
pilot wave and schroedinger equation from physical laws?

Well loosely speaking we are entitled to say that Schrodinger's equation is 'invented' (this terminology is widely used in the philosophy of science by the way) though from a strictly logical perspective it can still be deduced from the hamiltonian.But 'derivation' is not the most appropriate term to use here rather Schrodinger equation is equivalent with the equation of the quantum hamiltonian H which is constructed,invented,using an analogy with the classical situation.As about Bohm's equation it is the mere Schrodinger equation but rewritten in a form ressembling classical physics (having an extra term,the so called 'quantum potential').

 

[Rothiemurchus]

Max Born guessed that the wavefunction amplitude squared was proportional to the probability of finding a particle at a point in space.This guess works but it is not known why.If there is anything about QM that I would like clarified it is why Born's guess works.I prefer Bohm's pilot wave because it is a physical entity that moves particles around.

 

[marlon]

This idea was adopted from the theory of waves, i think. There you have that the squared amplitudo of the waves yields the intensity. Now in terms of particles one can say that intensity is equal to the number of particles that pass through a surface each second for example...

Once the concept of wave-function was constructed , the idea of intensity was used as the probability of finding a particle on a certain position.

Think also of the double-slit-experiment where the describing theory needed the concept interference of waves in order to fit the measured probabilities of finding a particle when both slits are open !

regards
marlon

 

[ZapperZ]

Well. (Better make this quick seeing as I'm only just keeping my head above the water here) No one is going to disagree with the fact the fact that quantum mechainics is a successful theory in that it dosen't make any predictions that do not fit the observed facts. However I am not alone (or at least I very much hope I am not alone ) in thinking that quantum mechanics as it stands contains some real conceptual problems. For example the only things in the theory that have real world counter parts are measurements. However no one can really say what a quantum mechanical measurement is, or exactly how it affects the wavefunction which in turn dosen't exist physically yet... arrggg. *brain gives out*

Well anyway suffice it to say that quantum mechanics is differen't to the other physical theories in that, the principle object, "the wavefunction" has no physical significance. I think that the conceptual problems with the theory are so extensive that they point to something being incorrect or something missing from the theory itself.

Again, this is an objection based on a matter of tastes.

Secondly, when you say something is "conceptually difficult", what does that mean? I know of MANY things which I found to be "conceptually difficult" to comprehend until I actually learned about those things. Then they became conceptually easy to understand! Are we basing it on intuition? If so, then this isn't a valid foundation to use since our intuition changes with our accumulated knowledge! So when you say something is conceptually difficult, how would you know the problem isn't with you or your understanding of it?

Again, this is something that most people either seem to overlook, or dismiss. There has been ZERO experimental observation so far that is inconsistent with QM. Now think about it for a minute. Of all the humongous number of experiments going on throughout the world, we have ZERO experimental observation that is inconsistent with QM. Regardless of anyone's level of education, this has GOT to be an astounding fact! If this does not convince anyone that there is something to QM, then nothing can. People have believed in things with significantly less convincing body of evidence than this!

There is, however, another issue involved here, and that is the a priori belief that our classical notion MUST work everywhere. The "conceptual problem" that you talked about are based on your insistance that things that you accept to be true in the classical sense, such as "position", "momentum", "wave", "particle", etc. MUST all still be true even at the sub-atomic scale. I would assert that THAT point of view is very narrow and prejudicial. I think it is highly arrogant of us to want to force nature to fit within our comfort level. Whether the CI version of QM is valid or not isn't the issue here. Rather, it is our unwillingness to realize that our cherished notion of the classical world may not be valid beyond a certain extent, inspite of the huge number of experimental evidence staring us in our face. I find this more troublesome than the "conceptual problem" of QM.

As far as Bohm's pilot wave formulation of QM, I have mentioned this before in another thread, but anyone who professes to be a fan of such a formulation seem to have swept under the carpet the zoo of problems that come with such formulation. This includes the still troubling inability to formulate a correct QFT-like formalism (meaning no creation/desctruction of particles) and the fact that the first attempt to do that resulted in a non-lorentz invariant form! Is this "conceptually easier" to accept?!

Zz.

 

[Rothiemurchus]

ZAPPER Z:
Again, this is something that most people either seem to overlook, or dismiss. There has been ZERO experimental observation so far that is inconsistent with QM. Now think about it for a minute. Of all the humongous number of experiments going on throughout the world, we have ZERO experimental observation that is inconsistent with QM. Regardless of anyone's level of education, this has GOT to be an astounding fact! If this does not convince anyone that there is something to QM, then nothing can. People have believed in things with significantly less convincing body of evidence than this!

Rothie M:
I don't doubt for a second that the maths of QM is right because the evidence for it being right is overwhelming.

ZAPPER Z:
As far as Bohm's pilot wave formulation of QM, I have mentioned this before in another thread, but anyone who professes to be a fan of such a formulation seem to have swept under the carpet the zoo of problems that come with such formulation. This includes the still troubling inability to formulate a correct QFT-like formalism (meaning no creation/desctruction of particles) and the fact that the first attempt to do that resulted in a non-lorentz invariant form! Is this "conceptually easier" to accept?!

Rothie M:
Bohm's theory might be Lorentz invariant if the pilot wave was made classical .
The current situation is that the pilot wave can change instantaneously along its length - that does not
go well with general relativity.Perhaps gravity will be quantized successfully when qft for a modified Bohmian QM is sorted.

 

[humanino]

gravity could not be quantized successfully until qft for Bohmian QM is sorted

I must say I disagree with such a statement. QM is perfectly fine the way it is right now. The axioms are well-defined, and only interpretation is debatable. Besides, loop quantum gravity could be about quantisizing gravity soon, without changing any thing to QM.

Do you really use QM everyday in your calculations ? Because if so, I do not understand where your problem with QM comes from. If you do not use QM, maybe that is the problem.

 

[Rothiemurchus]

Humanino:
QM is perfectly fine the way it is right now

Rothie M:
I will agree with you when gravity is quantized using standard qft!
Until then there could be a problem with standard QM. Bohmian QM definitely has problems but less work is done on Bohm's theory and solving the problems with that might be the way forward for quantizing gravity.I used to use QM every day in calculations - I don't doubt that it works for just about all practical purposes.
But gravity could be the downfall for standard QM.And bear in mind that Bohm's version of QM gives the same predictions as standard QM.

 

[Rothiemurchus]

There is a thread I opened on sci.physics.research about "solitons and water waves"
which has a link to the use of the Schrodinger equation in showing how water waves
take energy from one another.Apparently nobody thought the equation would work on this scale.There's not a lot of detail but I think you'll find the thread as a whole interesting.

 

[humanino]

I will agree with you when gravity is quantized using standard qft!

That is certainly not going to happen !
QM & QFT are two things. We know QFT failed with regard to gravity. Still QM must and will hold. You are confusing.

 

[humanino]

What you don't realize is that QM is a framwork much more general than when applied to a particular field. All the relevant theories attempting to quantize gravity today are very close to success, and none of them ever even tried to change QM. They have to fit in it. Don't tell me that is the reason why they did not successeded yet.

 

[Rothiemurchus]

The addition of a quantized electromagnetic field to Dirac's theory is qft isn't it?
And isn't Dirac's theory standard QM plus rest mass?

 

[humanino]

No it is not.