Just wondering, is the Copenhagen interpretation (along with Schrodingers cat and non-locality) widely accepted at this stage? Most quantum physics books Im reading are AT LEAST 5 years old and Im just wondering has the pendulum swung in another direction in recent times?

Originally posted by John Connellan
Just wondering, is the Copenhagen interpretation (along with Schrodingers cat and non-locality) widely accepted at this stage? Most quantum physics books Im reading are AT LEAST 5 years old and Im just wondering has the pendulum swung in another direction in recent times?

i would say no. i think most practicing physicists don t really think about it much, but i read a survey once that claimed that many modern physicists follow everetts interpretation.

and i think with all the research going on in quantum computing and decoherence, what constitutes a "measurement" is much less mysterious than it was 80 years ago.

in my opinion, the copenhagen interpretation, and the nonunitary wavefunction collapse, is dead.

oh:confused:
i was gonna discuss a certain danish hooker i met over there
oh well
you guys are boring

Here is a nice review of where things stand.

http://plato.stanford.edu/entries/qt-measurement/

Well, it is not really a review of "where things stand", it is just an historical overview of how the measurement proces has evolved in the early years of quantum mechanics. (It is a nice read though ;)).

I agree with lethe that the measurement proces has become less obscure, especially through research in quantum computing and the entanglement it relies on heavily, and the effects of decoherence. Many other mathematical models for measurements have been proposed, and they are not as drastic as the collapse of the wavefunction (I am thinking of Kraus measurements, continous measurements, ...). They do not constitute a unitary evolution as lethe seems to want, but they form completely positive maps, a way of "effectively" -- read: more easily -- describing a quantum system.

I am not sure that the wavefunction collapse is dead as lethe says. It leads to some philosophical problems (philosophical is not equal to physical!), but from a mathematical point of view, I think it fits very well in quantum theory. For this reasons I think that many theoretical physicists do not want to immediatelly let go of them.

So I think that it has all become quite clearer, but that these new ideas have not caught on yet in the general scientific community.

Bye!

Crisp

Some Quantum Cosmologists propose that when I make a measurement, I leap into a superposition of eigenstates.

Edit: I recently [about a year ago now] did an informal survey of the state of the measurement problem. From what I saw this is still an open ended question. The statements made lead me to believe that we have a number of schools of thought that simply don't recognized each other as valid.

wavefunction collapse is dead

what does that mean? do you mean you ignore it. that it does not happen?
tweak the data to fit the theory?

Originally posted by spookz
wavefunction collapse is dead

what does that mean? do you mean you ignore it. that it does not happen?
tweak the data to fit the theory?

it means that it does not happen. the evolution of the wavefunction is always unitary.

which of course is governed by the hamiltonian yes?

*jes pulling yer leg.

Originally posted by lethe
it means that it does not happen. the evolution of the wavefunction is always unitary.

Can you elaborate and give some links? I'm not disputing what you say, but at the least I am out of date...along with a number of highly accomplished physicists.

Originally posted by Ivan Seeking
Can you elaborate and give some links? I'm not disputing what you say, but at the least I am out of date...along with a number of highly accomplished physicists.

the notion that you described above, that when you "perform a measurement", you yourself "leap into a superposition of states", is a laymans description of everetts relative state formalism. the laymans title of that work is "many worlds", although there is more to everetts work than that. for example, in everetts formalism, no nonunitary evolution ever takes place. in essence, there is no measurement, there is only interaction.

point is, you and i are talking about the same thing.

Is Everett's relative state formalism a recent innovation? This is not the Many Worlds interpretation that I'm familiar with; its just the opposite. Perhaps you are saying that this interpretation can be deduced from the original theory?

Originally posted by Ivan Seeking
Is Everett's relative state formalism a recent innovation? This is not the Many Worlds interpretation that I' familiar with, its just the opposite. Perhaps you are saying that this interpretation can be deduced from the original theory?

No, its not recent. everett wrote his thesis on relative states in 1956.

What do you think about David Bohm's interpretation and the holographic principle?

Originally posted by ryans
What do you think about David Bohm's interpretation and the holographic principle?

Well, Bohmian mechanics isn t relativistic, therefore i m pretty much not interested.

AdS/CFT is extremely interesting, but it needs a lot of work, as do other holographic principles.

Following are some comments without much organization or agenda.

I have always thought that collapse of the wave function was jargon merely meaning that one particular event occurred.

The wave function provides probabilities corresponding to many possible events. When some particular event occurs, the other possibilities can no longer occur and the wave function is no longer applicable.

Note the terseness of the phrase collapse of the wave function in comparison with the second paragraph above.

I am not sure of everything that is included in the Copenhagen Interpretation, although I believe that it is still accepted by many of the experts. I suspect that some aspects of it are considered more dubious than other aspects, even by those who accept the Interpretation.

For myself, I think that any interpretation of Quantum Theory will be unsatisfactory because the phenomena behind the measurements cannot be described by models which make sense to the human mind. Millions of years of evolution have resulted in a mind wired to understand and cope with an environment modeled very well by classical physics. Our survival as a species is dependent on success in a classical reality. A mind wired to understand and visualize Quantum reality might be very helpless in the classical world. At least there is no evolutionary pressure pushing the development of a Quantum mind.

To me, the many worlds interpretation is easily understood nonsense. It is believed in due to the easily understood part, causing people to ignore the nonsense part. Belief in nonsense is often fueled by a desire for an understandable explanation. It is almost rational to accept a silly but understandable explanation when the alternative is no explanation or an explanation which is not comprehended (almost the same as no explanation).

No matter what interpretation you might prefer, the following observations seem valid. Depending on the experimental setup, the same quantum entity can be modeled very well by classical waves, and can also be modeled very well by classical particles. The two models are incompatible, so the reality must be something else. The data produced by various quantum phenomena is probabilistic. This strongly suggests (to me is conclusive) that the quantum world is not deterministic. Any measuring device must affect the object being measured. Photons (or waves) bounced (reflected) from an automobile do not have much effect on the automobile, and ignoring such effects does not introduce much error. Our perceptual image of the automobile is a damned accurate model of the reality. When dealing with quantum level phenomena, the measuring devices have considerable effects on the objects being measured. We have a knowledge of interactions between quantum entities and the measuring devices. We do not have direct knowledge of the quantum entities.In view of the above, no interpretation is likely to be much help. Those who ignore interpretations and consider only the measurements might be the wisest of all.

Neils Bohr once described a well known illusion. It was a drawing which could be viewed as either a black vase or as two white profiles facing each other. Either view was easy to see, but both could not be seen at the same time (apologies to relativity). Bohr said the following (paraphrase, not a quote).If you ask me what is really there, a vase or two profiles, I can only say that neither is really there. What is really there is black ink on white paper.

When asked if reality is waves or particles, I can only say that it is neither. Unfortunately, I cannot tell you what is really there for this phenomenon.Perhaps we will never know what is really there. Feynman once saidDo not ask me how it can be like that. Nobody knows how it can be like that.If those men did not have good explanations, I do not feel stupid for not being able to visualize what is happening at the quantum level.

I like the remark made by somebody: “Quantum entities arrive and depart as particles, but travel as waves.”

To me, the worst problem is that erroneous conclusions result from valid logic based on very reasonable assumptions about measurements not actually made.

Originally posted by Dinosaur

I have always thought that collapse of the wave function was jargon merely meaning that one particular event occurred.

The wave function provides probabilities corresponding to many possible events. When some particular event occurs, the other possibilities can no longer occur and the wave function is no longer applicable.

your description of what "collapse of the wavefunction" means is accurate, but i don t know why you dismiss it as jargon.

there is definitely something fishy about collapse. the wavefunction, in the absence of measurements, evolves unitarily and continuously. all of a sudden, some interaction with a "measuring apparatus" happens, and the wavefunction undergoes nonunitary and discontinuous evolution, the collapse.

the obvious question is: what makes one kind of interaction different from another, so that sometimes interactions yield continuous unitary evolution, and other kinds yield nonunitary discontinuous evolution?

the obvious answer is: there is no difference. all interactions are quantum, there is no such thing as an ideal, classical measurement (just as every measurement affects the system you are measuring, it also affects the measuring apparatus as well) and there is no collapse.

to believe in collapse is to believe in purely classical objects (as Heisenberg did). i think this notion is dead as a doornail, and most physicists believe that in fact all systems in our universe are quantum mechanical in nature.


I am not sure of everything that is included in the Copenhagen Interpretation, although I believe that it is still accepted by many of the experts.

i suppose it is bold of me to speak for the experts, but i don t agree.

I suspect that some aspects of it are considered more dubious than other aspects, even by those who accept the Interpretation.
i wouldn t really say you accept the Copenhagen Interpretation if you find aspects of it dubious.

For myself, I think that any interpretation of Quantum Theory will be unsatisfactory because the phenomena behind the measurements cannot be described by models which make sense to the human mind. Millions of years of evolution have resulted in a mind wired to understand and cope with an environment modeled very well by classical physics. Our survival as a species is dependent on success in a classical reality. A mind wired to understand and visualize Quantum reality might be very helpless in the classical world. At least there is no evolutionary pressure pushing the development of a Quantum mind.

i think the human mind is a very flexible device, capable of understanding many things that were not necessarily part of its evolutionary heritage.

i think probability is among the things the human mind can understand, and therefore i think you can find a satisfactory interpretation of quantum mechanics, as soon as you get comfortable with the world being in probability waves, instead of observable particles.

To me, the many worlds interpretation is easily understood nonsense. It is believed in due to the easily understood part, causing people to ignore the nonsense part. Belief in nonsense is often fueled by a desire for an understandable explanation. It is almost rational to accept a silly but understandable explanation when the alternative is no explanation or an explanation which is not comprehended (almost the same as no explanation).
many worlds is a label that is usually applied to everetts coherent state formalism. why do you find this nonsensical?

No matter what interpretation you might prefer, the following observations seem valid.[list] Depending on the experimental setup, the same quantum entity can be modeled very well by classical waves, and can also be modeled very well by classical particles.
i would say instead, that a quantum entity can be modeled very poorly by a classical wave, and very poorly by a classical particle, and very well by a quantum mechanical model.

the poorness of the model depends on the situation, of course, and perhaps you are more leniant in your acceptance of poor models than i.

What Quantum Mechanical Model do you have in mind?i would say instead, that a quantum entity can be modeled very poorly by a classical wave, and very poorly by a classical particle, and very well by a quantum mechanical model.Do you have a good explanation for wave/particle duality? Duality for photons is strange enough, but now they have experimental evidence for duality with electrons & atoms.

BTW: when i applied the term jargon to collapse, I did not mean to dismiss the concept. I only meant to suggest that the term might be misleading to non experts. Without some background, the term suggests something potentially catastrophic. Terms like spyware, software, motherboard, et cetera are meaningful to those with a little computer savy and very misleading to computer illiterates. Similarly, I consider collapse to be a potentially misleading term.

Am I missing something about collapse?there is definitely something fishy about collapse. the wavefunction, in the absence of measurements, evolves unitarily and continuously. all of a sudden, some interaction with a "measuring apparatus" happens, and the wave function undergoes nonunitary and discontinuous evolution, the collapse.I always considered it analogous to dice throws (ignoring the continuous nature of the wave function). The mathematics assigns probabilities to various totals which might be thrown. When one total is actually thrown, the mathematics is no longer applicable. The probabilities assigned to the other totals are now meaningless. The wave function assigns probabilities to all the places where an event might happen. The event happens at some particular place and the wave function is no longer applicable. The other probabilities are meaningless.

You have more faith in the human mind that I do, although I consider it the most remarkable object in the universe.i think the human mind is a very flexible device, capable of understanding many things that were not necessarily part of its evolutionary heritage.I agree within limits. I do not expect the human mind to be able to visualize the corner of a 4D or 5D hypercube. Similarly, I do not expect it to be able to visualize much (if anything) that occurs in the quantum world.

BTW: I am not uncomfortable with Quantum Reality. I am aware of many who are more uncomfortable than I. I merely have trouble visualizing quantum reality. Probability waves do not seem so bad. My only problem with them is that I cannot conjure up a satisfactory mental image of them. Electrons and photons as particles (wrong as the model might be) are easy to visualize. The solar system model of the atom is a terrible model, but I can build one and look at if I cannot visualize it.

Originally posted by Dinosaur
Am I missing something about collapse?
i think you ve mostly got it.

I always considered it analogous to dice throws (ignoring the continuous nature of the wave function). The mathematics assigns probabilities to various totals which might be thrown. When one total is actually thrown, the mathematics is no longer applicable. The probabilities assigned to the other totals are now meaningless. The wave function assigns probabilities to all the places where an event might happen. The event happens at some particular place and the wave function is no longer applicable. The other probabilities are meaningless.
you still need to know how the wavefunction evolves when you are not throwing the dice.

You have more faith in the human mind that I do

perhaps.

"the obvious question is: what makes one kind of interaction different from another, so that sometimes interactions yield continuous unitary evolution, and other kinds yield nonunitary discontinuous evolution?"

I am going to give a very lame answer, but an answer nevertheless: it is no longer a spontaneous evolution. The moment you measure something, you make the system, which you previously assumed to be closed (do you include the measurement device in your description?) an open system. I don't know how this answers the question, and I am not pretending it motivates the collapse of the wavefunction, but more than just "fundamental interactions" is happening here.

"to believe in collapse is to believe in purely classical objects (as Heisenberg did)."

Why ? Because you do not collapse your measurement apparatus ? Because you do not include it in your description ?

It is important in this issue to see that the measurement process in "regular" quantum mechanics is only an effective description, and not fundamental in any way; where is the apparatus in the description ? It is not included, so you use some sort of effective "interaction" to model the measurement. A similar procedure already exists in quantum mechanics: if you have the interaction of the system with a reservoir, then you can reduce the unitary dynamics on system+reservoir to only one on the system. By doing this you'll reduce the unitary dynamics to a non-unitary (but rather completely positive, or "effective") evolution.

Including the apparatus also does not (always?) work, I think you'll end up calculating decoherence rates and not measurement results. Well, I have seen at least one model where including the measurement apparatus did not yield the measurement process in any way (the curie-weiss model). But then again, if you have one of those effective evolutions I talked about in the previous paragraph, it is not possible to construct the "full" unitary evolution again.

I am not saying that going to this higher picture and seeing system and measurement apparatus as one will solve the problems, I am just saying that perhaps it is not surprising to see that collapses are non-unitary since they include something "external".

"i think this notion is dead as a doornail, and most physicists believe that in fact all systems in our universe are quantum mechanical in nature."

Hrmmmm... I am not sure. Perhaps this is geographical, but here in western Europe I have the impression that the collapse is still accepted as "something that might not be perfect, but due to the lack of something better, ... "

Bye!

Crisp

Originally posted by Crisp
"the obvious question is: what makes one kind of interaction different from another, so that sometimes interactions yield continuous unitary evolution, and other kinds yield nonunitary discontinuous evolution?"

I am going to give a very lame answer, but an answer nevertheless: it is no longer a spontaneous evolution. The moment you measure something, you make the system, which you previously assumed to be closed (do you include the measurement device in your description?) an open system. I don't know how this answers the question, and I am not pretending it motivates the collapse of the wavefunction, but more than just "fundamental interactions" is happening here.
in this context, what does spontaneous mean? can you give me an example of a spontaneous interaction, and how it differs from a nonspontaneous interaction? can you describe what makes spontaneous interactions evolve nonunitarily?

"to believe in collapse is to believe in purely classical objects (as Heisenberg did)."

Why ? Because you do not collapse your measurement apparatus ? Because you do not include it in your description ?
because the criterion for collapse is interaction with a classical apparatus, according to Heisenberg.

It is important in this issue to see that the measurement process in "regular" quantum mechanics is only an effective description, and not fundamental in any way;
whoa... i have not heard this opinion before. i believe quantum mechanics to be a fundamental description, and i though most people felt the same.

i suppose if you think that measurement is only an effective description... well then, what i said before, about needing classical objects, is false. you only need "effectively classical" objects, which quantum theory clearly allowes.


"i think this notion is dead as a doornail, and most physicists believe that in fact all systems in our universe are quantum mechanical in nature."

Hrmmmm... I am not sure. Perhaps this is geographical, but here in western Europe I have the impression that the collapse is still accepted as "something that might not be perfect, but due to the lack of something better, ... "
well, we are in agreement that we reject Heisenbergs delineation of the universe into classical systems and quantum systems, right? at least as a fundamental description.

i think every modern physicist rejects that idea, and believes that nature is quantum, or at least that the quantum description is more fundamental than the classical description.

it is my opinion that anyone that believes that there is no division of the universe into classical and quantum systems must reject the idea of collapse, but i may be mistaken.

i was also under the impression that decoherence was sufficient to replace measurement. can you say more about this curie-weiss model? i have not seen it.

I cannot help but think that there is too much emphasis on collapse as something mystical. Some process that nobody really understands is proceeding at the Quantum level. The process is disturbed by an interaction with some classical world apparatus (not necessarily a formal measuring experiment), causing a change in the quantum level process. The jargon for this disturbance/change in the quantum process is called collapse.it is my opinion that anyone that believes that there is no division of the universe into classical and quantum systems must reject the idea of collapse, but i may be mistakenFrom a theoretical point of view, there might be no division between the quantum and the classical world. I suspect that a true division would prevent a TOE, GUT, or unified field theory. I expect some such theory to be developed & require that theoretically there be no division between the quatnum & classical worlds.

In practical terms, I expect there will always be a division. Just as relativistic effects cannot be measured for many gravitational systems, quantum uncertainty and other weirdness cannot be measured for classical size objects. Measurement technology might never be able to distinguish between Quantum & classical theory or between Relativity & quantum theory for objects in certain size ranges.

Originally posted by Dinosaur
I cannot help but think that there is too much emphasis on collapse as something mystical. Some process that nobody really understands is proceeding at the Quantum level. The process is disturbed by an interaction with some classical world apparatus (not necessarily a formal measuring experiment), causing a change in the quantum level process. The jargon for this disturbance/change in the quantum process is called collapse.

you are saying that i emphasise collapse too much, and i think you deemphasise it too much... and by calling it jargon, you make it sound like it is a matter of semantics. i don t agree with this sentiment.

From a theoretical point of view, there might be no division between the quantum and the classical world. I suspect that a true division would prevent a TOE, GUT, or unified field theory. I expect some such theory to be developed & require that theoretically there be no division between the quatnum & classical worlds.

TOEs and GUTs do not have any input about interpretations of quantum mechanics. supposing string theory turns out to be a good theory of everything, well string theory is still a quantum mechanical way in the same way that regular nonrelativistic quantum mechanics is. so it doesn t look like these theories will have any answers for us in this department.

In practical terms, I expect there will always be a division. Just as relativistic effects cannot be measured for many gravitational systems, quantum uncertainty and other weirdness cannot be measured for classical size objects. Measurement technology might never be able to distinguish between Quantum & classical theory or between Relativity & quantum theory for objects in certain size ranges.

sure, in practical terms, none of this matters. but i expect quantum theory to be a consistent theoretical framework for describing nature on a fundamental level, so this is just not good enough.

"in this context, what does spontaneous mean? can you give me an example of a spontaneous interaction, and how it differs from a nonspontaneous interaction? can you describe what makes spontaneous interactions evolve nonunitarily?"

By spontaneous I meant "unattended", or "without human intervention", just plain unitary evolutions. The measurement involves a change of the system through external causes, or an interaction if you like, that is switched on at a given time (the moment you measure).

"whoa... i have not heard this opinion before. i believe quantum mechanics to be a fundamental description, and i though most people felt the same.."

If you refer to the unitary evolution, without collapse, then I totally agree, then I also consider QM as a fundamental description. However, I consider the non-unitary collapse as an effective part, originating from what you call an effective classical object:

"i suppose if you think that measurement is only an effective description... well then, what i said before, about needing classical objects, is false. you only need "effectively classical" objects, which quantum theory clearly allowes."

Exactly the point I was trying to make, only rephrased a lot better.

In that sense I do not consider quantum mechanics to be complete, but rather a unitary result of projecting a "higher" evolution on both the measurement device and the system to the level of only the system. This procedure is often used when describing open quantum systems btw, but then the situation is slightly different: you have a unitary evolution on a system and a reservoir, and you "trace out" the reservoir (reduce the description) to a non-unitary evolution on the system. Here I somewhat think the opposite: you have a higher non-unitary evolution on measurement apparatus + system and you trace out the measurement apparatus to be left with a unitary evolution on the system and an "effective" measurement process which is the non-unitary collapse.

I am not sure if you modelling both the measurement device and the system by QM at the same time will produce the phenomenon of "measurement" (i.e. a reproduction of one non-probabilistic quantity, the result of the measurement). At least I do not see any mechanism... You are probably thinking of decoherence here, but I got some words to say on that too :)... (see later).

I should add that this is a "feeling" I have, I have absolutely no maths to back this up. But then again, the copenhagen interpretation as a whole is just a "convention" or feeling, so debatable ;).

"well, we are in agreement that we reject Heisenbergs delineation of the universe into classical systems and quantum systems, right? at least as a fundamental description."

Ofcourse I agree with this.

"i think every modern physicist rejects that idea, and believes that nature is quantum, or at least that the quantum description is more fundamental than the classical description."

Ofcourse.

"i was also under the impression that decoherence was sufficient to replace measurement. can you say more about this curie-weiss model? i have not seen it.

The article I refered to was written by T. Van Nieuwenhuizen from the university of Amsterdam, I'll look it up tomorrow (it is 1 am here and I have to teach at 8 am tomorrow). Most of my colleagues and I disagree with his conclusions, where he states that he reproduces the measurement as a result of an interaction between a magnetic system and a reference spin, which is his "measurement apparatus" (it is a simplified model ofcourse). I think that what he calculates is exactly the decoherence of the model, but more is required for a measurement.

Decoherence reduces the quantum statistics to a classical statistics (the density matrix becomes diagonal in the eigenbasis of the observable you are measuring), but to have a measurement result, you need the densitymatrix to "collapse" (or nearly collapse) on one particular eigenstate. Decoherence does not single out one eigenvalue as a real measurement result would.

EDIT: Looked up the article anyway (I should go to sleep... really), it is on the preprint archives right <A HREF="http://arxiv.org/abs/cond-mat/0203460">over here</A>. I did not read the article yet, but I attended a talk by van Nieuwenhuizen on this work. I can asure you that there was quite an animated discussion afterwards :)



Bye!

Crisp

Once again, I want to emphasize that my calling collapse jargon is not meant to minimize the concept. Every discipline uses jargon to avoid excessive verbiage.

I merely am trying to express the notion that collapse is a shorthand phrase used by experts au lieu de a lengthier but more meaningful description. I think that the term tends to be misleading to those with minimal knowledge of quantum theory.

Originally posted by Dinosaur
Once again, I want to emphasize that my calling collapse jargon is not meant to minimize the concept. Every discipline uses jargon to avoid excessive verbiage.

I merely am trying to express the notion that collapse is a shorthand phrase used by experts au lieu de a lengthier but more meaningful description. I think that the term tends to be misleading to those with minimal knowledge of quantum theory.

the point is that i feel that collapse, as described in the Copenhagen interpretation, is something completely mystical, and it is exactly this point that makes us reject the Copenhagen interpretation.

collapse is shorthand for "state evolves nonunitarily and discontinuously and instantaneously to a an eigenstate upon interaction with a classical system".

look at your statement here:

Originally posted by Dinosaur
I cannot help but think that there is too much emphasis on collapse as something mystical. Some process that nobody really understands is proceeding at the Quantum level. The process is disturbed by an interaction with some classical world apparatus (not necessarily a formal measuring experiment), causing a change in the quantum level process. The jargon for this disturbance/change in the quantum process is called collapse.

it sounds like you think that collapse is something perfectly reasonable, and only sounds mystical because of the jargon that we have for it.

my position is that the collapse is just weird. so weird and unnatural, that i cannot accept the Copenhagen interpretation. i don t think the use of the word "collapse" makes it seem more mysterious than it is, in fact, quite the opposite. i think the description i gave above

"state evolves nonunitarily and discontinuously and instantaneously to a an eigenstate upon interaction with a classical system"

is infinitely more mysterious than the word "collapse", which sounds rather reasonable.

Lethe: Your view of collapse.there is definitely something fishy about collapse. the wavefunction, in the absence of measurements, evolves unitarily and continuously. all of a sudden, some interaction with a "measuring apparatus" happens, and the wavefunction undergoes nonunitary and discontinuous evolution, the collapse.My view.
Some process that nobody really understands is proceeding at the Quantum level. The process is disturbed by an interaction with some classical world apparatus (not necessarily a formal measuring experiment), causing a change in the quantum level process.I consider your description to relate to the wave function itself, not the actual quantum process occurring between measurements. Your description seems to me to be obfuscastion, implying an understanding of what is actually happening at the quantum level when no measurements are being made.

Note that my description explicitly denies any knowledge of the actual quantum level process.

What we know is that the probabilities indicated by the wave function are consistent with measurments made. Those measurments always indicate some quantum entity (or event) was detected at a specific place (or approximately at some specifc place).

We have no idea of what is going on between measurments. Why be disturbed because the model (wave equation) of an unknown process seems different from the measured result of the process?

The mystery is the unknown process between measurements. Emphasizing the so called collapse seems to me to direct attention away from the real mystery.

Maybe some already said something about this, i didn't read everything, but isn't the collapse of the wavefunction due to decoherence? I think i read something about that somewhere.

Dinosaur,

"We have no idea of what is going on between measurments. Why be disturbed because the model (wave equation) of an unknown process seems different from the measured result of the process?

The mystery is the unknown process between measurements. Emphasizing the so called collapse seems to me to direct attention away from the real mystery."

Why would "what happens between measurements" be any different than just the quantum unitary evolution ? Right after the measurement, the unitary evolution starts again, this time with a new initial coniditon, being a delta-peak around the measured value/variable.


Cephas1012,

"Maybe some already said something about this, i didn't read everything, but isn't the collapse of the wavefunction due to decoherence? I think i read something about that somewhere."

I would have a hard time believing that. Decoherence is something like the loss of the quantum statistical properties of a system after a certain time (i.e. the dynamics becomes an evolution on pure states only), while a measurement is a loss of all the statistical properties of a system (the collapse forces the system to a deterministic value).

Bye!

Crisp

Crisp: The following implies an understanding of what goes on between measurments.Why would "what happens between measurements" be any different than just the quantum unitary evolution ?Do you have a model of the processes occurring between measurements? Can your model be visualized? What does quantum unitary evolution mean?

The philosophical interpretation of quantum mechanics has absolutely no implications for the relevance nor completeness of the theory to describe nature, and the primary persons who feal it to be their obligation to discuss such issues, I have found to have little, if any mathematical appreciation of the theory.

"The following implies an understanding of what goes on between measurments.Do you have a model of the processes occurring between measurements? Can your model be visualized? What does quantum unitary evolution mean?"

Quantum unitary evolution is just the "regular" evolution that is predicted by the Schrodinger equation. Formally you can write (in the Heisenberg picture of QM) that the time evolution of an operator is given by:

A(t) = U*(t) A U(t)

where * is the Hermitian conjugate, and U(t) is exp( i t H / h-bar ). This U(t) is a unitary operator and is the solution of the Schrodinger equation.

Just for the sake of it, let us now define a unitary operator U(t1,t2) which describes the unitary evolution from time t1 to time t2, or:

U(t1,t2) = exp( i (t2 - t1) H / h-bar )

Now, assuming that the collapse describes the measurement (not everybody agrees), your time evolution suddenly gets stopped and a projection on the eigenspace of the measurement result takes place. After that, the unitary evolution just continues (what else would happen, after the collapse you can see the collapsed wavefunction again as an initial state for a now again not-monitored system). If we start the experiment at t0, we measure at t1 and we stop at t2, then the total time evolution becomes

U(t0,t2) = U(t1,t2) P U(t0,t1)

where P is the projection operator on the measurement result.

Oh BTW, I am not making this up as we go along, this has been described by von Neumann already in 1930 or something.

Bye!

Crisp

Originally posted by ryans
The philosophical interpretation of quantum mechanics has absolutely no implications for the relevance nor completeness of the theory to describe nature, and the primary persons who feal it to be their obligation to discuss such issues, I have found to have little, if any mathematical appreciation of the theory.

that is quite a viewpoint ryans.

i suppose you think that Crisp, dinosaur and i lack mathematical appreciation of quantum mechanics, then?

Lethe
i suppose you think that Crisp, dinosaur and i lack mathematical appreciation of quantum mechanics, then?

Your primary obligation is not the discussion and validation of the philosophical interpretation of QM is it Lethe. As I understand it, your primary focus is mathematical physics, with detours of interest playing in the background. I am talking about people who write books like

"The philosophical interpretation of Quantum Mechanics"

I forget the author. So get off that horse Lethe.

Do not give me any credit for understanding the mathematics of Quantum Theory. I have enough formal education and background knowledge to learn to understand it, but I have never read any serious books on the mathematics of Quantum Theory or Relativity. I took differential geometry using tensor notation which should be a fair background for GR mathematics. I have taken matrix algebra, set theory, calculus, differential equations, and various other courses, one or more of which should be a fair background for QT mathematics.

The formal mathematics does not seem all that interesting for its own sake, and I never had an interest in working in either field. In contrast, the interpretations and philosophy are very interesting.

I spent a lot of time trying to visualize 4D geometric objects. While I do not consider the time wasted, I have never been able to conjure up a mental image of even a simple 4D object. Trying to imagine what is really going on in the quantum world still fascinates me, although I have almost convinced myself that it is another example of the limits of human knowledge.

There is much mathematics applicable to 4D geometry, which is understandable in some formal sense, but a human mind does not seem able to make perceptual models of any of the objects described by the mathematics. QT seems to be an analogous discipline in this sense.

Originally posted by Dinosaur
The formal mathematics does not seem all that interesting for its own sake, and I never had an interest in working in either field. In contrast, the interpretations and philosophy are very interesting.Bite your tongue man! I do not believe you actually meant what you posted in the above statement. A Mathematician preferring Philosophy over Mathematics! Blasphemy! :eek:

By the way, what sub field(s) of Mathematics do you work/do your research in?

Dapthar: Shortly after graduating with a degree in mathematics, I became a computer programmer in the 1950's after a few years with the Air Force intelligence which had a very primitive computer-like device. Computer technology was just getting off the ground, and people like me did everything. I did a lot of partial and ordinary differential equations programming. I developed approximating functions and code for all the transcendental functions, matrix algebra, root finding software, and various other math applications. I wrote sorting, accounting, and data processing applications, programs for design of zoom lenses, a very primitive OS, two assemblers, various engineering design programs, and all sorts of miscellaneous applications before being forced to specialize. Specialization became a necessary evil as the industry matured, but it turned a hobby into a job.

Now I play with Visual Basic and MathCad.

Originally posted by ryans
So get off that horse Lethe.

you think that people who think about the philosophy of science have no appreciation, but you think it is i who am "up on my horse"?

i find that rather hypocritical.

It is an opininion Lethe, not a statement of absolute truth. If you do not like my point of view, that is fine, do not accept it.


Lethe
the point is that i feel that collapse, as described in the Copenhagen interpretation, is something completely mystical, and it is exactly this point that makes us reject the Copenhagen interpretation.

Not all physicists reject the Copenhagen interpretation.

At the end of the day, we are only able to view and measure the world with classical instruments, and so whether the wavefunction collapses discontinuously in time, or evolves according to some continuous Unitary transformation, the same results will be measured in our apparatus.

Quantum Mechanics and field theories major shortcomings are the fact that they are embedded within the harmonic paradigm and the principle of linear superposition. These approximations work fine for many problems, but fail when non-linesar effects are to be accounted for. Treating them perturbatively also has serious limitations.

Originally posted by ryans
It is an opininion Lethe, not a statement of absolute truth. If you do not like my point of view, that is fine, do not accept it.

and here is my opinion: the fact that you believe that people who think about interpretations of quantum mechanics are unable to appreciate quantum mechanics and you think it is i who am "up on a high horse" makes you a hypocrite.

that is my opinion, if you do not like it, you do not have to accept it.


Lethe


Not all physicists reject the Copenhagen interpretation.
how would you know? you have made it clear that you not only do not think about interpretations of quantum mechanics, but you think that people who do are worthy of ridicule.

so i must conclude that you haven t the foggiest idea about any interpretation of quantum mechanics.

i can have some confidence in my conclusion, when i go back and read this old post by you:

Originally posted by ryans

That the function Y(x) (Y=psi) when squared by its complex conjugate, represents a probabilty density, and more specifically that Y(x)^2 dx represents the probabilty of finding a particle between x and x+dx.


where you mistake the Born interpretation (where the wavefunction represents the probability distribution. a common feature of all interpretations of quantum mechanics) for the Copenhagen interpretation (where nonunitary wavefunction collapse happens upon measurement. something unique to the Copenhagen interpretation)

I am not sure why you posted a thread a while ago about the Copenhagen interpretation, considering you think that people who think about interpretations of quantum mechanics to be unable to appreciate quantum mechanics, but i guess maybe you still are able to appreciate quantum mechanics, considering you didn t even correctly state the copenhagen interpretation.




Quantum Mechanics and field theories major shortcomings are the fact that they are embedded within the harmonic paradigm and the principle of linear superposition. These approximations work fine for many problems, but fail when non-linesar effects are to be accounted for. Treating them perturbatively also has serious limitations.

oh!! this is an interesting turn of events. so you don t believe that quantum theory is even correct? the fact that many of the predictions of quantum theory have been experimentally verified to 12 decimal places, making it the most accurate theory in the history of science, but you feel the "harmonic paradigm" (whatever in the world that is) and the principle of superposition mean that it is wrong.

well, that gets you 10 points on the Baez crackpot index (http://math.ucr.edu/home/baez/crackpot.html). maybe you can get a job with MacM next.

by the way, i hope you aware that quantum theory is perfectly able to treat nonlinear systems, and there are lots of examples of nonlinear systems that are even exactly solvable nonperturbatively. this has nothing to do with the principle of linear superposition, of course.

Lethe I do not project myself as knowing everything about anything.

oh!! this is an interesting turn of events. so you don t believe that quantum theory is even correct? the fact that many of the predictions of quantum theory have been experimentally verified to 12 decimal places, making it the most accurate theory in the history of science, but you feel the "harmonic paradigm" (whatever in the world that is) and the principle of superposition mean that it is wrong.

It's not wrong, it's limited. Why are you now reverting to personal attacks Lethe. Go and read the first chapter in Zee "Quantum Field theory in a Nutshell", In formulating the Lagrangian of any field, then quantising this field, only quadratic and lower terms are kept and higher orger terms are discarded. It is clearly noted there is no method known to quantise higher order terms. DO YOU KNOW HOW TO DO THIS?
The non-local correlated motion of electrons in matter are seen as the primary cause of van der Waals forces in matter. This is seen to arise from instantaneous dipole-induced dipole interactions which are 1)Inherently non-linear and 2)Purely quantum mechanical in origin. This problem cannot be treated within the current framework of quantum mechanics. So do not tell me that quantum mechanics is not limited by the harmonic approximation.

by the way, i hope you aware that quantum theory is perfectly able to treat nonlinear systems, and there are lots of examples of nonlinear systems that are even exactly solvable nonperturbatively. this has nothing to do with the principle of linear superposition, of course.

Give me an example of a non-linear system that can be solved non-perturbatively.

and here is my opinion: the fact that you believe that people who think about interpretations of quantum mechanics are unable to appreciate quantum mechanics and you think it is i who am "up on a high horse" makes you a hypocrite.

You fu#$@#$% arrogant prick. You do not understand what I say. It is fine to think about philosophy and interpretation of QM, I have got no problem with that, I do it myself. But when you devote your entire life in arguing the validity of a theory without ever bothering to go and crunch the numbers, as some professionals do , that is when I have a problem!!

Originally posted by ryans

It's not wrong, it's limited. Why are you now reverting to personal attacks Lethe. Go and read the first chapter in Zee "Quantum Field theory in a Nutshell", In formulating the Lagrangian of any field, then quantising this field, only quadratic and lower terms are kept and higher orger terms are discarded.
i don t know this book, but i assure you, there is no reason to drop higher order terms other than ease of calculation. quantum mechanics is not based on the this approximation of everything as a harmonic oscillator, it just happens to be that the harmonic oscillator is one equation that we know how to solve. there are others.

for example, i m sure it didn t escape your notice that the Hydrogen atom in quantum mechanics is also exactly solvable. in fact, any central force is exactly solvable up to quadrature in quantum mechanics. so the harmonic oscillator is special, but not alone.

It is clearly noted there is no method known to quantise higher order terms. DO YOU KNOW HOW TO DO THIS?
of course i do. if Zee says it is not known how to quantize nonlinear systems, then he is just wrong. what he probably meant to say is that we don t know how to solve nonlinear differential equations (we don t), or equivalently diagonalize the interaction.

in other words, we don t know how to exhaustively come up with solutions to nonlinear equations. but we certainly know how to quantize. if you want me to quantize a nonlinear system for you, i can do that. do you need to know how to quantize QED, for example? this is a nonlinear system that we know how to quantize, and we have an approximation scheme for getting solutions, and this is the most accurate theory in the history of science.

if you had said doing perturbative calculations is a limitation because it can never know about nonperturbative physics, well that would have been correct. but instead you claim that quantum mechanics is wrong, because we don t know how to solve nonlinear equations. this is a ridiculous position.

The non-local correlated motion of electrons in matter are seen as the primary cause of van der Waals forces in matter. This is seen to arise from instantaneous dipole-induced dipole interactions which are 1)Inherently non-linear and 2)Purely quantum mechanical in origin. This problem cannot be treated within the current framework of quantum mechanics. So do not tell me that quantum mechanics is not limited by the harmonic approximation.
i think you are badly mixed up about what the superposition principle means in quantum mechanics. just because states in the Hilbert space can be added linearly, does not mean that the operators that govern the evolution of the system have to obey linear equations.

the linear superposition principle has nothing to do with the linearity of the equations of motion. i can understand why you have made this mistake, it is common for people learning quantum mechanics.



Give me an example of a non-linear system that can be solved non-perturbatively.

sine-gordon, or CFT

i should add, there are people who consider nonlinear quantum theories, but this is very dangerous stuff. as far as i know, nonlinear quantum mechanics violates the no-cloning theorem and therefore the second law of thermodynamics. so if you think QM should be nonlinear, i hope you are also going to reconsider the second law

but of course, this puts no constraint on the linearity of the interaction. this is my whole point.

Originally posted by ryans
You fu#$@#$% arrogant prick. You do not understand what I say. It is fine to think about philosophy and interpretation of QM, I have got no problem with that, I do it myself. But when you devote your entire life in arguing the validity of a theory without ever bothering to go and crunch the numbers, as some professionals do , that is when I have a problem!!

well that is not what you have previously stated. let me remind you what you said before:

Originally posted by ryans
The philosophical interpretation of quantum mechanics has absolutely no implications for the relevance nor completeness of the theory to describe nature, and the primary persons who feal it to be their obligation to discuss such issues, I have found to have little, if any mathematical appreciation of the theory.

so at one time you say that philosophy of quantum mechanics is irrelevant, and people who dicuss such issues don t know math.

now you say it is fine to think about philosophy and interpretations, as long as you do know math.

so it seems that whether you want to discuss philosophy or not is irrelevant to your criticism, it is just the knowing of math that you were discussing.

perhaps i can sum up your position this way: people who don t know math shouldn t talk about quantum mechanics!

if that is your position, then i agree, but what does it have to do with this thread? why did you come into the middle of a conversation about the Copenhagen interpretation to say this? was it meant to imply that someone in this thread does not know math?

So what if I am interested in the philosophy and basic ideas of quantum mechanics but I don't know the math on it yet? I am working towards learning it. In the mean time though I want to find out as much about it as I can. How it is interpreted is something I can try to understand.

Lethe and ryans, I want to let you both know I have a lot of respect for both of you (as well as many others on this forum) for your level knowledge and your willingness to help others learn.

Quoted by me

The non-local correlated motion of electrons in matter are seen as the primary cause of van der Waals forces in matter. This is seen to arise from instantaneous dipole-induced dipole interactions which are 1)Inherently non-linear and 2)Purely quantum mechanical in origin. This problem cannot be treated within the current framework of quantum mechanics. So do not tell me that quantum mechanics is not limited by the harmonic approximation

Do you understand what I said here Lethe?

Lethe

the linear superposition principle has nothing to do with the linearity of the equations of motion. i can understand why you have made this mistake, it is common for people learning quantum mechanics.


Give me an example of a non-linear inhomogenous DE that obeys the principle of superposition. You can't. In many-body interactions such as van der Waals interactions, you get a set of coupled DE's with coupled eigenmodes, which seemingly can only be treated numerically. Presently calculations within the Hartree-Fock or Density functional theory make the approximation that the system is weakly correlated and thus the particles assumed to be independant. Thus linear combination of orthonormal basis functions works. But the limitation is that effects due to correlation are unnaccounted for, and a series of successive, empirical approximations are used to recover this interaction energy.


of course i do. if Zee says it is not known how to quantize nonlinear systems, then he is just wrong. what he probably meant to say is that we don t know how to solve nonlinear differential equations (we don t), or equivalently diagonalize the interaction.

The rationale for diagonisation is due for reconstruction. It's whole preoccupation is for weakly coupled systems, which definately does not work in the field I'm in.


in fact, any central force is exactly solvable up to quadrature in quantum mechanics.

Exactly, within the harmonic approximation. Why don't you try and solve SE for the Lennard-Jones potential. This is also a central potential.

Originally posted by ryans
Do you understand what I said here Lethe?

i think i do. i think you claim that nonlinear equations are not solvable except by perturbative methods. this is of course not true, as i mention above.

Give me an example of a non-linear inhomogenous DE that obeys the principle of superposition. You can't.
i can, and in fact, already did give several examples, above

In many-body interactions such as van der Waals interactions, you get a set of coupled DE's with coupled eigenmodes, which seemingly can only be treated numerically. Presently calculations within the Hartree-Fock or Density functional theory make the approximation that the system is weakly correlated and thus the particles assumed to be independant. Thus linear combination of orthonormal basis functions works. But the limitation is that effects due to correlation are unnaccounted for, and a series of successive, empirical approximations are used to recover this interaction energy.
i don t know anything about the systems you mention, so i can t comment about them. all i can say is that if you believe that nonlinear systems are insolvable, or are counterexamples to the linearity of quantum mechanics, then you are wrong.

The rationale for diagonisation is due for reconstruction. It's whole preoccupation is for weakly coupled systems, which definately does not work in the field I'm in.

Exactly, within the harmonic approximation. Why don't you try and solve SE for the Lennard-Jones potential. This is also a central potential.
i don t know what the lennard-jones potential is.... so i cannot comment further on it.

The sine gordon equation is linear but inhomogenous, for which the superposition principle applies. I do not say that non-linear systems are non-analytically solvable, but rather I say that the principle of super-position does not apply, specifically due to there non-linear nature and eigenmode coupling!

Me

Do you understand what I said here Lethe? 3



think i do. i think you claim that nonlinear equations are not solvable except by perturbative methods. this is of course not true, as i mention above.

No, I did not say that at all, I say

Give me an example of a non-linear inhomogenous DE that obeys the principle of superposition.

sine-gordon is linear, I don't know what the CFT equation is?

Originally posted by ryans
sine-gordon is linear, I don't know what the CFT equation is?

why do you say sine-gordon is linear? this is obviously not true. are you unclear as to what nonlinear means?

let me give you a hint: sines and cosines are nonlinear

Can't we all just get along?

why do you say sine-gordon is linear? this is obviously not true. are you unclear as to what nonlinear means?

let me give you a hint: sines and cosines are nonlinear

I cannot believe I am hearing this. This is high school stuff.

Let's just start with ODE's. An ODE is said to be linear if it is of the form (a second order euation is given just as an example)

a(x)y''(x)+b(x)y'(x)+c(x)y(x)=F(x)


i.e. there are no terms such as y''(x)y(x) or y(x)^2

If F(x)=0 then the equation is also termed homogenous. A forced, damped harmonic oscillator is an example of a second order inhomogenous linear DE with constant coefficients. As I said earlier


The sine gordon equation is linear but inhomogenous, for which the superposition principle applies

I can't believe you said that this equation is non-linear. Man!

Originally posted by ryans
I can't believe you said that this equation is non-linear. Man!

perhaps your problem is that you just don t know what the sine-gordon equation is?

you don t have to tell me the definition of linear, i know it quite well, having learned it in high school, as you say. and i can assure you, the sine-gordon equation is nonlinear.

i promise you, you are wrong.

I certainly appreciate that this is a bit over my head atm but just what is a linear equation? Could it be as simple as an equation that draws straight graph? :) ( ie an equation with a constant first derivative)

Originally posted by AndersHermansson
I certainly appreciate that this is a bit over my head atm but just what is a linear equation? Could it be as simple as an equation that draws straight graph? :) ( ie an equation with a constant first derivative)

linear algebraic equations have loci that are linear spaces. for example, a linear algebraic equation in one variable is just a one dimensional linear space, in other words, a line, just as you say.

but we are really talking about differential equations here, not algebraic equations. linear differential equations have a lot of nice properties. for example, the solution spaces are always linear vector spaces. only they tend to be infinite dimensional, so we can t really draw them as lines or planes or something like that.

linearity can be summed up in a single equation: T(ax+by)=aT(x)+bT(y). linear (homogeneous) operators satisfy this in both the algebraic and the differential case (in fact, differential equations are nothing more than algebraic linear operators on infinite dimensional vector spaces)

and now to the disagreement between me and ryans. i claim that the sine-Gordon equation is nonlinear, since it contains a sine. for sines, instead of sin(A+B)=sin A + sin B, rather, i have sin(A+B)=sin A cos B + sin B cos A. this is clearly not linear.

This is easily sorted. Can one of you please post the sine-Gordon equation, and then we'll decide if it's linear or not.

Oooo, i know the sine-gordon equation. its

d^2(v)/dt^2-d^2(v)/dx^2+Sin(v)=0

where dt and dx are actaully partial derivatives but I dont know how to make the symbol and v is the function.


:):)

I looked it up at wolfram.

So i say its nonlinear.

Originally posted by cephas1012

So i say its nonlinear.

i agree with cephas

where dt and dx are actaully partial derivatives but I dont know how to make the symbol and v is the function.

You can make the symbol, &part;, with the HTML code & part ; (just don't leave any spaces in it).

AD1,

Where can one get a lits of the Html Codes?

Thank you.

A list of special characters can be found here:

http://www.chaos.org.uk/~eddy/bits/chars.html

AD1,

Thanks

Lethe I must apologise.

I obtained the form of the sine-gordon equation from a set of old lecture notes, which was of the same form as above but with the typo sin(x) instead of sin(v). I checked this against the wolfram site, and I am obviously in error. It was probably pointed out that the notes were incorrect but I did not change them, and since I do not use this equation at all in my work, I was unable to correct my mistake. That is why I made the statement


.The sine gordon equation is linear but inhomogenous

I am sorry for that

Originally posted by ryans
Lethe I must apologise.

apology accepted. no problem. it was a mixup. they happen.

would you like to return to the point under discussion? my position is that the linearity of the Hilbert space is unrelated to the linearity of the equations governing evolution of the Hilbert space, and therefore, at least in principle, nonlinearity of nature is perfectly at home with the linearity of quantum mechanics. the many examples of nonlinear quantum systems give evidence to support my case.

if the Hilbert space were itself nonlinear, this would violate the no cloning theorem and the second law of thermodynamics, this is what i have been told, although i cannot claim to know much about those proofs.

Are you familiar with Hartree-Fock theory? This is one of the first attempts at solving the quantum many body problem, whereby we proceed by making several assumptions in order to simplyfy and uncouple our system. I'll look at the case of interacting electrons within a crystal of nuclei

1)Born-Oppenheimer and clamped nucleaus approximation

2)Manybody wavefunction approximated as the product of single particle orbitals (Hartree or independant particle approximation)

3)Only 2-body interactions, and interactions with an external potential are accounted for.

How do you implement a unitary transformation on an n-body hilbert space without making the assumption that particle motion is uncorreltated, whilst correctly identifying all of the physics.

Ok. I hope we're all agreed the equation is non-linear.

Just to make it clear in the general case, if y is a function of x, then a differential equation of the form

f_n(x) Dⁿy + f_n-1(x) D^n-1y + ... + f₁(x) D¹y + f₀(x) y = g(x)

is a linear differential equation. Here, the functions f_n(x) are functions of x alone, and the notation Dⁿ y means the nth derivative of y with respect to x.

Note that the sine-Gordon equation is non-linear because it contains a term of the form sin(y), and so the d.e. doesn't fit the above general form. That would also be the case if any of the derivatives was raised to a power higher than 1.

Originally posted by lethe
if the Hilbert space were itself nonlinear, this would violate the no cloning theorem and the second law of thermodynamics, this is what i have been told, although i cannot claim to know much about those proofs.

I do not understand this. How could one make a statement about a law (the second law) that is not even rigoureously formulated for classical phasespace in a quantum case ? Do you have a reference on this ?

Bye!

Crisp

Originally posted by Crisp
I do not understand this. How could one make a statement about a law (the second law) that is not even rigoureously formulated for classical phasespace in a quantum case ? Do you have a reference on this ?

Bye!

Crisp

glurgh... i was hoping you wouldn t ask this, because i really don t know anything about it. but i should have expected that if i step into the turf of thermodynamics, i would get interested parties.

i can t provide a reference, its just something that i have read in passing, and i really don t know anything about it. so let me just regurgitate what i have read in a little more detail, and hopefully it will help.

the second law of thermodynamics would be violated if any of the following equivalent conditions were violated: if it were possible, in principle, to distinguish nonorthogonal quantum states, if the evolution of the quantum states were governed by a nonlinear operator, or if it were possible to clone an arbitrary quantum state.

i have heard shannon entropy mentioned in this context. its equivalence to thermodynamic entropy is important here (but i don t know what shannon entropy is. i know you told me once, Crisp, but i didn t get it)

the experimental validity of the second law of thermodynamics is used as confirmation that the space of quantum states must be linear.

and just to be clear, i will once again point out that this does not imply that the equations of motion must be linear.

sorry, i don t have a reference. if anyone can shed any light on why these things are true, or provide a reference, i would love to hear it.

Originally posted by ryans
Are you familiar with Hartree-Fock theory? This is one of the first attempts at solving the quantum many body problem, whereby we proceed by making several assumptions in order to simplyfy and uncouple our system. I'll look at the case of interacting electrons within a crystal of nuclei

1)Born-Oppenheimer and clamped nucleaus approximation

2)Manybody wavefunction approximated as the product of single particle orbitals (Hartree or independant particle approximation)

3)Only 2-body interactions, and interactions with an external potential are accounted for.

How do you implement a unitary transformation on an n-body hilbert space without making the assumption that particle motion is uncorreltated, whilst correctly identifying all of the physics.

unfortunately, i know next to nothing about Hartree-Fock. i believe that this method is where the notion of "effective charge" of the nucleus, or "shielding" comes from. am i correct about this? but other than that, i know next to nothing about this.

i would like to say, though, that many body problems are known to not be exactly soluble, even in classical mechanics, so i highly doubt that the problems of many body physics expose any flaws specific to quantum theory.

unfortunately, i know next to nothing about Hartree-Fock. i believe that this method is where the notion of "effective charge" of the nucleus, or "shielding" comes from. am i correct about this? but other than that, i know next to nothing about this.

Sort of. Don't worry to much about this. The thing I would like to point out is that it is the non-linear effects which are giving rise to current problems in atomic physics. To say that an n-body wavefunction is a ray in a linear Hilbert space, is I think incorrect, as the interacting bodies can give rise to things such as phonons and other such collective motion, whicj arise out of eigen mode coupling, and the fact that they do not obey the principle of superposition

"To say that an n-body wavefunction is a ray in a linear Hilbert space, is I think incorrect, as the interacting bodies can give rise to things such as phonons and other such collective motion, whicj arise out of eigen mode coupling, and the fact that they do not obey the principle of superposition"

I think the problem here is that the Schrodinger equation for each site become non-lineair becomes of some site-site coupling. Or do you have another mechanism of non-linearity in mind ?

Non-linear coupling between sites is "merely" a technical problem ofcourse, since we lack the mathematical tools to solve non-linear differential equations exactly. I think much can be said about the qualitative behaviour though, which is sometimes enough, no ?

Bye!

Crisp

I think much can be said about the qualitative behaviour though, which is sometimes enough, no ?

Qualitative is a start, but when accuracy required in the order of chemical binding energies, quantitative behaviour is extremely important as theoretical models often predict molecules such as fluorine (F2) not exist, which it clearly does.