Question on observer created reality

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vanesch

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Nicky said:
I was under the impression that Bohmian mechanics does not explain well the collapse of wavefunctions, i.e. the sudden discontinuity in Bohm's "quantum potential" after measurement.
I don't see what you mean (not pejorative: please explain). I thought that the quantum potential was a function of the actual positions of the particles (which are well-defined in Bohm's theory), and the wavefunction (the same as the one in quantum theory, and which continues to evolve according to unitary QM: that's why it has some MWI aspects). Both evolve smoothly and are not influenced particularly by a measurement, no ? A measurement (in Bohm's theory, there is only one kind of measurement possible, namely a position measurement ; hence the preferred basis problem is solved this way as there is explicitly a preferred basis: the position) just gives us information about what trajectory (from a statistical ensemble which had an initial Born rule given distribution: the deux ex machina in Bohm :-) we had.

The only objection one can have against Bohmian mechanics is its blunt violation of SR (the expression of the quantum potential) - ok, and the deux ex machina that the initial uncertainty on particle positions has to be given by the Born rule... If one is willing to let go SR, I think it is one of the finest resolutions of the measurement problem that exist. However, I find a violation of SR too gross to take it seriously.

cheers,
Patrick.
 
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vanesch said:
First of all, yes, I like the good wine and the smelly cheese (and lots of other stuff around here :-) ; one of the reasons to stick to a certain ontology :-)
: ))).

vanesch said:
I have indeed the impression that we are talking next to eachother, and I'm affraid I'm not clear on your viewpoint as well.
I'm *for the moment* a kind of MWI fan, with the hope that I'm wrong
I think so (for the impression). I am a fan of what I call the minimalist interpretation: the consistent logical content of QM formalism, especially the collapse postulate (hence, I hope it should be included in any consistent interpretation). I am sure it is not far from the logical content of MWI.

I think the main difference in our views seem to be your search for a reality behing the descriptive view given by QM. Personnally, I do not care if there is a concrete "reality", I can continue to appreciate good wine, and the rest, without that ; ))).

vanesch said:
I have difficulties with a purely epistemological view because *something* exists (a la Descartes, also from cheese and wine fame).
So you think we do not live in the matrix? :biggrin:
Ok, I understand what you mean (would like to have).

vanesch said:
Where does it lead us ?
If we take the wavefunction to be a representation of reality, clearly the collapse gives us a problem: how can a meager observation by a humble human change the state of the entire universe ?
So this "collapse" stuff must be something EPISTEMOLOGICAL.
I agree with that (all what I say is around this). QM formalism is, formally, a descriptive tool (in space and time). It does not require or assume other properties to work logically.
I would like to get a clear separation between the descriptive part (what we currently have) and the search of ontological properties or anything else (further improvements). The different posts we find in this forum highlights the confusion resulting from this mixture (especially concerning the collapse postulate and the HUP).
Moreover, most of the different existing interpretations participate to this confusion in mixing (by a lack of a clear separation in the formulation) the equivalent mathematical reformulation of the formalism (e.g. bohmian meachanics) together with the definition of external ontological properties (e.g. the path of a bohmian particle). MWI suffers from this default (e.g. especially by the choice of "many worlds", I would have chosen instead the ensemble interpretation ; ).


vanesch said:
But when you look at the "ontology" (if that's what the wavefunction represents), this measurement did in fact more to me than to the system, which is still with amplitude a in |u> and with amplitud b in |v> except that it's now entangled with my bodystates ; that's what I meant.
The main consistency problem with this interpretation relies on the , may be implicit, meaning "before" "after" and "still" words.
Collapse postulate just defines a property (description and not prediction of what a system is and not what it would be or may be, etc ...). Collapse postulate does not reject "ontology" nor it assumes it: (all properties may be assumed to be "real" as long as we keep the logical consistency of the description). The only interpretational problem comes from assuming the system had a different property "before" the property given by the measurement result (i.e. the state before the measurement and the state "after" the measurement).
Concerning this aspect, the consistent histories formalism allows one to understand better how the things work when we consider the measurement results on space and time: a measurement result is, formally, part of a collection of measurement results (which is also a "measurement result"). (A good introduction to general stochastic processes)

Seratend.
 

vanesch

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seratend said:
So you think we do not live in the matrix? :biggrin:
Even that is ontological. It is in fact pretty close to the view I'm advocating.

Concerning this aspect, the consistent histories formalism allows one to understand better how the things work when we consider the measurement results on space and time: a measurement result is, formally, part of a collection of measurement results (which is also a "measurement result"). (A good introduction to general stochastic processes)
Yes, exactly. When we learned something about a system at time t1, say: that the system was in state a|1> + b|2> and we found, at t1, that the result was "1", we are allowed to work with the state |1> alone (projection postulate) because all FURTHER measurement results are conjunct with "and at t1, the result was equal to 1". So a second measurement, at t2, with result U, is not an independent result but is a CORRELATION with "and at t1 we had result = 1". The possibility doesn't exist anymore of performing the measurement at t2, INDEPENDENT of what was the result at t1: we irreversibly HAVE that result.
This is in fact what allows us to apply unitary evolution and only apply the Born rule "at the end of the day", but this time with the measurement result "at t1 we had result 1 and at t2 we had result U". It will implicitly lead to the projection, first, of |1> and then of |U>
This is indeed the same as considering the entire history of measurement results as one single measurement.

cheers,
Patrick.
 
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vanesch said:
[...] I thought that the quantum potential was a function of the actual positions of the particles (which are well-defined in Bohm's theory), and the wavefunction (the same as the one in quantum theory, and which continues to evolve according to unitary QM: that's why it has some MWI aspects). Both evolve smoothly and are not influenced particularly by a measurement, no ? [...]
I am no expert in Bohmian mechanics, so maybe you are right. I read some criticism of Bohm's account of wavefunction collapse, but it was just informal commentary on the 'net, not a serious analysis.

On the other hand, I would point out where Bohm invokes some more "deus ex machina" as you call it. In second of two papers where Bohm lays out his theory (Phys Rev 85, p. 180 (1952)), the talk of "violent and extremely complicated fluctuations" in the wavefunction is a bit vague to me. Also he seems to take the Born rule as given:
Bohm said:
Because the probability density is equal to |psi|^2, we deduce that the apparatus variable [...] must finally enter one of the packets and remain with that packet thereafter [...] Thus, for all practical purposes, we can replace the complete wave function [...] by a new renormalized wave function
But maybe he explains this better elsewhere. I still have not understood the whole paper.
 

vanesch

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Nicky said:
I am no expert in Bohmian mechanics, so maybe you are right.
I found http://plato.stanford.edu/entries/qm-bohm/ a very useful and readable introduction. Don't mind Bohm's original papers, after all, the originator of an idea doesn't always, at that moment, have the clearest view and explication of it (it is still brand new and has to be digested: as well by himself as by others).

cheers,
Patrick.
 

George Jones

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What about the Kochen-Specker theorem? This says, roughly, that quantum systems do not, in general, possess properties.

Consider the measurement of S_z of a spin-1 particle. If the state is psi = |1> just before measurement, then, with certainty, the result of the measurement is S_z = 1, and it seems to make sense that S_z = 1 was a property of the particle before measurement. Now suppose that the state is psi = (sqrt(3)|-1> + |1>)/2 just before the measurement, and that the result of the measurement is S_z = 1. Is it possible, within the framework of standard quantum theory, to say that the particle had the property S_z = 1 before the measurent? In other words, is possible that to say that the observable S_z possessed the value 1 before the measurent?

Now generalize this. Let A be an observable for a quantum system. Is it possible that a value function V that takes as input an observable and a state of the system, and spits out, say V(A ,psi), for the value of the observable A when the system is in state psi?

No one has ever thought of a way of doing this. Is this because no one has been able to hit on the right idea, or is it because it is impossible? If it were possible to do this, many of the ontological "problems" associated with quantum theory would disappear, and realists, like Patrick (I think) and me, would be happier campers :-). The Kochen-Specker theorem says that for quantum systems having state spaces of dimension 3 or higher, no value function (with reasonable properties) exists. :-(

This one of the things that makes a "disturbance" interpretation of the uncertainty principle difficult. If a system doesn't possess a particular property, how is it possible to disturb that property.

Nice discussions of the Kocken-Specker theorem can be found in Lectures on Quantum Theory: Mathematical and Structure Foundations by Chris Isham, and in The Structure and Interpretation of Quantum Mechanics by R.I.G.Hughes. Hughes is a philosopher, and some philosophy he espouses in his preface is particularly interesting: "Having thus outlined my program and declared my allegiances, I leave the reader to decide whether to proceed further, or to open another beer, or both."

The Kochen-Specker theorem makes life somewhat difficult for realists, but positivists probably have no problem with it. Isham says "... very effective professionally without needing to lose any sleep over the implications of the Kochen-Specker theorem or the Bell inequalities. However, in the last twenty years there has been a growing belief among physicists from many different specialisations that even if modern quantum theory works well at the pragmatic level, it simply cannot be the last word on the matter."

Patrick, this is essentially your hope, isn't it? Maybe a subtle 4).

Isham says further "... there are now any any physicists who feel that any significant advance will entail a radical revision in our understanding in the meaning and significance of the categories of space, time and substance."

To me, this seems to point towards quantum gravity, which is no surprise coming from Isham. Many physicists would disagree this, but I am inclined to believe that a theory of quantum gravity might have interesting things to say about these issues. Unfortunately, despite claims by various camps to the contrary, I see no viable candidate for a quantum theory of gravity. Also, such a theory could bring forth new interpretational issues just as thorny as any it alleviates.

Regards,
George
 
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vanesch said:
In order to say so, you have to consider the measurement system "classical", in that you put a "classical" potential in the Schroedinger equation of the electron. In doing so, you've neglected the quantum nature of the microscope. Of course, in practice this works in many occasions (semi-classical models work very well in different fields) ; after all that is the correspondence principle.
The potential was introduced to give an idea of what might be happening.The wavefunction changes by the introduction of a potential(in this case it gets localized).This at least makes the measurement process less mysterious.In any case I also mentioned the interaction of the photon field with the electron field to be fully quantum--this is more difficult to calculate,but I think it is worth exploring(and my guess is that the 'potential' mimics this effect well).With my little knowledge of QFT,I am not in a position to do it---but you or other QFTists in the forum can certainly do a small calculation--may be localization comes out of such two interacting fields.

that's true for those who think that decoherence solves the problem (they use the density matrix, which itself is based upon the Born rule and the projection postulate), it is even true for Deutsch (who introduces some "reasonable assumptions" for his rational deciders which comes down to the Born rule).
I agree with this totally--when you sum up over the bath coordinates,you are in a way already using the Born rule.But what I am saying is different--you are introducing a quantum interaction(mimicked roughly by a potential)when you do a measurement and this localizes the wavefunction.I think sudden introduction of a potential has merit in the following sense---evolution is unitary in the absence of the potential as well as when you have the potential 'on',but in between these two phases is the evolution unitary?May be not,but I need to think about this.
 

vanesch

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gptejms said:
I think sudden introduction of a potential has merit in the following sense---evolution is unitary in the absence of the potential as well as when you have the potential 'on',but in between these two phases is the evolution unitary?May be not,but I need to think about this.
According to QM, of course it is unitary ! There is nothing else that can give you a time evolution. A time-dependent hamiltonian still gives you a unitary time evolution.
I think the problem in your approach is that you refuse to see the "thing that applies the potential" as a quantum object. I agree with you that FOR ALL PRACTICAL PURPOSES you can do that, by restricting your attention to the "system", and find a quite good approximation to the quantum evolution of the system alone ; also you wouldn't find much of a difference for a "full" quantum treatment IF AT ANY POINT YOU APPLY THE BORN RULE. But *applying the Born rule* means that you switch to a classical world, and unitary evolution doesn't apply there. That's the essential contents of the Copenhagen view: that there is a macroscopic, classical world, and that there is a microscopic quantum world and that both "communicate" by the Born rule (micro -> macro) and the projection postulate (macro -> micro). But this means that there is a barrier separating the two. So the macroscopic world is then NOT subject to quantum theory, which would be, in itself, an amazing statement: we don't have the same laws, and it is not exactly clear what objects obey what set of rules.
If you refuse that, and you say that EVERYTHING is ruled by quantum theory however, then you are stuck with your unitary evolution all the way up. This is the case I was considering.

cheers,
Patrick.
 
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vanesch said:
According to QM, of course it is unitary ! There is nothing else that can give you a time evolution. A time-dependent hamiltonian still gives you a unitary time evolution.
When I wrote that I had a step function kind of a potential in mind--a potential that's introduced all of a sudden.Say the particle is free to begin with and has a wavefunction [tex] \exp[\iota(k_1 x - \omega_1 t)] [/tex].Now a potential is introduced say at time [tex] t_1 [/tex] which to begin with we take to be constant.The wavefunction now readjusts(over some time) to [tex] \exp[\iota(k_2 x - \omega_2 t)] [/tex],where the new variables take care of the change in energy.What happens at [tex] t=t_1 [/tex]:-do the two wavefunctions merge?No they don't---there is some funny wavefunction during the readjustment phase which is a mixture of the two.Can you derive it from the Schrodinger equation?Can you give me the readjustment/response time?

Another problem:-Say the potential introduced is that of a harmonic oscillator,so the wavefunction not only goes from a plane wave to a Hermite Gauss function(if I remember correctly),but also the particle now falls into the nearest eigenstate i.e. it gains or loses some energy--can you describe this process thoroughly through unitary evolution/Schrodinger equation?May be you can,I need to think about these things.

I think the problem in your approach is that you refuse to see the "thing that applies the potential" as a quantum object. I agree with you that FOR ALL PRACTICAL PURPOSES you can do that, by restricting your attention to the "system", and find a quite good approximation to the quantum evolution of the system alone ; also you wouldn't find much of a difference for a "full" quantum treatment IF AT ANY POINT YOU APPLY THE BORN RULE. But *applying the Born rule* means that you switch to a classical world, and unitary evolution doesn't apply there. That's the essential contents of the Copenhagen view: that there is a macroscopic, classical world, and that there is a microscopic quantum world and that both "communicate" by the Born rule (micro -> macro) and the projection postulate (macro -> micro). But this means that there is a barrier separating the two. So the macroscopic world is then NOT subject to quantum theory, which would be, in itself, an amazing statement: we don't have the same laws, and it is not exactly clear what objects obey what set of rules.
If you refuse that, and you say that EVERYTHING is ruled by quantum theory however, then you are stuck with your unitary evolution all the way up. This is the case I was considering.
On the contrary I am taking everything to be quantum.What I am stressing is this:--what a measurement does is not to yield a particle with a definite x and a definite p.When you have made a measurement say by Heisenberg's microscope at one of the slits,you still observe a particle with some [tex] \Delta x [/tex] and a [tex] \Delta p [/tex] i.e. it is still a wavefunction--the orignal wavefunction corresponding to the two slits has now shrunk,but it's still a wavefunction or a quantum particle(without a definite x or p) that you observed when you made a measurement.I think the mistake one does is to think that 'since the particle has been observed,it must have a definite x and a definite p' and then one starts questioning how this mysterious collapse(of the wavefunction) comes out of unitary evolution.But if you keep in mind that all you do is to localize the wavefunction when you make a measurement,you can argue that the measurement process is akin to the introduction of a potential.
 

vanesch

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gptejms said:
Another problem:-Say the potential introduced is that of a harmonic oscillator,so the wavefunction not only goes from a plane wave to a Hermite Gauss function(if I remember correctly),but also the particle now falls into the nearest eigenstate i.e. it gains or loses some energy--can you describe this process thoroughly through unitary evolution/Schrodinger equation?May be you can,I need to think about these things.
First of all, I'd like to re-iterate my remark that there is no *interaction* between two systems that introduces a "potential" in the Schroedinger equation. You only introduce INTERACTION TERMS in the product hilbert space, and if the initial (product) state of both systems is not an eigenstate of that interaction term (which it most probably isn't), this will give rise to a time evolution of the total state which is an entangled state of both systems.

But now to come to your specific problem of the sudden introduction of a harmonic potential: clearly the particle will NOT evolve to an energy eigenstate, for the following simple reason:
at t0- (just before the potential is introduced), we assume that the particle is, say, in a plane wave state. Of course, we require continuity of the wavefunction (otherwise the schroedinger equation doesn't make sense: it is a second order differential equation), so at t0+ the particle is now still in this plane wave state. We now rewrite that plane wave state as a superposition of eigenstates of the harmonic potential (the Hermite functions). Clearly we do not have just one term !
Given that each individual term is a stationary state, the time evolution from t0+ onward is given by the hamiltonian of the harmonic potential, so each term (stationary state) just gets a phase factor exp(-i E_n t).
No term will "decay". So we remain in this superposition for ever.

cheers,
Patrick
 
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vanesch said:
First of all, I'd like to re-iterate my remark that there is no *interaction* between two systems that introduces a "potential" in the Schroedinger equation. You only introduce INTERACTION TERMS in the product hilbert space, and if the initial (product) state of both systems is not an eigenstate of that interaction term (which it most probably isn't), this will give rise to a time evolution of the total state which is an entangled state of both systems.
You introduce a 'nucleus' and say you your system of interest is an electron--don't the two things interact?You mean the nucleus just introduces a potential?!

But now to come to your specific problem of the sudden introduction of a harmonic potential: clearly the particle will NOT evolve to an energy eigenstate, for the following simple reason:
at t0- (just before the potential is introduced), we assume that the particle is, say, in a plane wave state. Of course, we require continuity of the wavefunction (otherwise the schroedinger equation doesn't make sense: it is a second order differential equation), so at t0+ the particle is now still in this plane wave state. We now rewrite that plane wave state as a superposition of eigenstates of the harmonic potential (the Hermite functions). Clearly we do not have just one term !
Again I take the example of an electron approaching a nucleus.Take for the moment the electron to be a classical particle.Depending on its speed,it should either fall into a circular orbit(rather spiral into the nucleus) or be scattered off.In a quantum scenario,if it is scattered off, it doesen't remain the same plane wave--its wavevector changes.If,as you say, the plane wave remains as it is,the electron should merrily pass on away from the nucleus with the same momentum and in the same direction as if nothing has happened--obviously this is not correct.In the case the electron gets trapped by the nucleus,the wavefunction again changes.

Let's however remain focused on the central issue.I stated that when you 'observe' a particle you are just observing a shrunk form of the wavefunction--the particle is still quantum and has a delta x and a delta p.The only thing a measurement does is to give you this shrunk form which comes out very well from quantum mechanics itself.Need to see some comments on this.
 
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gptejms said:
Again I take the example of an electron approaching a nucleus.Take for the moment the electron to be a classical particle.Depending on its speed,it should either fall into a circular orbit(rather spiral into the nucleus) or be scattered off.In a quantum scenario,if it is scattered off, it doesen't remain the same plane wave--its wavevector changes.
Attention, you are attaching a reality to the wave function (in the same way you may speak wrongly about the path of particle in a double slit experiment). Try to think with the heisenberg view where the wave function is a constant in time in order to avoid false deductions.
Note that in the quantum as well as in the classical scenario, a particle may be scattered of or captured by the scatterer (it depends on the interactions).

Seratend.
 

vanesch

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gptejms said:
Again I take the example of an electron approaching a nucleus.
Ok, it is an interesting example.

Take for the moment the electron to be a classical particle.Depending on its speed,it should either fall into a circular orbit(rather spiral into the nucleus) or be scattered off.
Let us more say that the speed is fixed, and that we work with the impact parameter b (the distance between the straight line on which the incoming electron moves, and the nucleus) in the classical case.

Indeed, for a large impact parameter, the particle almost goes straight through, while for smaller and smaller impact parameters, first the particle track bends lightly, a bit and finally, for small enough values, the particle scatters under large angles. Let us not consider the case of capture. The nucleus will undergo the appropriate recoil for momentum to be conserved.

In a quantum scenario,if it is scattered off, it doesen't remain the same plane wave--its wavevector changes.If,as you say, the plane wave remains as it is,the electron should merrily pass on away from the nucleus with the same momentum and in the same direction as if nothing has happened--obviously this is not correct.In the case the electron gets trapped by the nucleus,the wavefunction again changes.
In a semi-quantum scenario (incoming particle quantum, nucleus classical), if the particle's momentum is well known, it is an incoming plane wave, and hence it "contains" all impact parameters at once. The nucleus being considered classical, we can replace it by a potential in the electron's schroedinger equation. We now solve that Schroedinger equation, and out will come a rather complicated looking wavefront. If we apply the Born rule to it (meaning, we make a transition to classical), it gives us the probability of the particle to scatter under a certain angle. From that *classical* result, we can then find the corresponding probability of the nucleus to recoil with a certain momentum.

In the full quantum scenario, the initial state of the nucleus-electron system is a product state |nucleus> |electron> in which the nucleus corresponds to some localized lumpy wavefunction, and the electron an incoming plane wave. Now, there is no external potential, but an interaction term in the hamiltonian over the hilbert space of the two particles, which describes their Coulomb interaction. If we apply this hamiltonian's time evolution to
|nucleus> |electron>, this product state will evolve in an entangled state, which I intuitively think will take on something like:

|psi_final> = Integral db f(b) |nucleus-plane-wave-p(b)> |electron-plane-wave-p(b)>

Here, f(b) is some complex number as a function of a variable which I call b, because it more or less corresponds to some impact parameter, and the two states correspond to a product state of a plane wave with momentum p(b) for the nucleus and a plane wave with momentum - p(b) for the electron.

I'm not sure about this, I just guess it: it is my guess for the form of the solution of the schroedinger equation (in Schmidt decomposition).

So we end up with a global wavefunction for the nucleus and the electron, which is not a product state. If we now apply the Born rule to this system, measuring both momenta of nucleus and electron, we notice several things:

1) there is a very strong correlation between the momentum of the electron and the momentum of the nucleus (they are exactly opposite). That's what you obtained classically by imposing momentum conservation, but it will simply come out of the Schroedinger equation.

2) the probability distribution of the momenta of the electron-nucleus system will be VERY CLOSE to the probability distribution of the semiclassically obtained potential-scattering solution, where the nucleus was replaced by a potential.

I didn't claim that the wave function of the electron had to remain "as if nothing happened" ; after all there is an interaction with a nucleus ; and this entangles both wavefunctions, and transforms them unitarily. But you see again that no SINGLE scattering angle comes out of this: all angles are realised (that is: there is a term in the final wavefunction corresponding to each possible scattering angle for the electron) and to each angle for the electron, corresponds the matched angle for the nucleus.

Let's however remain focused on the central issue.I stated that when you 'observe' a particle you are just observing a shrunk form of the wavefunction--the particle is still quantum and has a delta x and a delta p.The only thing a measurement does is to give you this shrunk form which comes out very well from quantum mechanics itself.Need to see some comments on this.
If you mean that you got now entangled with that "shrunk part" (which is just a term in the overall wavefunction), then yes, I agree of course!
If you say that quantum theory by itself unitarily shrunk the wavefunction, I'm telling you that that is mathematically impossible for a unitary operator.

cheers,
Patrick.
 

vanesch

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I want to lift a potential confusion:

I said:

vanesch said:
We now rewrite that plane wave state as a superposition of eigenstates of the harmonic potential (the Hermite functions). Clearly we do not have just one term !
Given that each individual term is a stationary state, the time evolution from t0+ onward is given by the hamiltonian of the harmonic potential, so each term (stationary state) just gets a phase factor exp(-i E_n t).
No term will "decay". So we remain in this superposition for ever.
However, this doesn't mean that, after t0+, the particle remains in a PLANE wave! This wavefunction will evolve now according to the harmonic potential hamiltonian, and this will "dephase" the different terms in the initial plane wave decomposition in Hermite functions, so that after a finite time, their recomposition will give you something else than a plane wave. I only showed you that whatever that complicated function is, it cannot decay into the ground state of the harmonic potential.
 
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vanesch said:
If you say that quantum theory by itself unitarily shrunk the wavefunction, I'm telling you that that is mathematically impossible for a unitary operator.
Only for finite times. : ))

Seratend.
 

vanesch

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seratend said:
Only for finite times. : ))
I don't know exactly what you mean ? Do you allude to the result of DeWitt (I think) that all branches in which there is a serious deviation from the Born rule have a total (Hilbert) norm 0 for t-> Infinity ?
Or do you allude to the slightly imaginary slope given to the time axis to derive the LZH result (but that, to me, is just a sloppy trick) ?
Or still something else ?

cheers,
Patrick.
 
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vanesch said:
I don't know exactly what you mean ? Do you allude to the result of DeWitt (I think) that all branches in which there is a serious deviation from the Born rule have a total (Hilbert) norm 0 for t-> Infinity ?
Or do you allude to the slightly imaginary slope given to the time axis to derive the LZH result (but that, to me, is just a sloppy trick) ?
Or still something else ?

cheers,
Patrick.
: ).

We just need to take any translation generator. We have (with the good units) <x|exp(-ip.a)|psi> = psi(x+a) for all finite a and all states |psi> but not necessarily for infinite a (a group is closed under finite operations and not necessarily for infinite operations) where everything is possible (without external properties on this limit, e.g. sigma additivity).
The rest is just application of this property (e.g. classical limits, decoherence etc ...).

Some formal applications of this result for the time (i.e. the unitary evolution) are used in the formal study of the boundary limit between classical world and quantum world especially in the decoherence area . If you are interested on this aspect I recommend the excellent paper of of N.P. Landsman "Between Classical and quantum" quant-ph/0506082 (especially section 6.6 and 7 that deals with the results of infinite time on measurements).
(Note: it requires some knowledge on the C*-algebra formalism results, but it has a lot of pointers).

(I hope you will appreciate this paper as it may also help you in your search for an ontological reality :biggrin: ).

Seratend.

P.S. My comment does not invalidate what you say, just that we need to take some care with the unitary evolution for the sake of consistency.

P.P.S. I hope you may have a new view on the scattering matrix and more precisely the “limit at t --> +oO”.
 
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seratend said:
Attention, you are attaching a reality to the wave function (in the same way you may speak wrongly about the path of particle in a double slit experiment).
When you have shrunk the wavefunction to one slit,obviously the particle's path is certain to the the extent that the particle passes through this slit.

Try to think with the heisenberg view where the wave function is a constant in time in order to avoid false deductions.
Good suggestion.If the wavefunction is a constant,obviously no operator can cause the particle to pass through one slit only.However when you suddenly introduce a potential,you can no longer continue to work with the old wavefunction.

Note that in the quantum as well as in the classical scenario, a particle may be scattered of or captured by the scatterer (it depends on the interactions).
Of course, and I have mentioned the latter case also.
 
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vanesch said:
If we apply this hamiltonian's time evolution to |nucleus> |electron>, this product state will evolve in an entangled state, which I intuitively think will take on something like:

|psi_final> = Integral db f(b) |nucleus-plane-wave-p(b)> |electron-plane-wave-p(b)>

Here, f(b) is some complex number as a function of a variable which I call b, because it more or less corresponds to some impact parameter, and the two states correspond to a product state of a plane wave with momentum p(b) for the nucleus and a plane wave with momentum - p(b) for the electron.

I'm not sure about this, I just guess it: it is my guess for the form of the solution of the schroedinger equation (in Schmidt decomposition).
Ok,the nucleus is now entangled with the electron.You will use this to argue that the measuring apparatus also gets entangled with the system being measured etc. etc. and finally give only a 'conscious' observer the authority to break the superposition(by MW splitting) and that too to your own consciousness only and not that of a cat or for that matter not even any other human--this is a kind of wishful thinking that I don't subscribe to.

If you mean that you got now entangled with that "shrunk part" (which is just a term in the overall wavefunction), then yes, I agree of course!
If you say that quantum theory by itself unitarily shrunk the wavefunction, I'm telling you that that is mathematically impossible for a unitary operator.
When you are entangled with the shrunk part,at least the electron passed through a particular slit--you are entangled with one slit only.Regarding the unitarity,I like seratend's suggestion of the Heisenberg picture.Read my response to him.
 
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vanesch said:
I want to lift a potential confusion:

I said:



However, this doesn't mean that, after t0+, the particle remains in a PLANE wave! This wavefunction will evolve now according to the harmonic potential hamiltonian, and this will "dephase" the different terms in the initial plane wave decomposition in Hermite functions, so that after a finite time, their recomposition will give you something else than a plane wave. I only showed you that whatever that complicated function is, it cannot decay into the ground state of the harmonic potential.
Say you have a plane wave [tex] \exp[i(kx-\omega t)] [/tex] to start with.You Fourier analyze the x part of this to Hermite functions---the time dependence continues to be [tex] \exp(-i \omega t) [/tex] !If you insist that you'll Fourier analyze the t part also into different frequencies,even then that grand sum will sum up to act like [tex] \exp(-i \omega t) [/tex] only.Now you may say,you meant that the t part would change upon interaction--ok fine,but then you see that the energy has changed.In your model the x part has remained the same i.e. the momentum has not changed,whereas the energy has--I see no point in preserving the momentum.
 

vanesch

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seratend said:
If you are interested on this aspect I recommend the excellent paper of of N.P. Landsman "Between Classical and quantum" quant-ph/0506082 (especially section 6.6 and 7 that deals with the results of infinite time on measurements).
(Note: it requires some knowledge on the C*-algebra formalism results, but it has a lot of pointers).

(I hope you will appreciate this paper as it may also help you in your search for an ontological reality :biggrin: ).
Thanks for the reference ! I started (p 20) reading it (but it's LONG ! 100 p).

cheers,
Patrick.
 

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gptejms said:
Good suggestion.If the wavefunction is a constant,obviously no operator can cause the particle to pass through one slit only.However when you suddenly introduce a potential,you can no longer continue to work with the old wavefunction.
Of course you can ! The potential now just changes the evolution of the observables. "Suddenly introducing a potential" V(x) at time t0 just comes down to having a hamiltonian:

H(t) = H0 + h(t-t0) V(x)

(h(t) is the Heavyside step function)

U(t) is the solution to the equation:

hbar dU/dt = - i H(t)

and is a unitary operator

(the solution is not anymore exp(-i H/hbar t) because that only applies for time invariant hamiltonians).

Given your (time independent) waverfunction psi (which, say, coincides with the Schroedinger view at t = ta), we have now:
x(t) = U(t)U_dagger(ta) x U_(ta) U_dagger(t)
and the same for p(t),
with x and p the Schroedinger position and momentum operators, and x(t) and p(t) the Heisenberg position and momentum operators, evolving according to your hamiltonian with sudden potential.

psi remains always psi(ta).

cheers,
Patrick.
 
Last edited:
318
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gptejms said:
Good suggestion.If the wavefunction is a constant,obviously no operator can cause the particle to pass through one slit only.However when you suddenly introduce a potential,you can no longer continue to work with the old wavefunction.
?? What do you mean?
If the potential is bounded, you still have your constant state.

Seratend.

EDIT:

P.S. Damned too slow : ))
 

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gptejms said:
Say you have a plane wave [tex] \exp[i(kx-\omega t)] [/tex] to start with.You Fourier analyze the x part of this to Hermite functions---the time dependence continues to be [tex] \exp(-i \omega t) [/tex] !
No, it doesn't of course ! This time dependence is what you get for the free hamiltonian ; this time dependence was correct as long as the potential was not switched on (for t < t0, say). But when the hamiltonian changes (it is a time-dependent hamiltonian, because some potential is switching on at t = t0), all these different hermite functions get DIFFERENT omegas (namely E_n / hbar).

cheers,
Patrick.
 

vanesch

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gptejms said:
Ok,the nucleus is now entangled with the electron.You will use this to argue that the measuring apparatus also gets entangled with the system being measured etc. etc. and finally give only a 'conscious' observer the authority to break the superposition(by MW splitting) and that too to your own consciousness only and not that of a cat or for that matter not even any other human--this is a kind of wishful thinking that I don't subscribe to.
Apart from your last statement, yes, you captured my thought :-) HOW do you "break the superposition" ? My hope is that gravity will, somehow, do so, because I'd also rather not subscribe to this view, but I simply don't know how to avoid it if unitary QM is correct, and we're talking here about how to interpret unitary QM.

However, you missed a point: I'm not saying that *my* consciousness somehow breaks the superposition, as some objective thing it does to the state of the universe, as if I were somehow a special physical or devine construction. The superposition is still there ; my consciousness just only observes one term of it. Now, of course, TO ME, that comes down to exactly the same thing (and that's why the projection postulate WORKS FAPP) - epistemologically. The problem only exists if you say that there is a real world out there (ontologically) and that its objective state is given by the wavefunction ; if QM is correct on all levels, this wavefunction cannot do anything else but evolve according to a Schroedinger equation, and if it does, there's no way to break the superposition.
(except if one is going to introduce infinities and so on, but if we take it that the universe has only a finite, but very large, number of degrees of freedom, I don't see how this is going to solve the issue).
As not one single physical interaction in the universe (possessing a hamiltonian) can break the superposition, clearly something unphysical has to do it when our subjective observations are to be explained.
Again, IF unitary QM is correct.
 

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