Schrodinger's Wave Equation: Understanding Observability

[batabek]

Hi everyone,
I want to know why Schrödinger's wave equation is dependent to observation.If we look at the equation, I don't see any relation to dependent variable such as observer x. So if we think like looking down the wave function, are we just capturing some snapshots from the wave function of the system ? Isn't that what observability means ? If you can give some guidance, I'll be really appreciated.

 

[Demystifier]

Observer or a measurement apparatus consists of many particles, so a single-particle Schrödinger equation cannot include the observer. To include the observer, you must study the Schrödinger equation of a very large number of particles. Of course, in practice such systems cannot be studied exactly, but there are well developed concepts and techniques that allow to study such systems approximately. Of course, I cannot here present the details of this theory, but let me just give you a hint: The crucial concept related to the the quantum theory of measurement is - DECOHERENCE.

If you want to learn more, here are some good introductory reviews on decoherence, measurement and related stuff:
http://lanl.arxiv.org/abs/quant-ph/0312059
http://xxx.lanl.gov/abs/quant-ph/0306072
http://lanl.arxiv.org/abs/quant-ph/0505070
http://lanl.arxiv.org/abs/quant-ph/9908008
http://lanl.arxiv.org/abs/quant-ph/9803052

 

[Jolb]

The decoherence argument is only one possible view on the "observer effect" of quantum mechanics. In my opinion, decoherence is quite contrived and unnatural in quantum physics, and was only argued to deny the existence of wavefunction collapse. Wavefunction collapse is not only something you learn on the first pages of any quantum textbook, it also displays how observations change a system much more clearly.

Anyway, in quantum physics, the state of any system (whether it be one particle or many) is described by a ket |ψ> in Hilbert Space. Every measurable quantity is represented by an hermitian operator Q which acts on kets in Hilbert space. Any measurement of the quantity represented by Q must result in one of Q's eigenvalues, {Q1, Q2, Q3, ...}, and corresponding to these eigenvalues, Q has a set of eigenkets {|ψ_Q1>, |ψ_Q2>, |ψ_Q3>, ...} which form an orthonormal basis for the Hilbert space. These eigenkets are referred to as "pure states" of the observable Q. If you measure the Q on the state |ψ_Qm>, the measurement will result in Qm, with absolute certainty.

Here's where the observer effect comes in: in general, a ket is NOT in a pure state |ψ_Qm>, but is rather in a superposition of eigenkets. Here's an example: |ψ> = 2^(-1/2)(|ψ_Q1>+|ψ_Q3>). Because this is made in equal parts of the first eigenket and the third eigenket, a measurement on this state will result in either Q1 or Q3 with equal probability. [This is akin to an electron passing through two slits simultaneously or a cat being dead and alive at once.]

The thing is, the instant you measure 2^(-1/2)(|ψ_Q1>+|ψ_Q3>), you get either Q1 or Q3. If you were to get Q3, the state is instantly changed from 2^(-1/2)(|ψ_Q1>+|ψ_Q3>) into |ψ_Q3>. This is a discontinuous evolution of the state ket and is NOT consistent with the Schrödinger equation.

Thus the act of measurement causes a general quantum state to collapse into an eigenstate of the measurement operator--this discontinuous evolution of the state is not described by the Schrödinger equation and represents a wholly different quantum evolution than Schrödinger evolution.

A good example of this weirdness is called the Quantum Zeno Effect, or the Watched Pot Effect. As the rate of measurements on a quantum system approaches infinity, the quantum system's evolution stops. This is because the wavefunction is continually being collapsed back to an eigenstate. This has even been observed experimentally: if you observe whether a radioactive atom has decayed or not rapidly enough, its half life will get longer!

 

[lavinia]

Hi everyone,
I want to know why Schrödinger's wave equation is dependent to observation.If we look at the equation, I don't see any relation to dependent variable such as observer x. So if we think like looking down the wave function, are we just capturing some snapshots from the wave function of the system ? Isn't that what observability means ? If you can give some guidance, I'll be really appreciated.

The Shroedinger equation describes the evolution of the wave function. The wave function is not directly observable.

 

[batabek]

Observer or a measurement apparatus consists of many particles, so a single-particle Schrödinger equation cannot include the observer. To include the observer, you must study the Schrödinger equation of a very large number of particles. Of course, in practice such systems cannot be studied exactly, but there are well developed concepts and techniques that allow to study such systems approximately. Of course, I cannot here present the details of this theory, but let me just give you a hint: The crucial concept related to the the quantum theory of measurement is - DECOHERENCE.

If you want to learn more, here are some good introductory reviews on decoherence, measurement and related stuff:
http://lanl.arxiv.org/abs/quant-ph/0312059
http://xxx.lanl.gov/abs/quant-ph/0306072
http://lanl.arxiv.org/abs/quant-ph/0505070
http://lanl.arxiv.org/abs/quant-ph/9908008
http://lanl.arxiv.org/abs/quant-ph/9803052

thanks for the documents, I'll look over them

 

[Demystifier]

In my opinion, decoherence is quite contrived and unnatural in quantum physics,

Just the opposite, decoherence is something that can be derived from the Schrödinger equation itself, and in practice it is very difficult to avoid it whenever the system strongly interacts with the environment.

and was only argued to deny the existence of wavefunction collapse. Wavefunction collapse is not only something you learn on the first pages of any quantum textbook, it also displays how observations change a system much more clearly.

Decoherence alone cannot replace or explain the collapse. But it tells WHEN and IN WHAT BASIS the collapse will occur. What it does not tell is WHY and IN WHICH PARTICULAR STATE the collapse will happen.

A good example of this weirdness is called the Quantum Zeno Effect, or the Watched Pot Effect. As the rate of measurements on a quantum system approaches infinity, the quantum system's evolution stops. This is because the wavefunction is continually being collapsed back to an eigenstate. This has even been observed experimentally: if you observe whether a radioactive atom has decayed or not rapidly enough, its half life will get longer!

That's true, but a correct QUANTITATIVE description of Quantum Zeno Effect cannot be made without decoherence.

 

[lugita15]

Just the opposite, decoherence is something that can be derived from the Schrödinger equation itself, and in practice it is very difficult to avoid it whenever the system strongly interacts with the environment.

But can we really know that decoherence is really occurring, and not real wavefunction collapse? Here's a question I asked in another thread; maybe you can answer it for me.

A thought experiment in Ghirardi's "Sneaking a look at God's Cards" purports to provide a means of empirically distinguishing between actual wavefunction collapse and decoherence. (In fact Ghirardi apparently makes a bolder claim, that this is an empirical test of the Copenhagen interpretation!). Here's how it works: if A is the observable whose eigenstates form the pointer basis of an apparatus, Ghirardi proposes to perform a measurement on an observable Z of the apparatus which is incompatible with A. Does anyone know whether such an experiment has been performed? In practice our apparatus has a position pointer basis, because we have to read off the position of the pointer, so we would have to somehow perform a momentum measurement of the apparatus pointer or something.

Decoherence alone cannot replace or explain the collapse. But it tells WHEN and IN WHAT BASIS the collapse will occur. What it does not tell is WHY and IN WHICH PARTICULAR STATE the collapse will happen.

I thought decoherence tells us not only why a particular pointer basis is chosen for the appearance of collapse, but also how the particular choice of pointer state into which the collapse appears to have occurred is chosen, but that in practice we cannot know what that pointer state is going to be in advance, because of the classical statistical uncertainty we have concerning the initial quantum state of the system and the exact details of the Hamiltonian of the system.

That's true, but a correct QUANTITATIVE description of Quantum Zeno Effect cannot be made without decoherence.

Are you saying that it's impossible to get an exact quantitative description of the quantum zeno effect using real wavefunction collapse?

 

[kith]

I thought decoherence tells us not only why a particular pointer basis is chosen for the appearance of collapse, but also how the particular choice of pointer state into which the collapse appears to have occurred is chosen, but that in practice we cannot know what that pointer state is going to be in advance, because of the classical statistical uncertainty we have concerning the initial quantum state of the system and the exact details of the Hamiltonian of the system.

Mathematically, decoherence is the decay of the off-diagonal elements of the system density matrix in a specific basis. In the end, you always have a mixed state. So decoherence doesn't tell us which state will be chosen in a measurement.

I don't think this is due to statistical uncertainties or the Hamiltonian, because this basic feature is also present without dynamics and for well-defined initial states. If the initial state of system+environment is an entangled state, the system alone is in a mixed state already.

 

[lugita15]

Mathematically, decoherence is the decay of the off-diagonal elements of the system density matrix in a specific basis. In the end, you always have a mixed state. So decoherence doesn't tell us which state will be chosen in a measurement.

I don't think this is due to statistical uncertainties or the Hamiltonian, because this basic feature is also present without dynamics and for well-defined initial states. If the initial state of system+environment is an entangled state, the system alone is in a mixed state already.

But isn't any lack of knowledge concerning what quantum state the system+environment will appear to have collapsed into just statistical uncertainty about the initial quantum state of the system+environment, or ignorance about the precise details of the Hamiltonian?

 

[Demystifier]

But can we really know that decoherence is really occurring, and not real wavefunction collapse?

In some cases, yes.

I thought decoherence tells us ... how the particular choice of pointer state into which the collapse appears to have occurred is chosen, but that in practice we cannot know what that pointer state is going to be in advance, because of the classical statistical uncertainty we have concerning the initial quantum state of the system and the exact details of the Hamiltonian of the system.

Unfortunately, decoherence does not tell us that. If it did, there would be no need for various interpretations of QM.

Are you saying that it's impossible to get an exact quantitative description of the quantum zeno effect using real wavefunction collapse?

Yes, that's what I am saying. But still, a collapse may often serve as a good approximation.

 

[kith]

But isn't any lack of knowledge concerning what quantum state the system+environment will appear to have collapsed into just statistical uncertainty about the initial quantum state of the system+environment, or ignorance about the precise details of the Hamiltonian?

Maybe, but this is more of an interpretational question. The "missing information answer" leads probably to Bohmian mechanics or other hidden-variables theories.

 

[lugita15]

In some cases, yes.

Could you elaborate on this? Also, do you have any thoughts on the question I quoted in my post above, concerning measurements of an observable conjugate to that of the pointer basis?

Unfortunately, decoherence does not tell us that. If it did, there would be no need for various interpretations of QM.

I'm a little confused. I thought in the absence of wave function collapse, everything is completely determined by Schrödinger evolution, so the pointer state into collapse appears to have occurred can be predicted in advance, so that any doubt about what the specific pointer state will be must arise from classical ignorance concerning the initial state or the Hamiltonian.

Yes, that's what I am saying. But still, a collapse may often serve as a good approximation.

So are you saying that the Copenhagen interpretation can be disproven by the quantitative details of quantum Zeno effect experiments? That's a rather bold claim.

 

[Jolb]

I feel as though Demystifier has made tons of claims without any arguments to back them up. I gave a perfectly quantitative explanation of wavefunction collapse (which doesn't go very far beyond the basic postulates of quantum mechanics), and under the assumption that the Schrödinger equation evolves wavefunctions continuously through Hilbert space, it should be obvious that it immediately implies the Quantum Zeno effect. If you disagree that it constitutes a perfectly quantitative demonstration of the Quantum Zeno effect, then I refer you to Griffiths Intro to Quantum Mechanics Section 12.5 (second ed. p 431-433).

So please back up your following claims, which you have merely stated without any support, quantitative or otherwise.

1)

Just the opposite, decoherence is something that can be derived from the Schrödinger equation itself, and in practice it is very difficult to avoid it whenever the system strongly interacts with the environment.

2)

That's true, but a correct QUANTITATIVE description of Quantum Zeno Effect cannot be made without decoherence.

3)

But can we really know that decoherence is really occurring, and not real wavefunction collapse?

In some cases, yes.

4)

Are you saying that it's impossible to get an exact quantitative description of the quantum zeno effect using real wavefunction collapse?

Yes, that's what I am saying. But still, a collapse may often serve as a good approximation.

I have already addressed 2): I think my wavefunction collapse argument is a correct quantitative description, check Griffiths.

I am not so sure about 1). In practice, yes, there is always an environment that interacts with the system, and can be considered to cause "decoherence". However, in standard quantum theory, the entire universe can be considered to be described by a single wavefunction: there is no "environment" that causes decoherence. Please show me quantitatively how it can be derived from the Schrödinger equation without the assumption that there is some external "environment", since that that assumption is NOT one of the standard postulates of quantum theory.

I'd appreciate if you could back up 3) and 4) as well.

I think decoherence is a much more obscure and conceptually elaborate demonstration of the observer effect, whereas wavefunction collapse is one of the basic postulates of quantum mechanics. It seems as though you don't even disagree with wavefunction collapse: if you admit its existence, it's clearly a better example of the observer effect than any "decoherence" phenomena.


Let me ask another question: if decoherence were the definitive explanation for the observer effect, wouldn't Schrödinger's Cat be rendered moot? Based on the various claims you've made, I guess your argument would be something like: "once a quantum system interacts with some macroscopic system, like a cat, it decoheres." As a less philosophical example, let's suppose we put a geiger counter inside a box with a radioactive atom, whereby the geiger counter has two states "Not Decayed" or "Decayed" which correspond to the radioactive element not decaying or decaying. Does the presence of the macroscopic geiger counter inside the box collapse the radioactive element into either the decayed/not decayed state? This is what that argument seems to imply. I argue that's just an interpretation of quantum mechanics: there is an alternate interpretation that states that until the geiger counter is "observed," it too is in a superposition of states. Granted, the "box" is an idealization--no box can shield a system from the environment totally, but the idea of a universal wavefunction ensures that there is at least one system that is shielded from any sort of "environment."

 

[Demystifier]

Lugita15 and jolb, I have not intention to further elaborate my claims until you read at least one of the papers I mentioned in post #2. To make any further discussions on decoherence fruitful, we first need to have some minimal common background on it.

 

[kith]

I'm a little confused. I thought in the absence of wave function collapse, everything is completely determined by Schrödinger evolution, so the pointer state into collapse appears to have occurred can be predicted in advance, so that any doubt about what the specific pointer state will be must arise from classical ignorance concerning the initial state or the Hamiltonian.

Decoherence doesn't imply "absence of wave function collapse" and is a feature of quantum mechanics, not of specific interpretations. It simply arises if you look at open quantum systems instead of closed ones. It tells you on what time scale coherences decay (Demystifier's WHEN) and in what basis this decay occurs.

Whether this is already enough to explain collapse, depends on the ontology of mixed states. So for some interpretations (statistical, MW), decoherence explains collapse, but not for the Copenhagen interpretation.

 

[lugita15]

Lugita15 and jolb, I have not intention to further elaborate my claims until you read at least one of the papers I mentioned in post #2. To make any further discussions on decoherence fruitful, we first need to have some minimal common background on it.

Demystifier, is this good enough?
http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Zurek%20b.pdf[/PLAIN]

I read this paper a while back, but I think it's a pretty good explanation.

 

[Demystifier]

Demystifier, is this good enough?
http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Zurek%20b.pdf[/PLAIN]

I read this paper a while back, but I think it's a pretty good explanation.

Yes, that's good too.

 

[lugita15]

Yes, that's good too.

OK, then referring to that paper can you try and answer my questions?

 

[Demystifier]

I thought in the absence of wave function collapse, everything is completely determined by Schrödinger evolution,

Not everything, only the unitary time evolution of the quantum state (wave function).

so the pointer state into collapse appears to have occurred can be predicted in advance,

It cannot be predicted, because collapse into a one definite state is a nonunitary event, which Schrödinger equation does not describe.

so that any doubt about what the specific pointer state will be must arise from classical ignorance concerning the initial state or the Hamiltonian.

No. Or if you disagree, try to find a statement in the paper you mentioned which confirms your view.

So are you saying that the Copenhagen interpretation can be disproven by the quantitative details of quantum Zeno effect experiments? That's a rather bold claim.

A naive original version of Copenhagen interpretation can be disproven by quantum Zeno effect experiments, but it is still possible to introduce a refined version of Copenhagen interpretation which is compatible with quantum Zeno effect experiments.
See also
http://xxx.lanl.gov/abs/1009.4072

 

[martinbn]

Demystifier, I don't follow. He says clearly "in the absence of collapse", then you keep saying "no, because, collapse...".

Also, why is quantum Zeno a disproof of Copenhagen?

 

[lugita15]

It cannot be predicted, because collapse into a one definite state is a nonunitary event, which Schrödinger equation does not describe.

Yes, but isn't the whole point that we have the appearance of collapse into a pointer state without actually having real collapse, so that everything stays unitary?

No. Or if you disagree, try to find a statement in the paper you mentioned which confirms your view.

I have to find time to reread the Zurek paper, but if I recall it just discusses how the interaction Hamiltonian selects a preferred pointer basis, so I don't think I'll find any confirmation there.

A naive original version of Copenhagen interpretation can be disproven by quantum Zeno effect experiments, but it is still possible to introduce a refined version of Copenhagen interpretation which is compatible with quantum Zeno effect experiments.
See also
http://xxx.lanl.gov/abs/1009.4072

That's really remarkable if true. I guess I'll have to look into that as well.

 

[Demystifier]

Demystifier, I don't follow. He says clearly "in the absence of collapse", then you keep saying "no, because, collapse...".

Please quote the whole sentences, not part of sentences out of context!

Also, why is quantum Zeno a disproof of Copenhagen?

Copenhagen, or at least a naive version of Copenhagen, claims that there is a strict border between classical and quantum. In this sense, it claims that measurement cannot be described by quantum mechanics, and that there can be no partial collapse.

Quantum Zeno (or more generally, decoherence) demonstrates that there is no strict border between classical and quantum. More precisely, the density matrix evolves continuously during a finite (but typically very short) time from quantum to classical.
In this sense, measurement can at least partially be described by quantum mechanics. In particular, by certain physical definition of measurement apparatus, something closely related to "partial collapse" (that is, partial diagonalization of the density matrix) is possible.

 

[Demystifier]

Yes, but isn't the whole point that we have the appearance of collapse into a pointer state without actually having real collapse, so that everything stays unitary?

That is the point of many-world interpretation of QM (which to a large extent rests on decoherence) but that is not the point of decoherence per se.

I have to find time to reread the Zurek paper, but if I recall it just discusses how the interaction Hamiltonian selects a preferred pointer basis, so I don't think I'll find any confirmation there.

I agree, but then why did you say something which was not said in the paper?

 

[lugita15]

That is the point of many-world interpretation of QM (which to a large extent rests on decoherence) but that is not the point of decoherence per se.

So if many worlds has an explanation of which pointer state the wavefunction appears to collapse into, doesn't that mean that unitary time evolution is sufficient to tell you what the state is?

 

[Demystifier]

So if many worlds has an explanation of which pointer state the wavefunction appears to collapse into, doesn't that mean that unitary time evolution is sufficient to tell you what the state is?

But I didn't say that many worlds explains to WHICH pointer state the wavefunction appears to collapse. It definitely does not. At best, many worlds explains why the wave function appears to collapse in only one pointer state, but it definitely does not tell WHICH one.

 

[lugita15]

But I didn't say that many worlds explains to WHICH pointer state the wavefunction appears to collapse. It definitely does not. At best, many worlds explains why the wave function appears to collapse in only one pointer state, but it definitely does not tell WHICH one.

I'm a bit confused. In the absence of real collapse, the entire future history of the system is determined by unitary Schrödinger evolution, so if you know the initial quantum state and the Hamiltonian then the future is completely predictable, correct? So then how is it that these two pieces of information are insufficient to tell you which pointer state the wave function will appear to collapse into?

 

[Demystifier]

I'm a bit confused. In the absence of real collapse, the entire future history of the system is determined by unitary Schrödinger evolution, so if you know the initial quantum state and the Hamiltonian then the future is completely predictable, correct? So then how is it that these two pieces of information are insufficient to tell you which pointer state the wave function will appear to collapse into?

That's because in the many-world interpretation you need to distinguish two levels of reality:
(i) the multi-world, and
(ii) the single world we actually see
By knowing Hamiltonian, initial quantum state, and the Schrödinger equation, you can predict everything about the multi-world. In particular, you can predict that this multi-world will consist of MANY ordinary worlds. But it does not tell how to single out one particular single ordinary world out of this big set containing many ordinary worlds.

 

[lugita15]

That's because in the many-world interpretation you need to distinguish two levels of reality:
(i) the multi-world, and
(ii) the single world we actually see
By knowing Hamiltonian, initial quantum state, and the Schrödinger equation, you can predict everything about the multi-world. In particular, you can predict that this multi-world will consist of MANY ordinary worlds. But it does not tell how to single out one particular single ordinary world out of this big set containing many ordinary worlds.

Are you saying that in different worlds decoherence will lead to the wave function appearing to collapse into different pointer states, so that ignorance about what world you are going to branch into translates into uncertainty about what pointer state you will see the wave function appear to collapse into?

 

[Demystifier]

Are you saying that in different worlds decoherence will lead to the wave function appearing to collapse into different pointer states, so that ignorance about what world you are going to branch into translates into uncertainty about what pointer state you will see the wave function appear to collapse into?

Yes, that's more-or-less what the many-world interpretation claims.

See also this very readable, good, short recent paper on it:
http://lanl.arxiv.org/abs/1110.0549 [to appear as a Brief Review in Modern Physics Letters]