Questions about properties of wave-functions

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In summary, wave-functions are mathematical/probabilistic constructs that can modify/change the behavior/path of a photon. They exist within space-time, but the collapse of the wave-function is typically an instantaneous process.
  • #1
San K
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Had some questions about wave-functions. Some of the questions might be invalid due to limited knowledge of the OP.

1. Are wave-functions real? i.e. do they exist in reality?

Wave-functions are mathematical/probabilistic constructs.

However if they can modify/change the behavior/path of a photon then something real must exist, that is taking into account/calculating both/all the slits/paths.

2. Are wave-functions ("travelling") instantaneous?

7. When a photon is traveling from the sun towards the earth, is there a wave-function existing at all times between the photon and its final destination? and does that final destination keep changing to the first object of opaque obstruction?

a wave-function exists at all times when a particle is not being detected/measured.
regarding the length/reach of the wave-function I don't know, someone from/in the forum might have information.

3. Are wave-functions the same concept/thing that are used in Quantum Entanglement as well?

4. Do wave-functions have energy?

Pro argument - well if they can change the path of a photon/particle...

Con argument - if they have energy then it would be split at the slits, however when the photon is finally detected at the detector there is no loss in energy detected.

5. Is the collapse of the wave-function instantaneous?

a collapse happens when we try to detect the photon. at that point the photon is now "entangled" with the detector (?)

6. Are wave-functions (existing) within space-time?
"The new fad/fashion for reducing weight is Raspberry ketones..;)"
 
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  • #2
San K said:
4. Do wave-functions have energy?

We calculate the expectation value of the energy of a system by using the energy operator, in the same way that we can calculate the expectation value of momentum by using the momentum operator, or the expectation value of the position by using the position operator:

$$\langle E \rangle = \int {\Psi(x,t)^* \hat E \Psi(x,t) dx}
= \int {\Psi(x,t)^* \left( i \hbar \frac {\partial}{\partial t} \right) \Psi(x,t) dx}$$

$$\langle p \rangle = \int {\Psi(x,t)^* \hat p \Psi(x,t) dx}
= \int {\Psi(x,t)^* \left( -i \hbar \frac {\partial}{\partial x} \right)\Psi(x,t) dx}$$

$$\langle x \rangle = \int {\Psi(x,t)^* \hat x \Psi(x,t) dx}
= \int {\Psi(x,t)^* \left( x \right) \Psi(x,t) dx}$$

Whether this means the wave function "has" energy, or momentum, or position, depends on what you mean by the word "has".

6. Are wave-functions (existing) within space-time?

What does this mean?
 
  • #3
1. It's a mathematical concept that's a part of a theory that makes very accurate predictions. The question of whether it's also "real" doesn't seem to be a scientific question. Can you think of an experiment that can answer it?

2. No, they change with time as described by the Schrödinger equation.

7. I don't fully understand the question, so I'll just say that wavefunctions for massless particles is a tricky concept, and that typically, when a particle is emitted from a source, its wavefunction spreads out in all directions. The particle can't be described as localized until it interacts with something else (e.g. molecules in the air).

3. To discuss entanglement, you need a state vector for a two-particle system. You can think of it as a wavefunction that depends on twice as many variables.

4. A wavefunction represents the state of a particle. We will only say that the state has a well-defined energy if knowledge of the state is enough to predict the result of an energy measurement with certainty. This is the case e.g. after an energy measurement that doesn't destroy the particle. (A second measurement will have the same result). If the state doesn't have a well-defined energy, then it's a superposition (linear combination) of many states that do have well-defined energies.

5. The question only makes sense if we view collapse as a physical process. I don't.

6. They are functions from ℝ4 (="spacetime") into ℂ that satisfy the Schrödinger equation. Another way of looking at it is to say that they are functions from ℝ3 (="space") into ℂ, and given such a function, i.e. a point in the space of such functions, the Schrödinger equation determines a unique curve through that point.
 
  • #4
Great answers, Fredrik and jtbell, thanks!. You have clarified some difficult/deep questions.

first the following need to be understood:

what is - R4, R3, Hilbert space, relationship between time and space, 4D Vs 3D

Fredrik said:
5. The question only makes sense if we view collapse as a physical process. I don't.

sounds reasonable... perhaps it is a swap...
 
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  • #5
San K said:
Had some questions about wave-functions. Some of the questions might be invalid due to limited knowledge of the OP.

1. Are wave-functions real? i.e. do they exist in reality?

.-psi epistemic ontic:
reflect something of the reality.
.-psi epistemic epistemic:
nothing to do with reality.
.-psi ontic:
they are the real.


San K said:
5. Is the collapse of the wave-function instantaneous?

10^10 - 10^15 secs
around attoseconds, zeptoseconds, yoctoseconds.

as for macroscopy objects, time depend on the number of atoms.


San K said:
6. Are wave-functions (existing) within space-time?

if time is relational, it does not.
 
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  • #6
audioloop said:
time depend on the number of atoms.

And which mechanism do you suggest creates that time dependence? I saw in a different thread that you rightly stated that decoherence does not lead to collapse, so this can't be the decoherence time you're referring to. So what time scale do you mean?
 
  • #7
Jazzdude said:
And which mechanism do you suggest creates that time dependence? I saw in a different thread that you rightly stated that decoherence does not lead to collapse, so this can't be the decoherence time you're referring to. So what time scale do you mean?

a nonlinear schrodinger equation.
 
  • #8
audioloop said:
a nonlinear schrodinger equation.

There are several nonlinear modifications of the Schroedinger equation. Which one do you have in mind? GRW? In any case, I find added nonlinearities to be highly speculative and probably not suitable for a general authoritative answer to a beginner's question.
 
  • #9
Fredrik said:
6. They are functions from ℝ4 (="spacetime") into ℂ that satisfy the Schrödinger equation. Another way of looking at it is to say that they are functions from ℝ3 (="space") into ℂ, and given such a function, i.e. a point in the space of such functions, the Schrödinger equation determines a unique curve through that point.

I disagree, they are only defined on [itex]\mathbb R^3[/itex] (spatially speaking) for one particle systems. A general N particle wavefunction is defined on [itex]\mathbb R^{3N}[/itex]. And since the latter obviously cannot be interpreted as being defined on "space", it indicates that even for the one-particle the [itex]\mathbb R^3[/itex] should not be identified with space in any way.

Shortly put: no, the wavefunctions are not defined on spacetime

EDIT:

5. Is the collapse of the wave-function instantaneous?
I believe this question can even be answered without giving the wavefunction a physical meaning, again in disagreement with Fredrik (although note that due to my comment above I don't really see how you would even give the wavefunction a physical meaning, since it can't be defined on spacetime), all you need is that the wavefunction is an objective property.

Although Copenhagen treats it instananeously, there are other interpretations which describe that as a limit of the real process which takes a finite length of time. One interpretation that I know of that says collapse is not instantaneous, is the de Broglie-Bohm (which uses the concept of decoherence).
 
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  • #10
mr. vodka said:
I disagree, they are only defined on [itex]\mathbb R^3[/itex] (spatially speaking) for one particle systems. A general N particle wavefunction is defined on [itex]\mathbb R^{3N}[/itex]. And since the latter obviously cannot be interpreted as being defined on "space", it indicates that even for the one-particle the [itex]\mathbb R^3[/itex] should not be identified with space in any way.
I was talking about one-particle states. It didn't seem necessary to consider anything more complicated here. The space of complex-valued functions defined on ##\mathbb R^{3N}## is isomorphic to the the tensor product of N 1-particle Hilbert spaces. So a "wavefunction" for an N-particle system can also be thought of as an N-tuple of 1-particle wavefunctions, each of which is defined on "space".

mr. vodka said:
I believe this question can even be answered without giving the wavefunction a physical meaning, again in disagreement with Fredrik
My answer about the speed of collapse was a bit lazy. So maybe I should elaborate on it, or even change it. But I'm not going to spend any significant amount of time on that right now.

I think Dr Chinese has answered questions about the speed of collapse in several other threads, so the OP might want to search for those posts.
 
  • #11
I was talking about one-particle states. It didn't seem necessary to consider anything more complicated here. The space of complex-valued functions defined on R3N is isomorphic to the the tensor product of N 1-particle Hilbert spaces. So a "wavefunction" for an N-particle system can also be thought of as an N-tuple of 1-particle wavefunctions, each of which is defined on "space".

I do think that it's important to look at the more general case, since it indicates to me that even the 1-particle case wavefunction can not be interpreted as being defined on space. Sure, you might take a wavefunction for a system of N particles and integrate out N-1 variables, giving a one-particle wavefunction defined on [itex]\mathbb R^3[/itex] and do this N times, giving N one-particle wavefunctions, but this leaves out most information (entanglement!) and thus is in no way equivalent to the original wavefunction, which simply cannot be embedded in ordinary space in any sensible way.
 
  • #12
I don't follow your argument for why we shouldn't never think of wavefunctions as functions defined on ℝ3. Why shouldn't we do that when we're dealing with the non-relativistic theory of a single spin-0 particle under the influence of a potential?
 
  • #13
No sure, that is defined on [itex]\mathbb R^3[/itex], but I'm saying we shouldn't identify this with ordinary space but acknowledge that it is just a coincidence of the one-particle case. And my motivation for that assertion is the general N-particle case.
 
  • #14
Jazzdude said:
There are several nonlinear modifications of the Schroedinger equation. Which one do you have in mind? GRW? In any case, I find added nonlinearities to be highly speculative and probably not suitable for a general authoritative answer to a beginner's question.


hi jazz, i hate likewise, ad hoc propositions.
i mean inherently nonlinear like, Dobner-Goldin Class (not GRW/CSL) or from an invariance principle.
with testable predictions.




by the way, jazz,
what do you think about a derivation of the Born probability rule, from a nonlinear standpoint ?
 
  • #15
audioloop said:
hi jazz, i hate likewise, ad hoc propositions.
i mean inherently nonlinear like, Dobner-Goldin Class (not GRW/CSL) or from an invariance principle.
with testable predictions.

That's interesting. I haven't looked into nonlinearities due to non commutative geometry very deeply, but my impression was that there is no complete theory that would predict collapse including the Born rule. I could very well be wrong. Do you have an literature pointers?

by the way, jazz,
what do you think about a derivation of the Born probability rule, from a nonlinear standpoint ?

I'm quite certain that some kind of nonlinearity is needed to fully explain the observation of single unique measurement outcomes and as a consequence the Born statistic. The question is where such a nonlinearity might come from. As you said, something that doesn't have to be artificially added but is inherent to the theory is surely preferable.

But what if this nonlinearity is already present in the linear structure of quantum theory as we know it? If we eradicate the measurement postulate, the nonlinearity seems to arise as a subjective artifact of trying to reconstruct the state of the universe by interacting with it.

I know this sounds quite unlikely, but please check http://arxiv.org/abs/1205.0293 for the exact argument and a thorough discussion. The paper derives the measurement postulate including the Born rule and makes predictions for observable effects. I'd be happy to have someone to discuss it with.
 
  • #16
audioloop said:
.-psi epistemic ontic:
reflect something of the reality.
.-psi epistemic epistemic:
nothing to do with reality.
.-psi ontic:
they are the real.




10^10 - 10^15 secs
around attoseconds, zeptoseconds, yoctoseconds.
I think you meant 10^{-10} - 10^{-15} sec. What you wrote would run to several million years.

as for macroscopy objects, time depend on the number of atoms.




if time is relational, it does not.
 
  • #17
Jazzdude said:
I'm quite certain that some kind of nonlinearity is needed to fully explain the observation of single unique measurement outcomes and as a consequence the Born statistic.

The pilot-wave interpretation can also explain/deduce it. I would presume there are other (nonlinear) interpretations which can also do that.
 
  • #18
mr. vodka said:
The pilot-wave interpretation can also explain/deduce it. I would presume there are other (nonlinear) interpretations which can also do that.

The introduction of particles that are piloted by the field is a nonlinearity of the theory. The fields stay linear, but the properties of the particles are not subject to superposition. So it's the lack of a linear structure in the point particle representation that breaks the linearity of the theory.

As for other interpretations that are able to actually derive the Born rule, I'd be interested in hearing which you have in mind. The only derivations I know of are in the context of MWI (with artificial assumptions that are practically equivalent to stating the Born rule directly) or Zurek's envariance, which I consider circular.
 
  • #19
HallsofIvy said:
I think you meant 10^{-10} - 10^{-15} sec. What you wrote would run to several million years.
yes sir, i forgot the minus symboll.
thanks.
 
  • #20
Jazzdude said:
That's interesting. I haven't looked into nonlinearities due to non commutative geometry very deeply, but my impression was that there is no complete theory that would predict collapse including the Born rule. I could very well be wrong. Do you have an literature pointers?



I'm quite certain that some kind of nonlinearity is needed to fully explain the observation of single unique measurement outcomes and as a consequence the Born statistic. The question is where such a nonlinearity might come from. As you said, something that doesn't have to be artificially added but is inherent to the theory is surely preferable.

But what if this nonlinearity is already present in the linear structure of quantum theory as we know it? If we eradicate the measurement postulate, the nonlinearity seems to arise as a subjective artifact of trying to reconstruct the state of the universe by interacting with it.

I know this sounds quite unlikely, but please check http://arxiv.org/abs/1205.0293 for the exact argument and a thorough discussion. The paper derives the measurement postulate including the Born rule and makes predictions for observable effects. I'd be happy to have someone to discuss it with.


well, related arguments.

Schroedinger equation from an exact uncertainty principle
J. Phys. A 35 (2002) 3289
http://arxiv.org/abs/quant-ph/0102069

Exact uncertainty approach in quantum mechanics and quantum gravity
Gen. Relativ. Gravit. 37 (2005) 1505
http://arxiv.org/pdf/gr-qc/0408098v1.pdf

and
http://arxiv.org/pdf/gr-qc/0610142v2.pdf
 
  • #21
unifying quantum and thermodinamics

Nonlinear extensions of Schrödinger-von Neumann quantum dynamics: a list of conditions for compatibility with thermodynamics.
http://arxiv.org/pdf/quant-ph/0402180v1.pdfNonlinear Quantum Evolution Equations to Model Irreversible Adiabatic Relaxation with Maximal Entropy Production and Other Nonunitary Processes.
http://arxiv.org/pdf/0907.1977v1.pdf
...In contrast with the epistemic framework, where the linearity of the dynamical law is a requirement, the assumed ontic status of the density operator emancipates its dynamical law from the restrictive requirement of linearity...
 
  • #22
San K is still wrestling with these concepts if recent threads are anything to go by...
1. Are wave-functions real? i.e. do they exist in reality?
Depends what you mean by real - it is useful to think of them as mathematical tools for helping make predictions.
if they can modify/change the behavior/path of a photon then something real must exist, that is taking into account/calculating both/all the slits/paths.
They cannot do this. You do not get a photon interacting with a wavefunction - the photon and wavefunction are different descriptions of the same thing.[also see answer to #4]
2. Are wave-functions ("travelling") instantaneous?
Some wave functions describe traveling waves, so they can be thought of as travelling. This basically means that the time evolution of a wave-function means that the peaks and troughs are in different places at different times.
3. Are wave-functions the same concept/thing that are used in Quantum Entanglement as well?
Wavefunctions can be used to describe quantum entanglement.
4. Do wave-functions have energy?
No. This question was revisited in your recent thread here.
5. Is the collapse of the wave-function instantaneous?
The "collapse" is part of a description known as the Copenhagen Interpretation. It is not considered "instant" and there is some debate about how seriously to consider it.
a collapse happens when we try to detect the photon. at that point the photon is now "entangled" with the detector (?)
No - entanglement is a different process. A photon is usually detected by electromagnetism - it gets destroyed, and it's energy absorbed by an electron, which gets ejected. (There are other methods - but this gives you the idea: detection is not some magical effect of now you see it now you don't... it is a physical process.)
6. Are wave-functions (existing) within space-time?
There are wavefunctions which can be represented in terms of space-time...
7. When a photon is traveling from the sun towards the earth, is there a wave-function existing at all times between the photon and its final destination? and does that final destination keep changing to the first object of opaque obstruction?
If we treat the Sun as a source for photons, and the Earth as a detector, then we can use a wave-function method to work out the probability of the Earth detecting a photon from the Sun and so make predictions about the observed solar photon flux at the earth. We can also work out the probability that it, instead, finds itself at some other opaque obstruction. The final destination of the photon remains uncertain until it arrives and is detected.

... it may be useful people who have attempted to help San K with these earlier threads to visit some of the recent ones as it seems that some misunderstandings persist.
 

1. What is a wave-function?

A wave-function is a mathematical function that describes the quantum state of a particle or system. It contains information about the probability of finding the particle at a certain position or with a certain momentum.

2. What are the properties of a wave-function?

The properties of a wave-function include normalization, continuity, and differentiability. Normalization means that the total probability of finding the particle in all possible states is equal to 1. Continuity means that the function is smooth and has no abrupt changes. Differentiability means that the function is differentiable at all points, allowing for the calculation of momentum and other physical quantities.

3. How do wave-functions change over time?

According to the Schrödinger equation, wave-functions evolve over time in a predictable manner. They can be affected by external forces, such as potentials or interactions with other particles, causing them to change shape and position.

4. What is the significance of the wave-function in quantum mechanics?

The wave-function is a fundamental concept in quantum mechanics and is used to describe the behavior of particles on a microscopic scale. It allows for the calculation of probabilities and expectation values, which are essential in understanding the behavior of quantum systems.

5. Can the wave-function be observed or measured?

No, the wave-function itself cannot be observed or measured directly. However, the effects of the wave-function, such as the probability of finding a particle in a certain state, can be measured through experiments and observations.

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