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I think what you call "reshaping" is what the standard interpretation calls "collapse". Maybe you will find it easier if instead of thinking about it as "somet

by kgwgk 8y ago

I think what you call "reshaping" is what the standard interpretation calls "collapse". Maybe you will find it easier if instead of thinking about it as "something unlocalized becoming a point", you think about it as "the observable we measure is represented by an operator and after the measurement the wave-function will be the eigen-function of the operator correspoding to the eigen-value that we obtained as outcome of the measurement." This new wave-function is not necessarily "localized". By the way, the wave-function is not a probability wave evolving in space time. It is a (complex-valued) mathematical function "evolving" in parameter space.

kgwgk 8y ago
More precisely (Quantum Mechanics, McIntyre): "Postulate 5 (projection postulate): After a measurement of A that yields the result a_n, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement: ∣ψ′⟩ = P_n ∣ψ⟩ / sqrt( ⟨ψ∣ P_n ∣ψ⟩ )" I agree that there are reasons to prefer waves to idealized point-like particles as the more fundamental description (singularities are always difficult to handle). But that's essentially unrelated to the single-world / many-worlds debate, I think.
- scotty79 8y ago
  This sounds much more like something I imagine. Before collapse you have a system which evolution is governed by some equations, then interaction happens, then you get new system that is a transformation of old system into something dependent on result of interaction (measurement) but what we get in the end is still of the same nature and still evolves according to same equations as initial system.
  - kgwgk 8y ago
    That's QM in a nutshell. Of course, it's not completely clear why two different regimes are needed and when should one or the other be used. But as John Bell wrote in his "Against 'measurement'" article (https://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf https://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf): "ORDINARY QUANTUM MECHANICS (as far as I know) IS JUST FINE FOR ALL PRACTICAL PURPOSES". Quoting Everett's dissertation (http://www.weylmann.com/relative_state.pdf http://www.weylmann.com/relative_state.pdf): We take the conventional or "external observation" formulation of quantum mechanics to be essentially the following [1]: A physical system is completely described by a state function ψ, which is an element of a Hilbert space, and which furthermore gives information only to the extent of specifying the probabilities of the results of various observations which can be made on the system by external observers. There are two fundamentally different ways in which the state function can change: Process 1: The discontinuous change brought about by the observation of a quantity with eigenstates φ_1, φ_2, ... , in which the state ψ will be changed to the state φ_j, with probability |(ψ,φ_j)|^2. Process 2: The continuous, deterministic change of state of an isolated system with time according to a wave equation dψ/dt = Aψ, where A is a linear operator. This formulation describes a wealth of experience. No experimental evidence is known which contradicts it. [1] We use the terminology and notation of J. von Neumann, Mathematical Foundations of Quantum Mechanics, translated by R.T.Beyer (Princeton University Press, Princeton, I955).
scotty79 8y ago
> I think what you call "reshaping" is what the standard interpretation calls "collapse". Yes. But traditionally collapse is from a wave into result of measurement. My point is that it actually is from wave into a wave. > operator correspoding to the eigen-value This suspiciously for me sounds like a single value and something of different nature than operator you mentioned earlier. > It is a (complex-valued) mathematical function "evolving" in parameter space. I know. I called it probability function because its squared modulus can give you probability of getting measurement of given value.
- kgwgk 8y ago
  Traditional collapse is from wave-function to wave-function. The result of the measurement is something different (but related). The result of the measurement is one of the eigen-values of the observable operator (that's the essence of quantum mechanics, the set of outcomes is "quantized"). If the eigen-value observed is non-degenerate there is only one wave-function that corresponds to that pure state (i.e. the corresponding eigen-function). Immediately after the measurement, that will be the wave-function describing the system (and then will evolve according to Schroedinger's equation). In the general degenerate case, it will be the projection of the wavefunction before the measurement on the corresponding subspace. Depending on what you measure the probability density corresponding to the after-measurement wave-function may or may not be concentrated at some point. It will happen if you measure the position (you know where it is know!). But if you measure the momentum, the location becomes completely undetermined.
  - scotty79 8y ago
    That's great. So, pretty much I have correct intuitive understanding of quantum mechanics (except with that degenerate thing which I never encounterd before). Thank you for your comments. At least now I can be now at peace when I read pop-sci articles about quantum knowing that they are just using extreme simplification (to the point of being wrong) but the actual current understanding is same as I believe and I'm in no obligation to share my insights. ;-)
    - 8y ago

by kgwgk 8y ago

kgwgk 8y ago

More precisely (Quantum Mechanics, McIntyre): "Postulate 5 (projection postulate): After a measurement of A that yields the result a_n, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement: ∣ψ′⟩ = P_n ∣ψ⟩ / sqrt( ⟨ψ∣ P_n ∣ψ⟩ )" I agree that there are reasons to prefer waves to idealized point-like particles as the more fundamental description (singularities are always difficult to handle). But that's essentially unrelated to the single-world / many-worlds debate, I think.

scotty79 8y ago
This sounds much more like something I imagine. Before collapse you have a system which evolution is governed by some equations, then interaction happens, then you get new system that is a transformation of old system into something dependent on result of interaction (measurement) but what we get in the end is still of the same nature and still evolves according to same equations as initial system.
- kgwgk 8y ago
  That's QM in a nutshell. Of course, it's not completely clear why two different regimes are needed and when should one or the other be used. But as John Bell wrote in his "Against 'measurement'" article (https://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf https://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf): "ORDINARY QUANTUM MECHANICS (as far as I know) IS JUST FINE FOR ALL PRACTICAL PURPOSES". Quoting Everett's dissertation (http://www.weylmann.com/relative_state.pdf http://www.weylmann.com/relative_state.pdf): We take the conventional or "external observation" formulation of quantum mechanics to be essentially the following [1]: A physical system is completely described by a state function ψ, which is an element of a Hilbert space, and which furthermore gives information only to the extent of specifying the probabilities of the results of various observations which can be made on the system by external observers. There are two fundamentally different ways in which the state function can change: Process 1: The discontinuous change brought about by the observation of a quantity with eigenstates φ_1, φ_2, ... , in which the state ψ will be changed to the state φ_j, with probability |(ψ,φ_j)|^2. Process 2: The continuous, deterministic change of state of an isolated system with time according to a wave equation dψ/dt = Aψ, where A is a linear operator. This formulation describes a wealth of experience. No experimental evidence is known which contradicts it. [1] We use the terminology and notation of J. von Neumann, Mathematical Foundations of Quantum Mechanics, translated by R.T.Beyer (Princeton University Press, Princeton, I955).

scotty79 8y ago

> I think what you call "reshaping" is what the standard interpretation calls "collapse". Yes. But traditionally collapse is from a wave into result of measurement. My point is that it actually is from wave into a wave. > operator correspoding to the eigen-value This suspiciously for me sounds like a single value and something of different nature than operator you mentioned earlier. > It is a (complex-valued) mathematical function "evolving" in parameter space. I know. I called it probability function because its squared modulus can give you probability of getting measurement of given value.

kgwgk 8y ago
Traditional collapse is from wave-function to wave-function. The result of the measurement is something different (but related). The result of the measurement is one of the eigen-values of the observable operator (that's the essence of quantum mechanics, the set of outcomes is "quantized"). If the eigen-value observed is non-degenerate there is only one wave-function that corresponds to that pure state (i.e. the corresponding eigen-function). Immediately after the measurement, that will be the wave-function describing the system (and then will evolve according to Schroedinger's equation). In the general degenerate case, it will be the projection of the wavefunction before the measurement on the corresponding subspace. Depending on what you measure the probability density corresponding to the after-measurement wave-function may or may not be concentrated at some point. It will happen if you measure the position (you know where it is know!). But if you measure the momentum, the location becomes completely undetermined.
- scotty79 8y ago
  That's great. So, pretty much I have correct intuitive understanding of quantum mechanics (except with that degenerate thing which I never encounterd before). Thank you for your comments. At least now I can be now at peace when I read pop-sci articles about quantum knowing that they are just using extreme simplification (to the point of being wrong) but the actual current understanding is same as I believe and I'm in no obligation to share my insights. ;-)
  - 8y ago