4 ms·

If we measure, say, the spin of an electron along the z-axis the only possible outcomes are UP and DOWN. If we measure UP, the next measurement of the same thi

by kgwgk 8y ago

If we measure, say, the spin of an electron along the z-axis the only possible outcomes are UP and DOWN. If we measure UP, the next measurement of the same thing can only be UP. The system has a narrow wave function indeed. (On the other hand, if the system was originally prepared in a pure state along a perpendicular axis and has not been observed yet the probability of measuring UP or DOWN is 50/50. The difference between these scenarios is quite observable.)

scotty79 8y ago
Yes. Spin UP and DOWN are so "far" from each other that detector that can tell them apart reshapes the probability so much that from that point on this detector will always register UP. Think about location. If you send electron through half-mirror and put a screen behind it and see electron hitting the screen then you cut down your wave function to at most half of the size. You'll never see this photon in front of the mirror because that small interaction with the screen narrowed down the possibilities. You'll say that after it hit the screen it must go on in straight line and will hit any subsequent screen as if it was small ball just flying through space and screens. But imagine instead of the screen just a detector made from solenoid that will notice passing electron but not exactly where it is. This way location is still uncertain. Probability wave is not that narrow. The only thing your detection did is reshaping the wave so you no longer have any possibility of finding electron in front of a mirror. My point is that there's no difference in math between moving ball hitting things on the way and probability wave evolving in space time and exchanging momentum and energy with other waves and getting reshaped during these exchanges. We know equations that govern this evolution, we know what rules the exchanges obey and how the waves change shape when they interact. So why even have little balls cluttering our imagination if they are not needed to explain anything?
- 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).

by kgwgk 8y ago

scotty79 8y ago

Yes. Spin UP and DOWN are so "far" from each other that detector that can tell them apart reshapes the probability so much that from that point on this detector will always register UP. Think about location. If you send electron through half-mirror and put a screen behind it and see electron hitting the screen then you cut down your wave function to at most half of the size. You'll never see this photon in front of the mirror because that small interaction with the screen narrowed down the possibilities. You'll say that after it hit the screen it must go on in straight line and will hit any subsequent screen as if it was small ball just flying through space and screens. But imagine instead of the screen just a detector made from solenoid that will notice passing electron but not exactly where it is. This way location is still uncertain. Probability wave is not that narrow. The only thing your detection did is reshaping the wave so you no longer have any possibility of finding electron in front of a mirror. My point is that there's no difference in math between moving ball hitting things on the way and probability wave evolving in space time and exchanging momentum and energy with other waves and getting reshaped during these exchanges. We know equations that govern this evolution, we know what rules the exchanges obey and how the waves change shape when they interact. So why even have little balls cluttering our imagination if they are not needed to explain anything?

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).