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Collapse of a Wave

  1. Nov 29, 2015 #1
    How does the collapse of a wave function influence the results of an energy measurement or a position measurement taken immediately after another energy or position measurement?
     
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  3. Nov 29, 2015 #2

    vanhees71

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    There is no collapse of the state or wave function, although some flavors of the Copenhagen interpretation assume one, but it doesn't help to understand quantum theory nor has it any other merit. Thus the collapse assumption, which is not well defined anyway, should not be taken as a serious part of the physics of quantum theory. Thus your question does not make much sense. What happens to the measured system due to the interaction with the measurement apparatus depends on the choice of this measurement apparatus and thus cannot be answered in as general terms as you might expect.
     
  4. Nov 29, 2015 #3

    atyy

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    Contrary to what vanhees71 says, the collapse is a central part of the orthodox interpretation of quantum mechanics. It is used in the Schroedinger picture to calculate joint probabilities.

    Here is the rough statement: if a measurement of a particle's energy collapses its wave function into an energy eigenstate, then immediate repetition of the measurement will yield the same outcome. There are technical subtleties for continuous variables, but the rough statement is not misleading.

    If you would like to see the collapse stated for continuous variables, see http://arxiv.org/abs/0706.3526 (Eq 3).
     
  5. Nov 30, 2015 #4

    vanhees71

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    If you measure a photon's energy with a usual photodetector, it is absorbed. So there is not a single photon left with a state that's collapsed into an energy eigenstate (or a true state with a narrow energy spread). This is only one very simple example which shows that this collapse hypothesis is very misleading.

    Further, it is inconsistent with quantum theory to assume that there is a classical dynamics on top of quantum dynamics, leading to the collapse. So far nobody could find a clear physical distinction when classical dynamics should be applicable and not quantum dynamics. Nowadays, one can demonstrate the superposition principle, particularly also entanglement, for macroscopic objects, and there is no natural scale (in whatever kind of system size) where you can make this "quantum-classical cut".

    Last but most importantly an instantaneous nonlocal collapse of the state is incompatible with the very foundations of relativistic space time and its causality structure. If at all, you can take the collapse as a short-hand expression for "adapting ones knowledge because of new information on the system", i.e., in an epistemic instead of an ontological way of interpreting the state, but then you can as well simply adapt the minimal interpretation, taking Born's rule as another postulate rather than someting derivable from the other postulates, particularly quantum dynamics. As nicely demonstrated by Weinberg in his "Lectures on Quantum Mechanics" such a derivation is anyway still not known.
     
  6. Nov 30, 2015 #5

    bhobba

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    Vanhees is of course correct.

    Collapse is not a part of QM - its only a part of some interpretations:
    https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

    In some interpretations the wave-function is simply a state of knowledge. That it collapses is of zero concern.

    Thanks
    Bill
     
    Last edited: Nov 30, 2015
  7. Nov 30, 2015 #6

    atyy

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    This is wrong, because quantum mechanics is incompatible with the very foundations of relativistic spacetime and its causality structure. That is what Bell's theorem says.
     
  8. Nov 30, 2015 #7

    Nugatory

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    Uh, guys..... Anyone want to take a try at the original poster's question? Something about how two successive position or energy measurements behave?
     
  9. Nov 30, 2015 #8

    stevendaryl

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    I don't see how the knowledge interpretation is tenable, in light of Bell's theorem, unless you allow nonlocal interactions.

    Consider a spin-1/2 EPR experiment in which Alice and Bob both choose to measure the spins of their respective particles along the same axis, [itex]\vec{\alpha}[/itex]. Immediately after Alice measures spin-up (assuming some frame in which her measurement comes first), she knows with 100% certainty that Bob will measure spin-down (since the particles are perfectly anti-correlated). Bob's chances, as viewed by Alice, "collapse" from 50/50 chance of spin-up or spin-down to 0/100. If this change is simply a change in Alice's knowledge, then to me, that means that Bob's chances were 0/100 before Alice's measurement. That implies hidden variables determining the outcomes. And in light of Bell's theorem, any such hidden variables would have to involve nonlocal interactions.

    It seems to me that Bell's theorem is all about whether the wave function can be considered just a state of knowledge, and while it doesn't definitively rule out such an interpretation, it places very tight constraints on it.
     
  10. Nov 30, 2015 #9

    atyy

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    I did in post #3.
     
  11. Nov 30, 2015 #10

    stevendaryl

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    The prediction of the usual QM formalism is that immediately after a measurement, the system is in an eigenstate of whatever operator was measured. So immediately after an energy measurement, the system will be in an eigenstate of energy, and so a second measurement, immediately afterward, should give the same energy. Similarly, measuring the position results in an eigenstate of position, so a second position measurement immediately afterward will give the same position.

    If you alternate measurements of different, noncommuting operators, then you will get probabilistic results.
     
  12. Nov 30, 2015 #11

    stevendaryl

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    Well, you could split it into two different assertions:
    1. When you measure an observable, you get an eigenvalue, with probabilities given by the Born rule.
    2. After a measurement, the wave function is in the eigenstate corresponding to the eigenvalue measured.
    Only the second assertion is a "collapse". Of course, you might argue that you need collapse to give predictions for a sequence of measurements, but you don't, really; you can just view the sequence of measurements as a single, compound measurement.
     
  13. Nov 30, 2015 #12

    atyy

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    But would you call the compound measurement a "sequence" of measurements? The Born rule applies to measurements made at one slice of simultaneity. In the compound measurement, one does not measure A followed by B, rather one measures AA and B simultaneously, and p(AA,B) has the same form as p(A,B) so that measuring AA is taken to be the "same" as measuring A. However, the difference is that AA and B are real in the compound measurement, whereas A then B is real in the sequence of measurements.
     
  14. Nov 30, 2015 #13

    martinbn

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    Isn't a collapse needed for state preparatio?. If you need a system of half spin in the state |spin up>, how would one prepare it without collapse?
     
  15. Nov 30, 2015 #14

    bhobba

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    In modern times states are the equivalence class of preparation procedures. The concept of collapse need not even be introduced.

    Thanks
    Bill
     
  16. Nov 30, 2015 #15

    stevendaryl

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    In principle, you can get away with a single measurement (or a single application of the Born rule) in the following way:

    You have some system, [itex]S_1[/itex], that you'd like to make repeated measurements on. So you create a second system, [itex]S_2[/itex], which consists of a device that makes repeated measurements on [itex]S_1[/itex] and records the results (say, on a DVD). Then the creation of the DVD does not, in principle, need to involve the Born rule at all. The various possible sequences of measurement results correspond to orthogonal states of the resulting DVD, and the Schrodinger equation for [itex]S_1 \otimes S_2 \otimes DVD[/itex] would give you the probabilities for each sequence. So it's only necessary to invoke the Born rule once, and it's not necessary to consider what state the DVD is in after the measurement.
     
  17. Nov 30, 2015 #16

    stevendaryl

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    But it seems to me that the idea that a certain "preparation procedure" results in a specific state (mixed or pure) of the system being prepared is equivalent to the collapse hypothesis. Or another alternative approach (which might be equivalent--I'm not sure) is just to use decoherence and pretend that an improper mixed state is a proper mixed state (which I think is sort of a collapse-like assumption, but the collapse takes place off-stage, which might make it more palatable).
     
  18. Nov 30, 2015 #17

    DrChinese

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    Welcome to PhysicsForums, klw289!

    Position (q) and momentum (p) are non-commuting observables, as you are probably aware (Heisenberg Uncertainty Principle).

    a. A strong (tight) measurement of one makes the other indeterminate (regardless of any value it previously had). If you know a particle's momentum quite accurately, its position could be any of a range of places. It would not make sense to attempt to extrapolate position in this case (ie by using its previous position as a starting point).

    b. On the other hand, repeated measurements of the same quantum observable generally yields the same value over and over again.


    To understand the rules on this, it is often convenient to consider particle spin; as with spin you don't get hung up in trying to construct physical models in your head. Such models have a tendency to throw you off.
     
  19. Nov 30, 2015 #18

    atyy

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    Yes, but there is a slight difference, at least within Copenhagen. If you don't invoke the Born rule when you make the DVD, then the record on the DVD is not necessarily real. So there is no real sequence of measurement outcomes, where reality is attributed only to measurement outcomes (ie. when one invokes the Born rule).
     
  20. Nov 30, 2015 #19

    bhobba

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    It is - its just a change in how its viewed - but a very important one. Preparations do not happen instantaneously so you avoid this whole non-local instantaneous collapse with observation thing.

    Thanks
    Bill
     
  21. Dec 1, 2015 #20
    Imagine you measure position, and collapse the position wave function. Now imagine you keep measuring it over and over every second. You'll just keep getting the same result, since the wave function will remain collapsed. But if you stop and give it some time, the collapsed wave function will evolve back into a non-collapsed wavefunction. So the next time you measure it after waiting, you will get a different result, in general.
     
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