For instance, will attempts to predict comprehensively eventually prove contradictory?
I suppose it happens with measure 0 ??
If the effect is simple enough, and you include the full range of eigenvalues in your prediction, then I think it can happen; the single photon does or does not hit the target, or whatever.
It matters here not what the value of measurement predicts, just that the action of measurement yields an uncertainty interfering with a singular outcome. However, if a measurement leaves the system in a particular eigenstate, can we be 100% sure that remeasuring that observable immediately would reproduce the previous result?
Oh..ok u mean that (I thought you were speaking about the EPR argument)...
It's a good question....
Well...according to the rules, if you measure the energy, then remeasuring it will give again the same energy, because energy eigenstates are stationary, following Schroedingers equation (however neglecting things such as spontaneous emission, and other field interaction stuff).
However, if you measure the momentum or position, since those eigenstates after measurement are a superposition of different energy eigentstates, then there is a non trivial evolution after measurement, leading to other possible values if remeasured....
Maybe there exists some statistical fine structure from spacetime curvature that affects even consecutive measurements of compatible variables.
Back to maybe, Loren? Maybe the Easter Bunny collapses the wave function, too!
Wow...Loren seems to be very in advance upon my own time in physics...where are you studying ? Or are u just emitting hypotheses ?
What kind of outcomes are you talking about? If I have a coin that can come up heads or tails, then I can confidently the say that the event "on the next flip, this coin will come up either heads or tails"! I suspect that's not what you are talking about!
To Kleinwolf: for a simple, finite space like that, yes, prob 1 means the outcome MUST happen, prob 0 means it CAN'T happen. But that's not true for infinite outcome sets. If I have, say, a normal probability distribution for picking real numbers, then the probability of picking ANY specific number is 0- but obviously some number IS picked every time: probability 0 does NOT mean "impossible" and probability 1 does NOT mean "certain".
What I am first trying to say is that the determination from consecutively measuring observables identically and consistently, i. e. 100%, is allowed by quantum mechanics. I suggest secondly that gravitation or vacuum energy can introduce a statistical anomaly analogous to fine splitting of the hydrogen energy levels, but one whose asymmetry actually violates the discrete nature of quantum logic.
Actually Halls, that is also the type of outcome I include. (Just visit "Many Worlds.") The coin might land on its side, tunnel outside the room, or observers lose the meaning of the heads-tails duality. Aside from the spectacular, all classical arguments may be reduced to those quantum, and an accompanying duality. If I say that there is a chance of the universe being or not being, the outcome may be a superposition of both, with any prediction only 50% correct. Reduced to quantum probability, physics is essentially unpredictable, at least for some of the trials - yet there is another consideration where even probabilities are proved fallible by the disturbance of proof itself!
What I am trying to get at is that a given (classical or quantum) prediction's prophesy sways any anticipated outcome from absolute determinism. That is, the unitary state of the EPR setup is not truly isolated from the past, present and future of observer definiteness (which itself is singular in uncertainty). There is one "correct" result to agree with, but many infinities of alternative turnouts. I therefore propose that an observer's physical expectations evolve away from exactitude due to their very interference with their environment they attempt to measure, despite rather than considering specifically their quantum mechanics.
To be more precise, the probability of picking a number between x and x+dx is :P(x<X<x+dx)= f(x)dx, where f(x) is the probability density
It's clear, that in the limit dx->0, if f(x) is not too singular, then P(X=x)->0
But my original problem was : Suppose the trivial operator 1 on a single q-bit in a state |S>=(1,0).
Then the final state is |phi>=(cos(phi),sin(phi)) any normalized vector.
The probability of the system being in endstate |phi> after measuring with 1 is |<phi|S>|^2=cos(phi)^2 (*)
Hence |S> or -|S> are with probability 1 the endstates, but the other states also have non-vanishing probabilities..
Not here that the formula (*) gives the probability over a continuous set (the phis) (the probability which in your example was 0), and not the probability density.
So my question is : do the other final states appear ?
I think Max Planck summed it up very well when he said :
"Science cannot solve the ultimate mystery of nature. And it is because in the last analysis we ourselves are part of the mystery we are trying to solve."
In other words, 3rd person objective science cannot explain everything, because 3rd person objectivity is an ideal which cannot ultimately be realised.
It would seem there would need be no need to predict, were the out come known. Also, what prediction processes are at play in an equation. An alternative may develope an alternate form. Of course this may be one time it not only looks good on paper, but no where else,or everywhere else.!? Is it therefore posible therefore, to predict exceptions!? Curious question, though I suspect you have reason to give it an answer? I'd like to know.
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