A contradiction in quantum mechanics

[Killtech]

in an attempt to get a better understanding of what happens during a measurement i have constructed a gedanken-experiment with two photon interference that regardless of its outcome seems to contradict quantum mechanics in one way or another and i was hoping to get a clarification here where i am doing a mistake.

so consider the following setup: take two independent photon beams (let's call them A and B) and point A towards a half mirror (HA). direct one outgoing ray towards a detector (DA) while use the other to do conduct a HOM interference with the second photon beam B with another half mirror HB. my question now is what will i measure? the trivial idea behind this setup is to detect the presence of the wave function originating from beam A and therefore detect whether it has collapsed or not. if i understand the theory correctly then detector DA should collapse the wave function of A non-locally regardless which way the photon at HA chooses to go due to Renningers negative result wave collapse.

therefore i'd expect one of the following outcomes:
1) the wave function indeed collapses non-locally but only after it has reached DA. therefore moving DA further away from HA such that the beam travel distance is longer then the from HA to HB. in this case i will always measure HOM interference. but if the DA is moved closer then HOM interference will only take place in case DA detects nothing. therefore i would be able to measure the position of DA in a non local way and albeit just binary i would theoretically be able to transfer information instantly independently of the distance. this contradicts QM.

2) the collapse already occurs at HA and thus HOM interference is only there when DA detects nothing regardless of its position (or even if its removed). this however would seem to contradict the quantum bomb tester experiment and a whole lot of other self interference experiments.

3) the wave function does not collapse regardless of DAs position. in that case the axioms of measurement are simply incorrect and the wave function itself (probably never) collapses but instead always follows the time evolution given by Schrödinger (or later quantum equations of motions) and it's only the probability that collapses. but if the probability is not fully determined by the wave function it would require additional degrees of freedom in the theory. considering that the probability is specifically required only for particle detection it carries all the particle properties. so remembering von-Bohm's particle trajectories it might be convenient to just use this as a first attempt to describe these additional degrees of freedom theoretically. therefore this experiment could possibly be used to falsify or confirm Bohms idea. note that this outcome would not contradict Bells experiments as it only shows that this particular experiment non-local aspects are not required.

so in any case to my understanding i get into conflict with QM axioms. the general problem here is that i want to measure no particle property which QM axioms are build around - but instead i want to measure the wave function directly. if i interpret QM correctly this is strictly forbidden by the axiom that requires measurable observables to correspond to linear operators. therefore i wonder what operator this gedanken-experiment measurment corresponds to - since there is not know algorithm i know of to calculate the observable-operator from a experimental setup. on the other hand my attempts to find operators for wave function measurement always yielded that i would need to a non linear one (the quadratic operator family Tz: f(x,t) -> f(z,t)f(x,t) obviously commutes and each f satisfies a eigenvalue equation so it would collapse on itself without change during a measurement). then again i question what kind of experimental evidence is this axiom (observables=linear) build on anyway since it does seem rather difficult to verify such an exclusion postulate. on the other hand a whole lot of QM interpretation seems to be implicitly founded on it.

 

[PeterDonis]

two independent photon beams

What do you mean by "independent"? In the usual meaning of that term, it means the photon states are uncorrelated, which means you can't predict what the second beam splitter, B, will do--its effect will be random. I don't think that's what you intended.

i understand the theory correctly then detector DA should collapse the wave function of A non-locally regardless which way the photon at HA chooses to go

You don't understand the theory correctly. Wave function collapse is not part of the theory; it's part of one family of interpretations of the theory, the family that includes the Copenhagen interpretation. But there is another family of interpretations, which includes the many-worlds interpretation, in which there is no wave function collapse.

i want to measure the wave function directly. if i interpret QM correctly this is strictly forbidden by the axiom that requires measurable observables to correspond to linear operators

That's right. But that statement is true regardless of which particular operator describes the observable you're interested in; it's true because the wave function is not an operator at all, it's a different kind of thing.

there is not know algorithm i know of to calculate the observable-operator from a experimental setup

You need to spend some time with a good QM textbook, such as Ballentine.

my attempts to find operators for wave function measurement

Are not going to find any, because, as above, a wave function is simply a different kind of thing. Once again, some time with a good QM textbook will help you to better understand what the theory actually says.

 

[PeterDonis]

The OP is based on mistaken premises. Thread closed.