Quantum wave bursting out of 100 jars

[jartsa]

Let us consider a quantum wave of one very energetic particle, stored in 100 microscopic jars. There's some lock mechanism that keeps the jar lids on.

Then we open all the locks simultaneously. How many lids fly off? One? All?

 

[Simon Bridge]

If I understand you correctly, you mean that the particle's wavefunction is spread across all the "jars?
You mean that "open all the locks" amounts to a measurement of the position of the particle - you know which jar the particle was found in by noticing which of the lids flies off.

That would appear quite straight forward - so what did you come up with?

 

[jartsa]

If I understand you correctly, you mean that the particle's wavefunction is spread across all the "jars?
You mean that "open all the locks" amounts to a measurement of the position of the particle - you know which jar the particle was found in by noticing which of the lids flies off.

Yeah something like that.

That would appear quite straight forward - so what did you come up with?

well ... the jars and their contents can't be exactly the same, all the lids start to propagate, but not exactly the same way, the environment gets a hunch that one lid has moved, that causes a slight quantum-zeno effect on the other lids, the lids are more different now, the environment gets a little bit less uncertain information about the lids ... and so on.

If the lids are very sensitive, then the entropy of the lid population increases very easily ... in this case one lid flying off is probable.

If the lids are very in-sensitive, then the entropy of the lid population does not increases very easily ... in this case all lids flying off is probable.

 

[Simon Bridge]

"all the lids start to propagate" does not make sense.

I think you have under-described the system in question.
If the lid popping amounts to a measurement of the position of the particle, and there is only one particle, then it cannot be found to be an more than one jar at once.

But if the "lid popping" is in response to something else, then you have a different situation.
Please be clear. What do you intend the lids to respond to?

Another way of setting this is up with a single particle contained in an infinite square well - divided into 100 sub-wells (jars) by 99 thin membranes which the particle may tunnel through with some probability.

i.e. if the jar-state, "the particle is in jar n", is |n> and the particle is confined to a particular energy eigenstate |e>, then |e> = sum c_n(t)|n> ... i.e. the particles position wavefunction will span all the jars.

We can then ask what happens in measurements of energy, position, etc. without worrying about the exact physics of how a lid gets popped.

 

[jartsa]

Inertia tries to keep the lid quantum waves as standing waves at the jar opening.

Particle quantum wave tries to convert n1 standing lid waves to n2 propagating waves, where n1 and n2 are some numbers that are unkown to me.

Lid/lids is/are effected by inertia and force. I don't know any details about the force.Edit: And the lids are being seen by the environment at a normal degree. A finite amount of light is reflected by the lids. That light is seen... let's say by a conscious human observer.

 

[Simon Bridge]

Still not making sense to me - can you provide a link to where you got this problem from?

Sounds like you are overcomplicating things.

 

[jartsa]

Still not making sense to me - can you provide a link to where you got this problem from?

It's my own idea.


Let see ... the wave may kind of go through 100 holes. We might observe an interference pattern if we repeated this experiment many times. But there are some kind of detectors on the holes. If they are valid which-way detectors, then the wave is forced to choose one hole.

So we can conclude that this may be a question about what makes a valid which-way detector.

 

[Simon Bridge]

OK - so instead of the particle being distributed between 100 jars, as per the first post, you are now suggesting the setup of an interference experiment but where a detector is placed on each hole, and there are 100 holes.

If the "lid pops" when a particle goes through a hole, then only one lid will pop.

Your question is too vague to be about "what makes a valid which-way detector".
If you want to know how we would make such a detector, then that is quite a different question.
It sounds like you are working through wave-particle duality conceptually ... don't worry, it happens to all of us.
It helps to use the maths.

 

[Simon Bridge]

Breifly - here is how it helps to think about the maths:

When a particle passes through a row of 100 narrow slits, we may represent the wavefunction, after the slits, as
|Y>=(|1>+|2>...+|99>+|100>)/10

... this is basically a position wavefunction, and assumes that the particle is equally likely to pass through any of the 100 slits. The slits effectively measures the position of the particle in such a way as to narrow down the possibilities. Any particle emerging from the far side of the slits must have come through one of them.

Putting detectors on the slits basically measures the position of the particle, out of the 100 possibilities, collapsing the state vector further to whatever the outcome of the measurement was... |Y>-->|n> - where n is the number of the slit the particle was detected at.

We can represent the detector as an operator N: N|n>=n|n>, n=1,2,3...100

But left alone |Y> will evolve with time so an array of detectors some distance from the slits will see an interference patters. A quick way to see the interference pattern for simple geometries is to consider that such an initial restriction in position means that the momentum becomes uncertain. The superposition of the momentum states gives the interference pattern.

The other model, where you have 100 sub-wells (jars) in a loop - the state vector is basically the same form but now |n> is the state of being in the nth sub-well. Transition between sub-wells would be governed by some sort of operator, say:

T= a(|2><1|+|3><2|+ ... +|100><99| + |1><100|): a=constant carrying the units of T.

If we initially prepare the system so that the particle has probability 1 of being found in state n at tome t=0, then <N> (N|n>=n|n>, n=1,2,3...100 as before) will (unless I messed up) travel around the loop with time. (This is why I put them in a loop rather than an ISW.)

[note: for this to work, <n|m>=δ(n,m) and ∑|n><n|=1.]

To fit your needs, we want to initially prepare the system so the particle is equally likely to be an any of the 100 sub-wells, and then measure the position.

Whatever - the result of the measurement is that only one lid will "pop".

In none of the above has the details of how a measurement comes about been used, instead we are using our knowledge about the outcome of the experiment to deduce what must have happened to the state vectors and making assumptions about what it means to say "the lid pops".

What you seem to be doing is trying to work in a physical process for the lid to pop.
Maybe there is a situation where we can get a single particle to pop more than one lid? If so, then the lid has been rigged so that it measures something other than the |n> state of the particle.

i.e. maybe the interaction is that the lid carries off some energy from the particle, causing the system to decay - and the particle need not escape just because a lid has opened? As long as the system has another state to decay to, with energy difference higher than that needed to pop the lid, then many lids can be popped.

Whatever: to get a sensible answer you will have to bite the bullet and actually state what mechanism you are thinking of. I suspect it is this part, the bit you are not saying (maybe you are worried about looking silly?), that contains the heart of your question.