Some quantum queries

[some_dude]

I've had two QM courses but they've been pretty much exclusively about the mathematics of QM (which is okay since that is the most important in practice) so I'm still unsure about some really basic interpretative issues that I'd like to know even though they seem embarrassing to ask.

FIrst one:
When some quantity of a system hasn't been measured in a while it seems as if half the time someone says it's in a superposition so that it really has multiple values at the same time and the other half of the time I hear that its value is undefined. The first view seems to suggest that, say, some particle is in multiple places at the same time while the latter view suggests that it in fact is nowhere until measured. They seem mutually exclusive to me, so which explanation is right?

Second one:
At time t=t₀ Alice measures some quantity q of system A. At a later time t=t₁ Bob measures q in system A. If you ask Alice to give the value of q at time t=t₁ she will say it's either in a superposition or undefined (depending on the answer to the first question) while Bob will say that he has just caused wavefunction collapse and q has the value just measured. I remember reading somewhere that they're both correct, but it seems to me that Alice would be wrong because the conditions have changed since her measurement because of Bob's measurement. I think this question have different answers depending on the interpretation used, this might be a non-issue in MWI, but it would be interesting nonetheless to know how collapse interpretations deal with it. This is obviously all assuming a non-relativistic situation, if it was the situation would be entirely different because they wouldn't agree on when t=t₀ and t=t₁ and they might both be right. You can argue that they can never be certain whether they both really measured the same system (at least they couldn't be measuring exactly the same system because Alice's measurement would have changed it) but I'm not sure whether "they're not really talking about the same system so they can both be right" is the answer to the question.

Third one:
I measure, say, the momentum in the x-direction of some particle causing its momentum wavefunction to collapse. Then the momentum wavefuntion p_x is given by a dirac delta function and the Heisenberg uncertainty principle seems to suggest that the uncertainty in the particle's x-position tends to infinity so that its position wavefunction is a constant and if you then measure its position it might as well be somewhere in the Andromeda Galaxy. Obviously this doesn't happen but I can't see why not? (Again this is only an issue in collapse interpretations but I want to know how they deal with it)

[Xanthir]

I've had two QM courses but they've been pretty much exclusively about the mathematics of QM (which is okay since that is the most important in practice) so I'm still unsure about some really basic interpretative issues that I'd like to know even though they seem embarrassing to ask.

FIrst one:
When some quantity of a system hasn't been measured in a while it seems as if half the time someone says it's in a superposition so that it really has multiple values at the same time and the other half of the time I hear that its value is undefined. The first view seems to suggest that, say, some particle is in multiple places at the same time while the latter view suggests that it in fact is nowhere until measured. They seem mutually exclusive to me, so which explanation is right?

It's in a superposition, where it exists in multiple states at once.

What's undefined is which of those states you'll measure it to be in once you interact with it. That is really, truly random.

Second one:
At time t=t₀ Alice measures some quantity q of system A. At a later time t=t₁ Bob measures q in system A. If you ask Alice to give the value of q at time t=t₁ she will say it's either in a superposition or undefined (depending on the answer to the first question) while Bob will say that he has just caused wavefunction collapse and q has the value just measured. I remember reading somewhere that they're both correct, but it seems to me that Alice would be wrong because the conditions have changed since her measurement because of Bob's measurement. I think this question have different answers depending on the interpretation used, this might be a non-issue in MWI, but it would be interesting nonetheless to know how collapse interpretations deal with it. This is obviously all assuming a non-relativistic situation, if it was the situation would be entirely different because they wouldn't agree on when t=t₀ and t=t₁ and they might both be right. You can argue that they can never be certain whether they both really measured the same system (at least they couldn't be measuring exactly the same system because Alice's measurement would have changed it) but I'm not sure whether "they're not really talking about the same system so they can both be right" is the answer to the question.

If Alice measured q, then it has a particular value. As long as Bob measures it in a compatible way, they'll get the same answers.

Third one:
I measure, say, the momentum in the x-direction of some particle causing its momentum wavefunction to collapse. Then the momentum wavefuntion p_x is given by a dirac delta function and the Heisenberg uncertainty principle seems to suggest that the uncertainty in the particle's x-position tends to infinity so that its position wavefunction is a constant and if you then measure its position it might as well be somewhere in the Andromeda Galaxy. Obviously this doesn't happen but I can't see why not? (Again this is only an issue in collapse interpretations but I want to know how they deal with it)

You can't measure the momentum with infinite certainty, so this is luckily a non-issue. The momentum is never described by a dirac delta; it's always a more "standard" curve with amplitude spread over a possibly-small region.

[Charlie!]

Third one:
I measure, say, the momentum in the x-direction of some particle causing its momentum wavefunction to collapse. Then the momentum wavefuntion p_x is given by a dirac delta function and the Heisenberg uncertainty principle seems to suggest that the uncertainty in the particle's x-position tends to infinity so that its position wavefunction is a constant and if you then measure its position it might as well be somewhere in the Andromeda Galaxy. Obviously this doesn't happen but I can't see why not? (Again this is only an issue in collapse interpretations but I want to know how they deal with it)

There are some extra intricacies from relativity (can't get to Andromeda faster than the speed of light, so you need relativistic QM), but that's pretty much what happens. You measure the momentum fairly precisely, and you find that the position is now spread out. Pretty strange and nonclassical. Why would this be any less strange and nonclassical (i.e. "an issue") for a no-collapse interpretation?

[mfb]

To the first one:
>> When some quantity of a system hasn't been measured in a while it seems as if half the time someone says it's in a superposition so that it really has multiple values at the same time and the other half of the time I hear that its value is undefined.
It can be well-defined (because you prepared it to be for example), otherwise it is in a superposition and it is not known in advance what "you" will measure if you perform a measurement. Or, all possible measurements will be obtained for some "you".


To the second one: If they measure the same object (e.g. some particles), it is always clear which interaction is done first, as this object moves time-like in space-time.
>> this might be a non-issue in MWI
Very true. With Copenhagen interpretation, the first measurement (by Alice) will cause a collapse to a state with a specific value of quantity q.


To the third one: The precise momentum measurement will also need a lot of time. Enough time for the particle to leave your lab (or hit the wall of your vacuum)), if you want to get too many digits in your measurement .

[some_dude]

Thanks a lot for the answers. I think I've misunderstood what the collapse of a wavefunction implies. I thought that when you measure some quantity its wavefunction collapses so that the quantity takes on a definite value so that the wavefunction is given by a dirac delta function in a short time span before it spreads out again which leads to question three being a problem specific to collapse interpretations. This is wrong cf Charlie!'s answer, but what, then, is wavefunction collapse?

[SU3SU2U1]

I thought that when you measure some quantity its wavefunction collapses so that the quantity takes on a definite value

Yes

so that the wavefunction is given by a dirac delta function

Only sometimes. If you measured the energy, its now in an energy eigenstate. If you measured spin, its now in a spin eigenstate,etc.

before it spreads out again

It evolves via Schroedinger/Heisenburg. If you measured the energy and the system is isolated, it will remain in the energy eigenstate (remember, energy states are "stationary states"). Everyone measuring will get the same answer, over and over. If you measured position, now it will be in a superposition of energy states and evolve via Schroedinger.