Description of Wave Function Collapse

[Couchyam]

Hello, I'm new to Physics Forums, so I apologize if this question seems somewhat uninformed, but I have recently started studying quantum mechanics, and was curious about the idea of the wave function collapse, and how, from what I can tell, it seems to be approached completely independently from the solutions to the time-variant schroedinger equation.

Firstly, I don't seem to fully understand the idea of an "observation." From my own intuition, I would guess that a measurement of position would be very difficult to make without constraining the particle to wherever you wanted it to be. The simplest approach seems to be "shooting" out millions of equally spaced deep potential wells at the wave function at the same time to "trap" the particle between two adjacent wells. However, even then the initial state of the quantum system would change, and so the resulting wave equation would still not contain exact numbers, and would have completely no relevance to the initial wave equation.
So the potential wells would need to be "gradually" grown up around the particle. But in that case, when would the wave equation ever need to "choose"? Wouldn't everything would be defined by solving the wave equation as a function of time?

Sorry for not being more specific, but any insight would be welcome.

 

[Eynstone]

I wonder how one could 'shoot' potential wells at,say,photons.
Consider the double slit experiment. We don't really 'restrict' particles to measure the position - they can hit the screen anywhere.An observation is the position of a particle thus measured.
Since the subsequent evolution after an observation doesn't depend on the precedent state, the wavefunction is said to collapse. ( This is analogous to Huygen's principle of wave propogation.In fact, Feynman's path integral formulation resembles this.)

 

[meopemuk]

Firstly, I don't seem to fully understand the idea of an "observation." From my own intuition, I would guess that a measurement of position would be very difficult to make without constraining the particle to wherever you wanted it to be. The simplest approach seems to be "shooting" out millions of equally spaced deep potential wells at the wave function at the same time to "trap" the particle between two adjacent wells. However, even then the initial state of the quantum system would change, and so the resulting wave equation would still not contain exact numbers, and would have completely no relevance to the initial wave equation.
So the potential wells would need to be "gradually" grown up around the particle. But in that case, when would the wave equation ever need to "choose"? Wouldn't everything would be defined by solving the wave equation as a function of time?

If you want to measure particle position, you can simply use the old-fashioned ruler.

Eugene.

 

[Fredrik]

If you want to measure particle position, you can simply use the old-fashioned ruler.

Yes, but only after the position states of the particle have become correlated with the states of a system that's approximately classical, like a photographic plate. I'd say that the "measurement" is the interaction that produces those correlations, not what you're doing with the ruler.

 

[Tomsk]

The problem of interpreting wave function collapse is one of the least well understood things in physics, there isn't really a definite answer - look up the interpretation of QM.

I like the idea that wave function collapse represents an exchange of information between the system being measured and the observer. If you don't take into account the state of the observer, it looks like the wavefunction changes discontinuously. But if you take into account the state of the observer it looks like the observer and system are now correlated in some way. But you don't know which state the observer has measured because you are taking the point of view of a second observer who has not gained any information about either the system being measured or the first observer - he just knows the two are now correlated.

This is probably explained better elsewhere, look up relational quantum mechanics (that's just my preferred interpretation, there are loads of others).

As for your idea about shrinking potential wells to trap the particle - aren't you completely free to trap the particle wherever you like? So it doesn't really tell you where the particle was before you trapped it.

 

[Frame Dragger]

Hello, I'm new to Physics Forums, so I apologize if this question seems somewhat uninformed, but I have recently started studying quantum mechanics, and was curious about the idea of the wave function collapse, and how, from what I can tell, it seems to be approached completely independently from the solutions to the time-variant schroedinger equation.

Firstly, I don't seem to fully understand the idea of an "observation." From my own intuition, I would guess that a measurement of position would be very difficult to make without constraining the particle to wherever you wanted it to be. The simplest approach seems to be "shooting" out millions of equally spaced deep potential wells at the wave function at the same time to "trap" the particle between two adjacent wells. However, even then the initial state of the quantum system would change, and so the resulting wave equation would still not contain exact numbers, and would have completely no relevance to the initial wave equation.
So the potential wells would need to be "gradually" grown up around the particle. But in that case, when would the wave equation ever need to "choose"? Wouldn't everything would be defined by solving the wave equation as a function of time?

Sorry for not being more specific, but any insight would be welcome.

Whether or not collapse even occurs is a matter of debate among Interpretations of QM. If you're actually working in a related field, and not a theoretician, it's less of an issue. Decoherence seems to be a fair road to something beyond collapse, so maybe you'll have an answer in a decade... or ten. That's science for yah!

By the way, you don't need ANY interpretations to actually use or understand QM; there are people who just see it as a partial theory, but a terribly sucessful one to be used until it evolves or is replaced. If you work on the theory end of things, sadly I suppose that is a luxury you're not afforded.

Anyway, don't think about wavefunction collapse... bad for the BP!

 

[meopemuk]

there are people who just see [quantum mechanics] as a partial theory

That's what quantum mechanics is, in fact. QM is not designed to give a "comprehensive" picture of the world. It's purpose is to predict/explain results of specific measurements. In particular, quantum mechanics makes a clear separation between (measured) physical system and measuring apparatus or observer. States of the physical system are described by wave functions or vectors in a Hilbert space. The measuring apparatus is described by a Hermitian operator in the same Hilbert space. There is nothing wrong in this separate treatment of these two (seemingly equivalent) pieces of the physical world. In each particular experiment the distinction between the physical system and the measuring device can be easily made. The formalism of QM is just a reflection of this fact.

It is wrong to put states of both the electron and the ruler in one Hilbert space and hope to get a description of the electron-position-measurement experiment in this fashion. Of course, you are allowed to consider the enlarged physical system electron+ruler and put its states into one Hilbert space. But then you need to define who is observing it, what are the observables, what are the corresponding Hermitian operators. In this case you are not considering the experiment of the electron-position-measurement. You are considering a different experiment, so its quantum-mechanical description is understandably different.

Eugene.

 

[Couchyam]

Ok so what I guess I'm not understanding is that mathematically, the existence of a particle within a certain bounded region is equivalent to saying that the probability of finding the particle in that region is 1. If I want to measure a particle within a certain space interval "exactly," then I would need a potential curve that looks like a delta-function to constrain the wave equation. The idea that probability density can be represented by a wave function sounds almost like a truism, and so it seems strange that the wave function should ever disappear at all rather than just shrink to a smaller interval.
I was thinking that a hypothetical "measurement" could take place by constraining the particle's wave function to a set of discrete intervals, by "growing" a grid of potential energy around the particle that could trap it in a set of discrete locations. Then after the particle (assumed to have non-zero charge) has been trapped, shoot other particles into the "bins" (and then release them) to see where they go to find out where the particle is. Shooting up many different potential wells would ideally keep the particle from getting locked into a single location.

I guess this is sort of messed up because the potential well could never be well-defined in a small enough interval, but then that would just go into the math as well; there's now an associated probability and expected value of the energy of the delta functions.

Most "observations" include chemical processes/reactions, and so I was thinking that these reactions are basically the same thing as trapping the particle in a randomly spaced lattice of potential wells.