Uncertainty Principle And Collapse Wavefunction

[Waxterzz]

Upon a measurement of the position, the wavefunction collapses to a spike centered at x0

https://farside.ph.utexas.edu/teaching/315/Waveshtml/img3240.png

I encounter similar spike pictures numerous times, but there is an uncertainty in position , it can't be a spike right.

First thing I see on google:

1. 
   In quantum mechanics, wave function collapse is said to occur when a wave function—initially in a superposition of several eigenstates—appears to reduce to a single eigenstate (by "observation").
   

I thought position was not an eigenstate?

See similar thread now:

https://www.physicsforums.com/threads/position-eigenstates.764912/

Pls explain, thanks in advance. I have a lot of other questions but I don't want to cram them all in on thread, BUT I don't want to spam either, I'll just do it one by one.

 

[jfizzix]

The notion of a position measurement is special compared to measurements of other observables, because there is a continuum of possible position measurement outcomes (the same holds true for momentum, and other continuous observables).

At the foundations of quantum mechanics, we say that an observable has a spectrum of possible measurement outcomes (eigenvalues of that observable).

For an observable with a discrete spectrum, like the energy levels of the hydrogen atom, or components of the spin of a particle, it's reasonable to say that when we measure a discrete observable, we find a single eigenvalue, and the subsequent state of the system is in the single corresponding eigenstate of that observable.

When we measure a continuous observable (position), it's not so much that we find a single eigenvalue of position, but rather that the information we've gained has allowed us to narrow down the set of possible measurement outcomes to be close to some particular value. For example, we might know a particle definitely hit a particular pixel on an array of detectors. As a result, the position wavefunction would "collapse" to be zero outside of a narrow range of positions (defined by the pixel size), but not to be zero everywhere except at an infinitesimal point.

Indeed, the position-momentum uncertainty principle would say that if we were to measure the position of a particle with infinite precision (finding it in a "single" position eigenstate), then the momentum of that particle would be overwhelmingly likely to be near-infinite. Working backwards, it's simply not possible to measure the position of a particle so precisely because of the energies required.

 

[Waxterzz]

The notion of a position measurement is special compared to measurements of other observables, because there is a continuum of possible position measurement outcomes (the same holds true for momentum, and other continuous observables).

At the foundations of quantum mechanics, we say that an observable has a spectrum of possible measurement outcomes (eigenvalues of that observable).

For an observable with a discrete spectrum, like the energy levels of the hydrogen atom, or components of the spin of a particle, it's reasonable to say that when we measure a discrete observable, we find a single eigenvalue, and the subsequent state of the system is in the single corresponding eigenstate of that observable.Indeed, the position-momentum uncertainty principle would say that if we were to measure the position of a particle with infinite precision (finding it in a "single" position eigenstate), then the momentum of that particle would be overwhelmingly likely to be near-infinite. Working backwards, it's simply not possible to measure the position of a particle so precisely because of the energies required.

Discrete spectrum of eigenvalues; the corresponding operator commutes with Hamiltonian, zero uncertainty, values with 100% certainty that are always the same like energy states in Hydrogen, because energy is basically a scalar and a scalar commutes with every operator.

So, u have a wavefunction, u measure for example the total energy, the Hamiltonian is Hermitian, hence the eigenvalues are a real number, hence a scalar, and a scalar commutes with an operator, so when u measure, wavefunction collapses to a single spike, with a value corresponding to one of the possible eigenstates, E1 , E2, E3 etc

Continuous spectrum of eigenvalues: operator does not commute, hence always finite uncertainty, case for position and momentum

When we measure a continuous observable (position), it's not so much that we find a single eigenvalue of position, but rather that the information we've gained has allowed us to narrow down the set of possible measurement outcomes to be close to some particular value. For example, we might know a particle definitely hit a particular pixel on an array of detectors. As a result, the position wavefunction would "collapse" to be zero outside of a narrow range of positions (defined by the pixel size), but not to be zero everywhere except at an infinitesimal point.

This I don't get. Zero everywhere, except at an infinitesimal point?? Is this a Dirac function? But this is a spike. Then it's not continuous but discrete and that's impossible with position?

I thought it collapses, but not into a spike, although spiky-ish, but with a spread from the center of the spiky-ish function aka uncertainty around the point?

http://afriedman.org/AndysWebPage/BSJ/WaveCollapse.jpg

/head explodes

 

[bhobba]

This I don't get. Zero everywhere, except at an infinitesimal point?? Is this a Dirac function? But this is a spike. Then it's not continuous but discrete and that's impossible with position?

You need to study Rigged Hilbert Spaces to get the full resolution to this, but this requires a strong background in functional analysis.

But basically things like that are simply introduced for mathematical convenience and don't actually exist in practice. Its done in many areas of applied math eg if you bang something with a hammer that is modeled as a Dirac Delta function. Obviously that's not what really happens - it just helps with the math.

As a start to understanding the math get a hold of the following:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

Thanks
Bill

 

[andresB]

I don't remember the name of the article, but I once read that measuring with perfect accuracy the position of a particle is theoretically impossible (and not just impossible in practice) because the apparatus needed for it would violate energy conservation.

 

[bhobba]

I don't remember the name of the article, but I once read that measuring with perfect accuracy the position of a particle is theoretically impossible (and not just impossible in practice) because the apparatus needed for it would violate energy conservation.

It is.

Without going into the technical details this is all resolved in the Rigged Hilbert Space formulation. Only some states are physically realizable - the rest are introduced purely for mathematical convenience. For example an exact position measurement isn't really exact - its simply exact FAPP.

Thanks
Bill

 

[Waxterzz]

You need to study Rigged Hilbert Spaces to get the full resolution to this, but this requires a strong background in functional analysis.

But basically things like that are simply introduced for mathematical convenience and don't actually exist in practice. Its done in many areas of applied math eg if you bang something with a hammer that is modeled as a Dirac Delta function. Obviously that's not what really happens - it just helps with the math.

As a start to understanding the math get a hold of the following:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

Thanks
Bill

So the wavefunction can collapse to a spike, thus like a Dirac Delta Function, with 100% certainty of position?

 

[bhobba]

So the wavefunction can collapse to a spike, thus like a Dirac Delta Function, with 100% certainty of position?

Well first there is no collapse in QM - only in some interpretations.

But yes in theory it can - in practice no.

Like I said things such as the Dirac Delta function are just things to make the math easier - they don't represent anything physical.

Thanks
Bill