Mathematically what causes wavefunction collapse?

[Nugatory]

And yet another thing. If we include observational apparatus into the system then individual systems include the same observational apparatus (yet at different times and in different states) and are not really separate.

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared through the same procedure to conduct the same experiment.

 

[bhobba]

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared through the same procedure to conduct the same experiment.

Exactly. But its a slight blemish you have to do this.

And yet another thing. If we include observational apparatus into the system then individual systems include the same observational apparatus (yet at different times and in different states) and are not really separate.

Yea - that's an issue with that interpretation - not much of an issue IMHO (I agree with Nugatory) - but an issue.

That's one reason (there are others - but it is one - if it was only this issue I probably wouldn't worry about it) why I hold to the ignorance ensemble interpretation with decoherence.

Not that Ballentine agrees that decoherence is of any value interpretatively - he doesn't - but I respectfully disagree with him on that point.

Thanks
Bill

 

[OCR]

(as entering "random walks" in Google has not proved particularly fruitful!).

That's odd...

https://www.google.com/#q=Random+walk

It seems to be the first "hit", then merely a jaunt to...

http://en.wikipedia.org/wiki/Random_walk

Then to...

http://en.wikipedia.org/wiki/Category:Variants_of_random_walks

We then arrive to view the... you guessed it...

http://en.wikipedia.org/wiki/Wiener_process

And, look at all the processes at the bottom of the page ... wow!



OCR...

 

[Sugdub]

I certainly agree that observational apparatus should be included into the system. But ...
then it would seem that you have to include preparation apparatus too ... and manipulation apparatus. And we end up at the same thing that Sugdub was saying earlier in discussion that the state is property of the whole experimental setup...

May be it is wise to also relate this to my last input as #88 in the other thread related to Ballentine's ensemble-interpretation:

(in a slightly different context, ...)... I would agree with your statement if the word “systems” was replaced with a proper concept. Paraphrasing one of your previous inputs I would say: "It is undeniable that the state vector can and should be thought of as a representation of the statistical property of an iterative run of a uniquely prepared experiment which delivers, at each run, one amongst a set of possible outcomes". Stating that the statistical property relates to the flow of qualitative pieces of information produced by an experiment is the only true minimal position that cannot be challenged. Stating that the distribution relates to some physical system or stating that each individual piece of information relates to an individual physical system, that already goes beyond the bare minimum since it cannot be proven experimentally...
Thanks

 

[Superposed_Cat]

imaginary time?

 

[Jilang]

That's odd...

https://www.google.com/#q=Random+walk

It seems to be the first "hit", then merely a jaunt to...

http://en.wikipedia.org/wiki/Random_walk

Then to...

http://en.wikipedia.org/wiki/Category:Variants_of_random_walks

We then arrive to view the... you guessed it...

http://en.wikipedia.org/wiki/Wiener_process

And, look at all the processes at the bottom of the page ... wow!



OCR...

Ha Ha, that's really funny - that Wiki Page is one of my bestest favourites!
What I should have said "not particulary fruitful when looking for published articles pertaining to QM"
Thanks for pointing all out the processes though - I'd forgotten just how much it applies to! Interesting that they don't seem to mention Quantum Mechanics though...even the Wiener part only talk about fluid dynamics.

 

[Jilang]

imaginary time?

Cat, It sounds like science fiction doesn't it? Hawking and physicist James B. Hartle have applied the concept of imaginary time in their research on the origin of the universe, including their efforts to develop a unified theory derived from Einstein’s theory of relativity and from Richard Feynman’s concept of multiple possible histories of the universe.

 

[stevendaryl]

If you have time could you expand on this a bit more. I'm very interested in the Born postulate and would love to have a better understanding of it. As it's defined it looks like a joint probability to me rather than a probability of a single entity. The similarity in its form to probability of transitions between the initial and final states and interactions has an implication that I'm trying to understand.

This is just fooling around with symbols, but...

The probability amplitude to go from state $| A\rangle$ at time $t$ to state $|B\rangle$ at time $t + \delta t$ is given by:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle$

If we assume that this formula works when $\delta t < 0$, then the probability amplitude for going from state $| B \rangle$ at time $t + \delta t$ to state $|A\rangle$ at time $t$ is given by:

$\langle B | e^{+i H \delta t/\hbar} | A \rangle$

So the amplitude for going from $| A\rangle$ to $|B\rangle$ and back in time to $| A\rangle$ would be the product:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle\langle B | e^{+i H \delta t/\hbar} | A \rangle = |\langle A | e^{-i H \delta t/\hbar} | B \rangle|^2$

which is the Born expression for the probability of going from $| A\rangle$ to $|B\rangle$.

So, mathematically, the probability of going from $| A\rangle$ to $|B\rangle$ is the probability amplitude of making a "round-trip" back to the starting point (and starting time).

 

[Jilang]

So the amplitude for going from $| A\rangle$ to $|B\rangle$ and back in time to $| A\rangle$ would be the product:

$\langle A | e^{-i H \delta t/\hbar} | B \rangle\langle B | e^{+i H \delta t/\hbar} | A \rangle = |\langle A | e^{-i H \delta t/\hbar} | B \rangle|^2$

which is the Born expression for the probability of going from $| A\rangle$ to $|B\rangle$.

So, mathematically, the probability of going from $| A\rangle$ to $|B\rangle$ is the probability amplitude of making a "round-trip" back to the starting point (and starting time).

Thanks for this. I can see what you mean. It's given me a lot to think about. I think Swinger introduced circles in time in 1960. I wonder what determines whether a particle travels clockwise or anti-clockwise. Logically it would need to go both ways so phase factors would cancel I think.

 

[zonde]

Nope - each element of the ensemble includes its own measuring apparatus. You could think of it as an ensemble of laboratories, all prepared through the same procedure to conduct the same experiment.

Can you elaborate what is your statement?
It's your belief? Or do you mean that it's Ballentine's interpretation? Or maybe you think it's experimentally verified fact?