Mathematically what causes wavefunction collapse?

  • #126
1,116
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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... :biggrin:

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

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



OCR... :smile:
Ha Ha, that's really funny - that Wiki Page is one of my bestest favourites!:tongue2:
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.
 
  • #127
1,116
72
imaginary time?
Cat, It sounds like science fiction doesnt 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.
 
  • #128
stevendaryl
Staff Emeritus
Science Advisor
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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 [itex]| A\rangle [/itex] at time [itex]t[/itex] to state [itex]|B\rangle[/itex] at time [itex]t + \delta t[/itex] is given by:

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

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

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

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

[itex]\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[/itex]

which is the Born expression for the probability of going from [itex]| A\rangle [/itex] to [itex]|B\rangle[/itex].

So, mathematically, the probability of going from [itex]| A\rangle [/itex] to [itex]|B\rangle[/itex] is the probability amplitude of making a "round-trip" back to the starting point (and starting time).
 
  • #129
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72
So the amplitude for going from [itex]| A\rangle [/itex] to [itex]|B\rangle[/itex] and back in time to [itex]| A\rangle [/itex] would be the product:

[itex]\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[/itex]

which is the Born expression for the probability of going from [itex]| A\rangle [/itex] to [itex]|B\rangle[/itex].

So, mathematically, the probability of going from [itex]| A\rangle [/itex] to [itex]|B\rangle[/itex] 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.
 
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  • #130
zonde
Gold Member
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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?
 

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