Wave Function Collapse: Quantifying Quickness

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Discussion Overview

The discussion revolves around the concept of wave function collapse in quantum mechanics, specifically focusing on the timing of repeated measurements and how quickly they must be performed to yield the same result. Participants explore the implications of measurement uncertainty and the behavior of wave functions over time.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a quantitative understanding of how quickly repeated measurements must occur to observe the same result, referencing Griffith's quantum mechanics.
  • Another suggests using the Heisenberg uncertainty principle to calculate the momentum uncertainty, which could provide insights into the particle's speed and the time required for it to move a distance comparable to the measurement uncertainty.
  • A participant challenges the previous argument by stating that while momentum uncertainty gives a range, it does not specify the exact momentum value.
  • One contributor proposes that the time between measurements should be very small, potentially around ##10^{-10}## seconds or less, depending on the wave function's characteristics.
  • Several participants discuss the content of Griffith's text regarding wave function collapse and its implications for measurement timing, with some expressing uncertainty about the interpretations presented.
  • A later reply provides a specific example, stating that if an electron is localized at an atomic scale, the time for the wave packet to double in width is approximately ##10^{-16}## seconds.

Areas of Agreement / Disagreement

Participants express various viewpoints on the timing of measurements and the implications of wave function behavior, indicating that there is no consensus on the exact timing or the interpretations of the measurements discussed.

Contextual Notes

Some statements rely on assumptions about the wave function and measurement techniques, which are not fully elaborated. The discussion includes references to specific texts and interpretations that may not be universally accepted.

Thejas15101998
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I read in Griffith's quantum mechanics that in a particular system, the second time measurement of the position (say) would yield the same result (the same collapse or the same spike)given that the measurement is done quickly (since it soon spreads out).
I don't understand how quick this is supposed to be. Could somebody give a quantitative feeling for this quickness?
 
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Say that your first measurement is done with an uncertainty ##\Delta x##. Using the Heisenberg uncertainty principle, calculate the corresponding ##\Delta p##. That will give you an approximation of how fast the particle is moving. You can then calculate how long it will take for the particle to move a distance of ##\sim \Delta x##, such that there is a strong probability that it will not be found at the same place as the first measurement.
 
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DrClaude said:
Say that your first measurement is done with an uncertainty ##\Delta x##. Using the Heisenberg uncertainty principle, calculate the corresponding ##\Delta p##. That will give you an approximation of how fast the particle is moving. You can then calculate how long it will take for the particle to move a distance of ##\sim \Delta x##, such that there is a strong probability that it will not be found at the same place as the first measurement.

Nice argument but ##\Delta p## is about p belonging to and interval of the form ##[p-\Delta p,p+\Delta p]## has a very high probability but doesn't give us info about what p is.
 
ehm ok I guess we ll have a first value for p that goes along with the first measurement of x.To the OP: (I am a mathematician like you that I took only one introductory course in Quantum Mechanics during my undergraduate studies, so I am not quite sure about this): I THINK it depends on the wave function of the system, that how fast the two measurements should be done so that the probability to get the same measurement is high enough. If the wave function is such that quickly spreads out then the time between the two measurements should be very small ( I would say something like ##10^{-10}## seconds) or even smaller.
 
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Thejas15101998 said:
I read in Griffith's quantum mechanics
In the 1995 edition I find two mentionings of collapse and only the comment 'immediately repeated measurement' . First one is in connection with non-commuting spin operators, the second in the 'afterword'.What exactly did you read ?
 
BvU said:
In the 1995 edition I find two mentionings of collapse and only the comment 'immediately repeated measurement' . First one is in connection with non-commuting spin operators, the second in the 'afterword'.What exactly did you read ?
Well I read it in the first chapter
 
Delta² said:
Nice argument but ##\Delta p## is about p belonging to and interval of the form ##[p-\Delta p,p+\Delta p]## has a very high probability but doesn't give us info about what p is.
The fact that p can be non-zero will depend on the measuring method. If ##p > \Delta p##, then it is of course ##p## that will give an upper limit to the time interval.

My idea is simply to give the OP a sense of how fast the spatial wave function evolves.
 
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Thejas15101998 said:
Well I read it in the first chapter
Ah, sorry. I skipped that one on page 4. It also says 'immediately', and as far as I can judge that means something in the sense of 'before time development changes the situation'. Not much use in this stage, I agree. Read on to learn about QM and leave this interpretation business for later (and then find another source than Griffiths if you plan to become a theoretician :wink: ). This is advice from an experimentalist, so I'll gladly trade it in for something better fitting your stage in the curriculum.

Obviously, your question is a good one: kudos !
 
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Thejas15101998 said:
I don't understand how quick this is supposed to be. Could somebody give a quantitative feeling for this quickness?

For instance, if an electron is originally localized in a region of atomic scale, Δx ~ 10-10 m, then the characteristic time for a wave packet of original width Δx to double in spatial extent is only about 10-16 sec. (from: http://farside.ph.utexas.edu/teaching/qmech/Quantum/node26.html#exp)
 

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