Wave Function Collapse: Quantifying Quickness

In summary, according to Griffiths, the second measurement of the position of an object will always yield the same result, given that the measurement is done quickly.
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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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  • #3
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.
 
  • #4
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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  • #5
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 ?
 
  • #6
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
 
  • #7
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)
 

Related to Wave Function Collapse: Quantifying Quickness

1. What is wave function collapse?

Wave function collapse is a phenomenon in quantum mechanics where an object or particle exists in multiple states simultaneously, known as a superposition, until it is observed or measured. The act of observation causes the wave function to collapse, resulting in the object or particle being in a single, definite state.

2. How is wave function collapse quantified?

Wave function collapse can be quantified using mathematical equations, such as the Schrödinger equation, which describe the probability of a particle being in a certain state at a certain time. The more measurements are taken, the more accurately the wave function collapse can be quantified.

3. What factors affect the quickness of wave function collapse?

The quickness of wave function collapse is affected by several factors, including the size and complexity of the system being observed, the type of measurement being taken, and the environment in which the measurement is being performed. Generally, larger and more complex systems take longer to collapse, and more precise measurements result in quicker collapse.

4. How does wave function collapse tie into the uncertainty principle?

The uncertainty principle, a fundamental principle in quantum mechanics, states that it is impossible to know both the position and momentum of a particle with absolute certainty. This is due to the wave-like nature of particles and the fact that measurement disturbs the system. Wave function collapse plays a role in this principle, as the act of measurement causes the wave function to collapse and results in uncertainty in the state of the particle.

5. Can wave function collapse be controlled or manipulated?

There is currently ongoing research into manipulating or controlling wave function collapse, as it has potential applications in fields such as quantum computing. However, this remains a challenge as it goes against the fundamental principles of quantum mechanics and the randomness of wave function collapse. Some theories propose using quantum entanglement or advanced measurement techniques to influence the collapse, but further research is needed to fully understand and utilize this phenomenon.

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