I Wave Function Collapse: Quantifying Quickness

[Thejas15101998]

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?

 

[DrClaude]

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.

 

[Delta2]

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.

 

[Delta2]

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.

 

[BvU]

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 ?

 

[Thejas15101998]

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

 

[DrClaude]

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.

 

[BvU]

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 ). 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 !

 

[Lord Jestocost]

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)