Question regarding sudden approximation

[azztech77]

Hey everyone - just a bit of a conceptual question regarding the sudden approximation for a particle in an infinite square well. In theory, if we were to suddenly decrease the width of the potential from say L, to L' << L, in a very quick period of time - wouldn't this in some sense constitute a "measurement" of the particle's position in that you would now have a very good idea of where it was (especially if L' is very small) - at what point does the "collapse" of the wavefunction occur? would it be at the point at which the well length L was infinitely small (ie, a delta function potential?). When does collapsing of the wavefunction occur - what I don't understand is that since any measurement of position in the lab has some level of error (We can never pinpoint the particle's position exactly) how could a wavefunction ever truly "collapse" to a delta function? I imagine it doesn't and if anything this is as usual just an approximation of reality.

Many thanks for help clearing up the confusion.

 

[Jano L.]

how could a wavefunction ever truly "collapse" to a delta function? I imagine it doesn't and if anything this is as usual just an approximation of reality.

Although the formalism leads one to think so, the idea that the wave function can collapse to a delta function is not consistent with the rest of the theory, namely Born's rule. The reason is that delta function is not a function, but a distribution which does not follow Born's rule that |psi|^2 is density of probability.

Instead of delta function, wave function can be thought to (due to special interaction) localize into a small wave packet with limited extension. In practice wave functions have spatial extension of orders of Bohr radius, 10^{-10}~m, or bit smaller for nuclei, but there is no experiment currently known that could find position of the electron exactly.

Your example is correct, and would work for a macroscopic well, say 1mm long; after shrinking the well rapidly, the particle will be still inside (if it is infinite) and the wave function will be altered correspondingly; this can be calculated from Schrödinger equation in principle.

However, at the level of atomic distances, such measurement is too hard to perform in practice; to decrease the length of the well rapidly would mean faster than the natural oscillation of the wave function, but this is of order 10^{-15}~s, or period of visible light and this is too fast for manipulating some borders of the potential well. The atom is already very short potential well; shrinking it further is hard.

Please do not take this as argument that the electron does not have a position; that is a very different question.