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Question regarding sudden approximation

  1. Oct 18, 2012 #1
    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.
     
  2. jcsd
  3. Oct 18, 2012 #2

    Jano L.

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    Gold Member

    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, [itex]10^{-10}~[/itex]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 Schrodinger 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 [itex]10^{-15}~s[/itex], 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.
     
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