Yes, it's ironic that for the two most commonly used operators used in quantum mechanics--the position operator and the momentum operator--neither of them has eigenstates (for the free-particle case) that are actually in the Hilbert space. The momentum eigenstates are sinusoids, and as you say, non-normalizable (and therefore not physically allowed). While the position eigenstates are Dirac delta distributions, which are not even functions! (They're just the limit of a series of functions.) So neither of them are actually in the space you're supposedly working in. And neither of them ever happen in the real world.
There's a fairly intuitive way to understand why they never happen in the real world though. You can't localize a particle to be at an exact location, because then the uncertainty in the position would be zero, which would yield an infinite uncertainty in momentum. Conversely, you can't have a perfect sinusoid that stretches out all the way to the boundary of spacetime (infinite uncertainty in position) which is what would be required in order to perfectly measure the position of something.
The moral of the story is just that you can never precisely measure position or momentum, you always have to measure some combination. The position and momentum operators are mathematical idealizations of real measurements that you can do which consist of combinations of them.
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timeasmymeasure
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Date: 2010-09-15 06:14 pm (UTC)There's a fairly intuitive way to understand why they never happen in the real world though. You can't localize a particle to be at an exact location, because then the uncertainty in the position would be zero, which would yield an infinite uncertainty in momentum. Conversely, you can't have a perfect sinusoid that stretches out all the way to the boundary of spacetime (infinite uncertainty in position) which is what would be required in order to perfectly measure the position of something.
The moral of the story is just that you can never precisely measure position or momentum, you always have to measure some combination. The position and momentum operators are mathematical idealizations of real measurements that you can do which consist of combinations of them.