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BHL 20

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One thing I cannot understand no matter how much I think about it, is momentum uncertainty. In classical mechanics a specific kinetic energy implies a specific momentum. And solving the time-independent Schrödinger equation gives specific energy levels for the system.

Now for systems with a mix of potential and kinetic energy there is of course no reason for the Hamiltonian eigenvalues to imply a specific momentum. But if you take a system like the particle in a potential well, where the Hamiltonian eigenvalues are essentially kinetic energies, and you force that system to collapse to a specific eigenvalue, how can there be any uncertainty in momentum at all? And in a system like the particle in a well there's always some amount of certainty in position so the momentum should never be 100% certain, right?

Also I'm always thrown off by comments like "the ground state energy can not be zero due to the uncertainty principle". How would collapsing to a zero energy ground state be any different from collapsing to any other eigenstate, in terms of reducing certainty?

I hope my question is clear, thank you.