Can Quantum Mechanics Explain the Uncertainty Principle?

[kanato]

If the wave function has definite quantum numbers n,l,m then the trajectory is a circle with the angular velocity proportional to m. In particular, when m=0 (which is the case when n=0) then the particle is at rest.

Uh.. how does that work? If the electron in a 1s state is at rest, then why does it have a nonzero expectation value for its kinetic energy? (By the virial theorem, the electron should have a kinetic energy expectation value of <p^2/2m> = +13.6 eV.) I can understand that the particle could be at rest but have uncertainty in position, but I don't understand how the particle can be at rest and have kinetic energy at the same time.

 

[Demystifier]

So in the ground state of the hydrogen atom, the electron just sits there hovering at a fixed distance from the proton?

Yes.

 

[Demystifier]

Uh.. how does that work? If the electron in a 1s state is at rest, then why does it have a nonzero expectation value for its kinetic energy? (By the virial theorem, the electron should have a kinetic energy expectation value of <p^2/2m> = +13.6 eV.) I can understand that the particle could be at rest but have uncertainty in position, but I don't understand how the particle can be at rest and have kinetic energy at the same time.

That is a good (and very frequent) question. When the electron is in the ground state, then its kinetic energy is zero. However, when you MEASURE the kinetic energy of the electron, then the electron becomes entangled with the measuring apparatus (which is also made up of quantum particles), so the electron is NO LONGER in the ground state. Instead the electron starts to move and attains a definite kinetic energy. It turns out, and this is THE CENTRAL part of Bohmian mechanics, that whatever you MEASURE, the probabilities of obtaining particular measurement outcomes are exactly the same as those given by standard QM.

The crucial point to remember is the fact that measurement changes the properties of the system. This fact is called - contextuality.

 

[Demystifier]

I don't want to take this thread into the area of one interpretation or another, but you might gain some information from this:

http://arxiv.org/abs/0903.3878

This paper (just as many similar ones) is wrong for a simple reason that it does not take into account the theory of quantum measurements involving entanglement between the measured system and the measuring apparatus. When this is taken into account, then, as Bohm proved GENERALLY in his 1952 paper, the Bohmian QM and standard QM have exactly the same predictions in all circumstances, as long as one measures observables defined by hermitian operators in the Hilbert space.

But for some reason, people tend to have an opinion on Bohmian mechanics without styding its crucial part - the theory of quantum measurements. Without that, Bohmian mechanics cannot be properly understood. It is the crucial part, even more important than particle trajectories themselves.