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I'm still absorbing some basic ideas about quantum physics and now I think I have to reconsider the Uncertainty Principle.

Here is what I understand, in summary:

  • a "configuration" specifies the position, momentum, spin, and type, of every particle in the universe
  • there is a complex amplitude distribution over the full space of configurations
  • each configuration evolves into all possible subsequent configurations, and the associated amplitude transforms in some way for each of these
  • the amplitude for a future configuration is the sum of amplitudes over all inbound evolutions
  • this is basically what the Shrödinger Equation represents
  • the probability that we are in any particular configuration is the relative squared amplitude.

I think I get it.

But now I've started to think about the Uncertainty Principle all over again. Long before I understood the above points, I understood that there was no exact joint (position,momentum) value that could ever be measured.

If I go back over it all now, I get the impression that, within each configuration, the position and momentum is exactly defined and in fact the Uncertainty Principle comes from the fact that the amplitude distribution itself is fuzzy.

That is, if I make a measurement of momentum, I am projecting many configurations, encoding many velocities, onto a preferred basis. The information from the momentum measurement is "you are in one of these configurations, for which the corresponding position is one of these values".

Is this interpretation correct?

Follow-up question: does this imply that, if the amplitude distribution was pointlike, then uncertainty would be eliminated? (Not that this ever actually happens.)

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Your point #1 has a flaw: coordinate + momentum is phase space, the configuration space space is "smaller" - just momentum OR just coordinate. Amplitude distribution is over the configuration space, it is precisely the uncertainty principle that precludes us to have the ampl. dist. over the phase space (Although it is possible to develop a phase-space language for QM - search for "Wigner function") –  Slaviks Sep 2 '13 at 14:41
    
The Wigner Function is waaay over my head. Can you please turn "it is precisely the uncertainty principle that precludes us to have the ampl. dist. over the phase space" into an answer which I can actually accept? –  spraff Sep 2 '13 at 17:43
    
Just to generalize to the fullest extent the point Slaviks is making: a configuration is described by a complete set of commuting operators (CSCO). At this level you can replace "commuting operators" by the less jargony "compatible obervables," which really means the same thing: these are a full set of quantities which it is possible to specify at the same time. So for example position and momentum are not compatible. Neither are the components of angular momentum in different direction. But position is compatible with spin: you can measure both at the same time. –  Michael Brown Sep 3 '13 at 7:07
    
Another ex. of a CSCO is the value of quantum fields at every point in space at one time. But the values at different times are in general incompatible (because one can influence another that comes later). There are many choices of CSCO you can make, depending on the system and what properties you are after, but the physics is unchanged at the end of the day by what choice you make for the description of the system. –  Michael Brown Sep 3 '13 at 7:11
    
What you cannot do is mix descriptions willy nilly so that you try to measure incompatible observables at the same time. As long as you stay consistent in this way you can use any CSCO that you want. –  Michael Brown Sep 3 '13 at 7:11

1 Answer 1

Your bullet list is a very good summary except for the point #1 which has a flaw leading to confusion.

Coordinate + momentum is phase space, while the configuration space space is "smaller" - it is just the momentum OR just the coordinate. Amplitude distribution is over the configuration space, it is precisely the uncertainty principle that precludes us to have the amplitude distribution over the phase space.

Why so? Let's look at a specific amplitude distribution over the coordinate. The same distribution already contains as much information about momentum as possible (without violating the basic principles)! Information about the moment is "encoded" in the relative phases between the amplitude for different spatial coordinates. You can "translate" (t is called "change of representation") and amplitude distribution over the coordinate into an equivalent distribution for momentum (in this particular case the change of representation turns out to be the Fourier transform).

For example, if the quantum state is such that there is only one coordinate component (localized particle with a position-representation wave function proportional to Dirac delta-function), then the momentum is completely "uncertain" - the corresponding amplitude in the moment space will have equal probabilities for all possible momenta.

The opposite extreme is a plane wave - smear over all positions but with complex phases of the amplitudes that repeat themselves as you move by $\lambda$ in the direction of the wave $\hat{\mathbf{n}}$. The corresponding momentum distribution is a delta-fucntion of the momentum $\mathbf{p}$ centered at $\mathbf{p} = h \hat{\mathbf{n}}/\lambda$. (Here $\hat{\mathbf{n}}$ denotes a unit vector.)

Note that it is possible to develop a phase-space language for QM - search for "Wigner function".

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