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Old Sep 8th 2016, 09:00 PM   #1
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Understanding the Uncertainty principle

Hi everybody,
I was reading about the Uncertainty principle and I came across the statement: "the position-space wavefunction of a minimum-uncertainty state is a Gaussian. Indeed, because it’s a minimum-uncertainty state, its momentum-space wavefunction is also a Gaussian." Unfortunately it was stated without any mathematical proof or diagrams so I am a bit puzzled!
Also, what does it mean exactly that: "The first energy level of a system is minimum uncertainty." My guess it that it means for lower energies the uncertainty in the energy is minimum. However, due to the quantized nature of the energy levels, how is it possible to be uncertain about the energy of a particle?

Last edited by Luxion; Dec 24th 2016 at 11:42 AM.
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Old Sep 9th 2016, 01:26 AM   #2
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Perhaps Dan or Chip have better ideas by my guess is.

Note that the first statement refers to one expression of the UP
That of momentum and position.

The second statement refers an alternative expression that of energy and time, and is simply observing that given more than one available energy state, the time of a particle being in that state is uncertain.

More context might produce a better answer.

Last edited by studiot; Sep 9th 2016 at 02:32 AM.
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Old Sep 9th 2016, 03:56 AM   #3
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A minimum-uncertainty state is one in which σ x σ p = ℏ 2 σxσp=ℏ2, where σ x σx is the uncertainty in position and σ p σp is the uncertainty in momentum. ℏ / 2 ℏ/2 is the smallest possible value for the product of uncertainties in position and momentum by the Heisenberg uncertainty principle. I don't know if every minimum-uncertainty state is a gaussian, but it is certainly true that gaussians are minimum-uncertainty states.
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Last edited by NicoleJS; Dec 24th 2016 at 11:48 AM.
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Old Sep 10th 2016, 02:06 AM   #4
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The minimum uncertainty refers to the product: σxσp=ℏ/2, as stated above by Lucas SV. There is no uncertainty in energy in a stationary state, as the energy is known exactly.

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Last edited by adamjohnson; Sep 12th 2016 at 03:57 AM.
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