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"the wisdom of the wise will perish, the intelligence of the intelligent will vanish."
Welcome to Fairfax Underground, a project site designed to improve communication among residents of Fairfax County, VA. Feel free to post anything Northern Virginia residents would find interesting.
The Uncertainty Principle
Posted by:
Pancks
()
Date: May 02, 2009 08:38AM
In quantum physics, the Heisenberg uncertainty principle states that certain physical quantities, like position and momentum, cannot both have precise values at the same time. The narrower the probability distribution for one, the wider it is for the other.
In quantum mechanics, a particle is described by a wave. The position is where the wave is concentrated and the momentum is the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.
The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength. Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there are no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.
The uncertainty principle can be restated in terms of measurements, which involves collapse of the wavefunction. When the position is measured, the wavefunction collapses to a narrow bump near the measured value, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement. The amount of left-over uncertainty can never be reduced below the limit set by the uncertainty principle, no matter what the measurement process.
This means that the uncertainty principle is related to the observer effect, with which it is often conflated. The uncertainty principle sets a lower limit to how small the momentum disturbance in an accurate position experiment can be, and vice versa for momentum experiments.
A mathematical statement of the principle is that every quantum state has the property that the root-mean-square (RMS) deviation of the position from its mean (the standard deviation of the X-distribution):
\Delta X = \sqrt{\langle(X - \langle X\rangle)^2\rangle} \,
times the RMS deviation of the momentum from its mean (the standard deviation of P):
\Delta P = \sqrt{\langle(P - \langle P \rangle)^2\rangle} \,
can never be smaller than a fixed fraction of Planck's constant:
\Delta X \Delta P \ge {\hbar \over 2}.
Any measurement of the position with accuracy \scriptstyle \Delta X collapses the quantum state making the standard deviation of the momentum \scriptstyle \Delta P larger than \scriptstyle \hbar/2\Delta x.
In quantum mechanics, a particle is described by a wave. The position is where the wave is concentrated and the momentum is the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.
The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength. Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there are no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.
The uncertainty principle can be restated in terms of measurements, which involves collapse of the wavefunction. When the position is measured, the wavefunction collapses to a narrow bump near the measured value, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement. The amount of left-over uncertainty can never be reduced below the limit set by the uncertainty principle, no matter what the measurement process.
This means that the uncertainty principle is related to the observer effect, with which it is often conflated. The uncertainty principle sets a lower limit to how small the momentum disturbance in an accurate position experiment can be, and vice versa for momentum experiments.
A mathematical statement of the principle is that every quantum state has the property that the root-mean-square (RMS) deviation of the position from its mean (the standard deviation of the X-distribution):
\Delta X = \sqrt{\langle(X - \langle X\rangle)^2\rangle} \,
times the RMS deviation of the momentum from its mean (the standard deviation of P):
\Delta P = \sqrt{\langle(P - \langle P \rangle)^2\rangle} \,
can never be smaller than a fixed fraction of Planck's constant:
\Delta X \Delta P \ge {\hbar \over 2}.
Any measurement of the position with accuracy \scriptstyle \Delta X collapses the quantum state making the standard deviation of the momentum \scriptstyle \Delta P larger than \scriptstyle \hbar/2\Delta x.
Re: The Uncertainty Principle
Posted by:
Gravis
()
Date: May 02, 2009 08:56AM
"the wisdom of the wise will perish, the intelligence of the intelligent will vanish."
Re: The Uncertainty Principle
Posted by:
Billy Bowlegs
()
Date: May 02, 2009 09:02AM
Fascinating!
Re: The Uncertainty Principle
Posted by:
Spunky's a Dumbass
()
Date: May 02, 2009 12:08PM
I developed my own uncertainty principle, its based off years of observing Hispanics in the NOVA area. Depending on what area you live in, there is a large percentage that the Hispanic that's cleaning the toilet or sweeping the floor at the McDonald's is either from El Salvador, Puerto Rico, or Mexico. I still have to work out the formula, but suffice to say, it's going to change the way NOVA discriminates against the Hispanics.
Re: The Uncertainty Principle
Posted by:
Funky Spunky
()
Date: May 02, 2009 12:11PM
I wasn't aware NOVA discriminated against Hispanics. The police seem to leave them alone, and plenty of people recognize that Hispanics keep the economy moving.