This applet models the Heisenberg Uncertainty Principle for the "free particle".
This applet demonstrates the impossibility of determining the position of a particle on the x-axis and its momentum in the x-direction simultaneously to infinite precision (regardless of experimental errors). A "free particle" (a particle under no potential) moving in the positive x-direction has a unidimensional wavefunction Ψ=exp(ikx), where k is some constant, and i is the imaginary number (known as "j" to engineers). Its momentum is hk/2π. Click on w1+ to increase wavelength #1, and w1- to decrease it. Likewise, to increase or decrease wavelengths #2 and #3.
Wave number 1 is colored red, wave 2 pink, and wave 3 yellow. The violet wave at the bottom of the graph is the probability density (Ψ2) of the sum (the superposition) of waves 1, 2, and 3, all weighted equally.
When wave 1 = wave2 = wave 3, the momentum is exactly the momentum associated with wave 1. However, the probability density of the particle (the purple curve) is greater than zero periodically from -∞ to +∞, meaning that the particle can be anywhere - its position is completely indeterminate.
If you start changing the wavelengths, you will see that the purple curve starts to change shape, and then it is possible to make larger regions of the curve equal to zero, meaning that the particle is now more "confined". However, the price of this success is that now the momentum varies. It is equally probable that the momentum will be measured as the momentum of wave 1 as of wave 2 as of wave 3, which are different when their wavelengths are different.
With an infinte number of waves, you could localize the particle in exactly one location, but then there would be an infinite number of possible momenta. Therefore, it is impossible to simultaneously determine the momentum and the position in the same dimension.
© 2004-2011 by Lawrence T. Sein. All rights reserved.
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