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Uncertainty in the momentum of an electron and a proton

(a) Estimate the uncertainty in the momentum of an electron whose location is uncertain by a distance of 2 Angstrom. What is the uncertainty in the momentum of a proton under the same conditions?
(b) What can one conclude about the relative velocities and energies of the electron and proton in the last problem? Are wave phenomena apt to be more apparent for light particles than for heavy ones?

Solution Preview

I will be using rounded numerical values, so you will need to redo the final calculations yourself.

Both of these questions are about Heisenberg's uncertainty principle, which occurs in a few forms. For part a, the relevant uncertainty principle is:
1. Dx*Dp >= hbar/2,
where Dx is the uncertainty in the x-coordinate of a particle, Dp is the uncertainty in its momentum in the x-direction, >= means greater than or equal to, and hbar is Planck's constant divided by (2 x pi). In the calculations below, we will use an equals sign between the two sides of the equation.

What the uncertainty in position means is that if a particle is expected to be at x = x0, then its true position will be somewhere between x = x0 - Dx and x = x0 + Dx. Similarly, if a particle is expected to have momentum p = p0, the true momentum will lie between p = p0 - Dp and p = p0 + Dp. So the uncertainty in a variable gives a measure of the range or spread of ...

Solution Summary

The uncertainty in the momentum of an electron and a proton are examined. The expert concludes about the relative velocities and energies of the electron and proton.

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