# 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?

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#### 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.

Questions Using Planck's Constant

Part 1

An electron microscope operates with a beam of electrons, each of which has an energy of 20 KeV. Use the uncertainty principle in the form delta(x)delta(p) (greater or equal to) h/2 to find the smallest size that such a device could resolve. Planck's constant is 1.0552 × 10^-34 J · s. Answer in units of pm.

Part 2

What must the energy of each neutron in a beam of neutrons be in order to resolve the same size object?

Answer in units of eV.

Part 3

A beam of neutrons with a kinetic energy of 0.0002 eV falls on a slit of width 0.0001 m.

The Planck's constant is 6.63 × 10 ^-34 J · S.

What will be the angular spread of the beam after it passes through the slit?

Answer in units of radian.

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