Heisenbergy uncertainty principle

Question

What is the velocity of electrons that form the same pattern as 450 nm light when passes through a double slit? What is the kinetic energy of each?

Answer 1

Your question doesn't have much to do (directly anyway) with the uncertainty principle.  The title to your question should have been--DeBroglie hypothesis--the idea that particles have a wave whose length is inversely proportional to their momentum.  The constant of proportionality is Planck's constant, h.

p = h / lambda (=mv for a non-relativistic particle)

Solve for your speed:

v = h / (m lambda)

The kinetic energy (non-relativistic) is:

T = p^2 / 2m = h^2 / (2 m lambda^2)

The energy of the photon (which is all kinetic, since a photon has no rest mass) is:

E = hf = hc / lambda

They give you the wavelength you want to match--lambda.  You need to look up Planck's constant (h), the speed of light (c), and the mass of the electron (m).  Then plugnchug.

Answer 2

the interference pattern is determined by the wavelength

so the electron must have the same wavelength as the photons

λ = 450 nm = 450E-9 m

h = 6.63E-34 J.s

m = 9.1E-31 C

c = 3E8 m/s

v = h/(mλ) = 1600m/s

E(electron) = ½mv² = h²/(2mλ²) = 1.2E-24 J = 7.5μeV

E(photon) = hf = hc/λ = 4.4E-19 J = 2.8eV

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