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Section 20.5 Working with Duality
Subsubsection Activities
Activity 20.5.1 . Double-slit Electrons.
Suppose electrons with speed
\(v = 10^4 \mathrm{~m/s}\) are sent through a double-slit experiment with
\(d = 1.5 \mathrm{~mm}\text{.}\)
(a) Find the wavelength of each electron in nanometers.
(b) Find the angle to the first constructive point?
(c) How would each of your previous answers change if the speed of each electron were
\(v = 10^6 \mathrm{~m/s}\)
Activity 20.5.2 . Momentum of a Photon.
A particle with mass has a wavelength
\begin{equation*}
\lambda = \frac{h}{p}
\end{equation*}
Suppose that the this equation were also valid for a photon, which is massless and has energy given by
\begin{equation*}
E = \frac{hc}{\lambda}
\end{equation*}
(a) Use these equations to write a relationship between the energy and momentum of a photon. Check the fundamental units of this equation to confirm its accuracy.
(b)
The special theory of relativity expresses the following relationship between energy and momentum for all particles:
\begin{equation*}
E^2 = (pc)^2 + (mc^2)^2
\end{equation*}
Show that this equation agrees with the relationship you found above for a massless photon.
(c) Investigate the energy-momentum relation above for a particle with mass
\(m\) that is
at rest (that is, with no momentum). Is the result a familiar one?
Activity 20.5.3 . Wavelength and Energy.
You conduct an experiment using two slits separated by
\(2500 \mathrm{~nm}\) and observe a two-slit interference pattern. You measure the angle to the first maximum to be
\(0.164 \mathrm{~rad}\text{.}\)
(a) If the experiment was conducted with photons, determine the wavelength, frequency, and energy of each individual photon.
(b) If the experiment was conducted with electrons, determine the wavelength, velocity, and energy of each individual electron.
(c) Compare the wavelengths and the energies of the two particles.
