Example from Physics, 21.3 The Dual Nature of Light

Electron and Photon Momentum Compared

(a) Calculate the momentum of a visible photon that has a wavelength of 500 nm. (b) Find the velocity of an electron having the same momentum. (c) What is the energy of the electron, and how does it compare with the energy of the photon?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Finding the photon momentum is a straightforward application of its definition: $p = h / λ .$ If we find the photon momentum is small, we can assume that an electron with the same momentum will be nonrelativistic, making it easy to find its velocity and kinetic energy from the classical formulas.
Solution
Solution for (a)

Photon momentum is given by the de Broglie relation.

$p = \frac{h}{λ}$
21.12

Entering the given photon wavelength yields

$p = \frac{6.63 \times 10^{−34} J \cdot s}{5.00 \times 10^{−7} m} = 1.33 \times 10^{−27} k g \cdot m / s .$
21.13
Solution
Solution for (b)

Since this momentum is indeed small, we will use the classical expression $p = m v$ to find the velocity of an electron with this momentum. Solving for v and using the known value for the mass of an electron gives

$v = \frac{p}{m} = \frac{1.33 \times 10^{−27} k g \cdot m / s}{9.11 \times 10^{−31} k g} = 1,459.9 m / s \approx 1,460 m / s .$
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Solution
Solution for (c)

The electron has kinetic energy, which is classically given by

$K E_{e} = \frac{1}{2} m v^{2} .$
21.15

Thus,

$K E_{e} = \frac{1}{2} (9.11 \times 10^{−3}^{1} {kg)(1,456m/s)}^{2} = 9.64 \times 10^{−25} J .$
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Converting this to eV by multiplying by $\frac{(1 eV)}{(1.602 \times 10^{- 19} J)}$ yields

$K E_{e} = 6.02 \times 10^{−6} eV .$
21.17

The photon energy E is

$E = \frac{h c}{λ} = \frac{(6.63 \times 10^{−34} J \cdot s) (3.00 \times 10^{8} m/s)}{5.00 \times 10^{−7} m} = 3.98 \times 10^{−19} J = 2.48 eV,$
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which is about five orders of magnitude greater.
Discussion
Tips For Success

When determining energies in particle physics, it is more sensible to use the unit eV instead of Joules. Using eV will help you to recognize differences in magnitude more easily and will make calculations simpler. Also, eV is used by scientists to describe the binding energy of particles and their rest mass, so using eV will eliminate the need to convert energy quantities. Finally, eV is a convenient unit when linking electromagnetic forces to particle physics, as one eV is the amount energy given to an electron when placed in a field of 1-V potential difference.

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