Calculate momentum

Question

An electron's wavelength is 3.0×10^−7m. What is its momentum?

Answer 1

Given the wavelength of the electron, λ = 3.0 x 10^-7 m

Find the electron's momentum, p = ?

Use the de Broglie formula λ = h/p , where Planck's constant h = 6.63 x 10^-34 J⋅s

Re-arrange to isolate p: p = h/λ

Substitute and solve: p = (6.63 x 10^-34 J⋅s) / (3.0 x 10^-7 m) = 2.21 x 10^-27 J⋅s/m

Note: a more common combination of units for momentum within the SI system is kg⋅m/s (p=mv tells you that), so you could also correctly answer that p = 2.21 x 10^-27 kg⋅m/s .

Interesting aspect to the problem is that you were not given the speed of the electron, and you did not even need to look up and use the mass of the electron. In fact, a proton, a baseball, or a covid virion with that same wavelength would have the same momentum calculated above.

Answer 2

Given the wavelength of the photon, we can use a couple of expressions to help ensure we have the right calculation of momentum. Energy for a photon is given as Planck's constants times its frequency (υ)

E = hυ

We also know the the speed of light for a photon is constant (c). We also know that frequency and wavelength are "inverses" of one another via the speed squared.

υλ = c²

Rearrangement yields

υ = c²/λ

E = h(c²/λ)

We also know via Einstein's equation --> E = pc if it is a massless particle.

h(c²/λ) = pc

Upon solving for p (momentum) we see that p = h/λ

Now plugging in what we know h = 6.626 x 10^-34 J·m and λ = 3.0 x 10^-7 m

E = 6.626 / 3.0 * 10^-34/10^-7 ≈ 2.21 x 10^-27 J

Conversion to electron Volts (eV) --> 2.21 x 10^-27 J / 1.602 x 10^-19 J/eV --> 1.38 x 10^-8 eV