Deriving photon energy equation using E = m <msup> c 2 </msup> and de Br

znacimavjo

znacimavjo

Answered question

2022-05-03

Deriving photon energy equation using E = m c 2 and de Broglie wavelength
So I was learning de Broglie wavelength today in my physics class, and I started playing around with it. I wondered if it was possible to calculate the energy of a light wave given its wavelength and speed. After rearranging a bit, I plugged it into E = m c 2 , and realized I had found the equation for the energy of a photon that I learned at the beginning of my quantum mechanics unit, E = h f
λ = h p
p = h λ
E = m c 2
E = p c c 2
E = h c λ = h f
I have a few questions about this. Firstly, I do not understand how it can make sense to do p c in this context, because, as I understand it, light has no mass. How can I come to E = h f using the mass of a massless object?
Secondly, as I was doing those steps, I thought I would be calculating the energy of the entire light ray. I now realize I was finding the energy of a single photon. In hindsight, this makes sense, because the energy of the entire light ray must depend on some length value, correct?
This led me to two other questions: do light rays have some finite length? how do I calculate the energy of a light ray, not just a single photon?

Answer & Explanation

Ann Mathis

Ann Mathis

Beginner2022-05-04Added 11 answers

"Firstly, I do not understand how it can make sense to do p/c in this context, because, as I understand it, light has no mass."
That's correct. Light has no mass so using m = p c is technically incorrect.
"How can I come to E=hf using the mass of a massless object?"
From the relation
E 2 = p 2 c 2 + m 2 c 4
given that the mass of a photon is indeed zero, then
E = p c E = h c λ       since        p = h λ
You also know that c = f λ so
E = h f
"the energy of the entire light ray must depend on some length value, correct?"
Yes, it depends on the wavelength, λ. It's not clear what you mean by "entire". The energy of a light ray is characterized by its frequency, and therefore wavelength. Note that a single photon is a particle, and the "light ray" classical wave character comes from many photons.

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