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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 $E = mc^2$, and realized I had found the equation for the energy of a photon that I learned at the beginning of my quantum mechanics unit, $E = hf$.

$\lambda = \frac{h}{p}$

$p = \frac{h}{\lambda}$

$E = mc^2$

$E = \frac{p}{c}\cdot c^2$

$E = \frac{hc}{\lambda} = hf$

I have a few questions about this. Firstly, I do not understand how it can make sense to do $\frac{p}{c}$ in this context, because, as I understand it, light has no mass. How can I come to $E = hf$ 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?

Níck Aguiar Alves
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dwib
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1 Answers1

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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 $m=\frac{p}{c}$ is technically incorrect.

How can I come to E=hf using the mass of a massless object?

From the relation $$E^2=p^2c^2+m^2c^4$$ given that the mass of a photon is indeed zero, then $$E=pc\rightarrow E=\frac{hc}{\lambda}\ \ \ \text{since }\ \ \ p=\frac{h}{\lambda}$$ You also know that $c=f\lambda$ so $$E=hf$$

the energy of the entire light ray must depend on some length value, correct?

Yes, it depends on the wavelength, $\lambda$. 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.

joseph h
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