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I have been trying to understand absorption of a photon by a massive object, specifically the transfer of momentum and kinetic energy between the two.

Starting from energy-momentum relation $E^2 = p^2c^2 + m_0^2c^4,$ or rather, for simplicity, setting $c = 1$:
$$E^2 = p^2 + m_0^2,$$

I can't understand how both the energy and momentum can be conserved in case of the full absorption of the photon. If we add the energies of a photon and massive object, then the resulting momentum is clearly not sum of the two momenta. For example, let's set the photon's momentum to be $p_1 = 1$ then photon's energy $E_1 = 1$, and let's set the massive object's momentum $p_2 = 0$ and object's energy to be $E_2 = m_0 = 1$.

By conservation of energy I would expect the final energy to continue to be $E = E_1 + E_2 = 2$. However, if after the absorption the object has momentum $p_2=0+p_1= 1$, then it has to have energy of $E_2 = \sqrt{p^2 + m_0^2} = \sqrt{2}$, which is not $2$ as I would have expected.

What am I missing?

Philip
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Salamander86
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  • Just to be clear, you have a free particle (say, an electron) and a photon, and you're trying to examine the situation where the photon is completely absorbed by the electron? – Philip Sep 10 '20 at 11:40
  • @Philip yes it can be for example electron – Salamander86 Sep 10 '20 at 11:42
  • Is there a reason why you expect the rest mass to be the same? – NDewolf Sep 10 '20 at 12:11
  • @NDewolf Because the rest mass of a fundamental particle cannot change – Jim Sep 10 '20 at 12:17
  • @Jim I wasnt assuming it to be a fundamental particle. – NDewolf Sep 10 '20 at 13:28
  • @NDewolf the reason is simplicity, if these laws have to hold in the simple case then there is no reason to introduce complexity. However it does seem to be that absorbtion is impossible in this simple case. – Salamander86 Sep 10 '20 at 13:33
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    Exactly, but for composite systems this is the parameter that will have to change. Otherwise you obtain the same contradiction. – NDewolf Sep 10 '20 at 13:41

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Congratulations! You have found a very interesting result: a free particle (like an electron, for example) cannot completely absorb a photon, since energy and momentum cannot be simultaneously conserved in such an interaction.

You can see this (as you have) by explicitly calculating the four-momentum before and after the interaction, as you have. Another way is to consider Compton Scattering, which is the general case of the interaction of an electron with a photon. It can be shown that the change in wavelength of the photon will be:

$$\Delta \lambda = \frac{h}{m_e c}\left( 1 - \cos{\theta}\right) = \lambda_c \left(1-\cos\theta\right),$$

where $\theta$ is the angle between the outgoing and incoming photon, and $\lambda_c$ is called the Compton wavelength of the electron. (It's a nice exercise, you should do it.) Clearly, $\Delta \lambda$ is a bounded quantity, with a maximum when $\theta = \pi$, or when $\Delta \lambda = 2\lambda_c$. In other words, you can never make the photon completely "disappear". (Interestingly, since the Compton wavelength varies inversely as the mass of the particle, the bound becomes tighter for more massive particles.)

It turns out, however, that if the electron isn't free (but is bound, for example, to at atom) then such an absorption can indeed occur. There's a very nice description of it in this answer here. Of course your may ask to same question for such a "composite" particle: from the outside it might "look" as though the atom has "absorbed" the photon.

The reason for this is that atoms (and other such composite particles) have internal structure: they have internal states with different energies, and absorbing a photon changes the atom's internal state. The reason that you can't model an atom as you have in your analysis above is that you haven't taken into account these "internal" degrees of freedom.

Philip
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  • Oh ok then but i dont see how the full absorbtion is possible for any mass except one particular mass where these values just happen to be the same by accident. – Salamander86 Sep 10 '20 at 11:55
  • @Salamander86 I'm not sure I understand what you mean, in my answer I've explained why full absorption cannot occur for any mass. Which values are you talking about? – Philip Sep 10 '20 at 12:02
  • You mention that it cannot occur for any free particle, and then lastly you mention that the full absorbtion can occur in more heavy object - like an atom. I do not understand how it is possible for any massive object, as the energy increase is always smaller than momentum increase for any object with m>0. – Salamander86 Sep 10 '20 at 12:19
  • @Salamander86 ah! No, that's not what I meant, I meant that electrons bound to atoms can absorb photons, not free electrons. Of course your question is valid: from outside it might "look" as though the atom has "absorbed" the photon. The reason for this is not the atom's mass, but the fact that atoms have internal structure: they have internal states with different energies, and absorbing a photon changes the atom's internal state. The reason that you can't model an atom in your analysis is that you haven't taken into account these "internal" degrees of freedom. – Philip Sep 10 '20 at 12:49
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    Ah ok, so you are saying that the atom will get the full momentum, and $\sqrt(2)$ of energy, and the missing $2-\sqrt(2)$ energy will be somehow hidden inside the atom ? – Salamander86 Sep 10 '20 at 13:02
  • I would cautiously say yes but, to be honest, I haven't actually done the calculation myself to be certain of the numbers. As @NDewolf points out in the comment on the question, you can't hold the same assumptions that you do for "fundamental" particles (i.e. particles with no internal structure) when you're working with composite particles like atoms. – Philip Sep 10 '20 at 13:44
  • Correct! Good job on exploring ideas yourself and deriving consequences from it! – Prahar Sep 10 '20 at 13:45