Solution

A photon has momentum $p_{\gamma} = mc$, and after absorption, this momentum must be conserved. Let the atom move with momentum $p$ after absorbing the photon, then:

$$p = mc$$

The initial energy of the system is the rest energy of the atom plus the energy of the photon:

$$E_{\text{initial}} = Mc^2 + mc^2$$

After absorption, the atom will have a new mass $M_1$ and kinetic energy:

$$E_{\text{final}} = M_1 c^2$$

Since total energy must be conserved:

$$Mc^2 + mc^2 = M_1 c^2$$

Solving for the new mass of the atom:

$$M_1 = M + m$$