# Is there a probability that an electron in an atom change its energy level due to emitting gravitational wave?

Is there a probability that an electron in an atom change its orbital by emitting a quantum of gravitational radiation instead of photon?

Yes. There is a nonzero probability for such a process, however it is extremely small in comparison with the probability of transition with emission of a photon.

To understand how small this probability is let us check two formulas (both are from Landau & Lifshitz' The Classical Theory of Fields).

First the intensity of quadrupole electromagnetic radiation: $$I=\frac{1}{180 c^5} (\dddot{Q}_{\mu\nu})^2,\tag{1}$$ where $Q_{\mu\nu}$ is an electric quadrupole moment.

Analogous equation for an intensity of quadrupole radiation: $$I=\frac{G}{45 c^5} (\dddot{D}_{\mu\nu})^2,\tag{2}$$ with $G$ the gravitational constant an $D_{\mu\nu}$ is an inertial quadrupole moment.

As we see, the formulas are quite similar, and if electric and inertial quadrupole moment change due to the motion of a single type of particle (electron in our case) with a constant charge to mass ratio then $\dddot{D}_{\mu\nu}$ would be proportional to $\dddot{ Q}_{\mu\nu}$.

In quantum mechanics both equations (1) and (2) could be used to write probability of transition. (1) gives us the probability for the emission of photon $P_\gamma$, (2) probability of emission of graviton $P_g$. Therefore the ratio of probabilities would be: $$\frac{P_g}{P_\gamma}=\frac{4 \,G\, m_e^2}{e^2} = 9.6×10^{-43}.$$ This is incredibly small (but finite) number.

In actuality, since the quadrupole EM radiation is not the most dominant type of transitions, the actual ratio would be even smaller.