Sanjoy Mahajan's Back of the Envelope column included the 2019 paper “Fleeing the Quadratic Formula,”^{1} which argued that students gain more insight by approximating solutions than by direct application of an exact formula. He illustrated the use of this technique to estimate the depth of a well, given the time required to hear the splash after a stone is dropped into it. Since the result is dominated by the time for the stone to fall, an approximation is clearly useful.

For instructors who would like to teach this approximation method in modern physics courses, the following problem is suggested.^{2}

An atom with rest mass *m* is in the excited electronic state of an energy gap equal to Δ units. The atom, initially at rest, emits a photon and comes to the ground electronic state. Estimate the energy of the photon. Assume that the motion of the atom can be treated non-relativistically.

*p*and energy $ E ph = p c$. Since the initial momentum of the atom was zero, the atom must recoil with momentum

*p*and kinetic energy $ E = p 2 / 2 m = E ph 2 / 2 m c 2$. From conservation of energy, $ E + E ph = \Delta $, we obtain a quadratic equation in $ E ph$, with a solution

*p*and so an improved estimate of its recoil energy is $ E = \Delta 2 / 2 m c 2$. Using energy conservation again, we revise our estimate of the photon energy as

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts of interest to disclose.

## REFERENCES

*Atoms, Molecules and Photons: An Introduction to Atomic-, Molecular- and Quantum-Physics*