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.

The photon is emitted with momentum 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 = Δ, we obtain a quadratic equation in E ph, with a solution
E ph = m c 2 [ 1 + 2 Δ m c 2 1 ] Δ Δ 2 2 m c 2 + Δ 3 2 m 2 c 4 ,
where the above Taylor series expansion holds as we work in the non-relativistic limit ( Δ / m c 2 1). The above problem is well mapped to the stone-in-the-well problem, so we may alternately apply the same method to estimate the energy of the photon. In the first approximation, we assume the recoil energy to be negligible and assign the whole excitation energy to the photon ( E ph = Δ). Thus, a first-order estimate of the photon momentum is p = Δ / c. Respecting conservation of momentum, we infer that the atom carries the momentum p and so an improved estimate of its recoil energy is E = Δ 2 / 2 m c 2. Using energy conservation again, we revise our estimate of the photon energy as
E ph = Δ Δ 2 2 m c 2 .
The method may be continued to obtain more accurate approximations by iterating the algorithm
E ph ( n ) = Δ [ E ph ( n 1 ) ] 2 2 m c 2 , n = 1 , 2 , 3 , ,
where E ph ( 0 ) = 0 may be regarded as the state with no emitted photon. These successive iterations agree (as they must) with the Taylor series approximation carried out to the same order, but with the benefit of helping students develop an intuition into the form of the solution, and also teaching an estimation method that will serve them well in other situations.

The author has no conflicts of interest to disclose.

, “
Fleeing from the quadratic formula
Am. J. Phys.
Atoms, Molecules and Photons: An Introduction to Atomic-, Molecular- and Quantum-Physics
Berlin Heidelberg
), Chap. 7, p.
, problem 7.