1: Is obvious true?

Is something obvious true? What I mean is that sometimes something we take to be true just because it is obvious turns out to be something not true. As Sir Arthur Conan Doyle says through the voice of Sherlock in ‘The Boscombe Valley Mystery’, “There is nothing more deceptive than an obvious fact”.

In this series of articles, I will present my analysis on a simple case of absorption and emission of light which was written in form of a report and show you how something obvious leads to a paradox. In the starting articles, I attach my report on this paradox. And later on, I will end the series with an explanation. You might want to skip to the last article in the series if you want to skip all the math part below, although it takes an only basic understanding of few theories of Modern Physics to go through the math. I copy-pasted my report, so it is in the form of sections and the equation numbers continue to start here.

On Photon Absorption and Emission

Laser cooling uses the process of absorption of a photon by atoms to cool down atoms. In doing so, we consider the Doppler effects. They arise when there is relative motion between the observer and source. In laser cooling case, the atoms and lab are in relative motion. Atoms are observers and source is fixed in the lab. We reach an interesting result when we try to look at the energy and momentum conservation equations from Atom’s frame and lab’s frame separately. In both frames, we will see that Energy is not conserved assuming momentum is conserved.

This non-conservation would imply that absorption didn’t take place. It implies laser cooling isn’t working in the way we know. But it is not so since the conditions we induce in lab-based on our understanding of the absorption process can produce expected results.

In this record, I try to look at the equations of conservation closely and try to understand what is giving rise to this paradox. This paradox has been studied once by Carlo Maria Scandolo in his paper ‘A paradox about an atom and a photon [1]’. I came across his paper only after having found the problem in laser cooling and studied totally in an independent manner from him.

BASIC UNDERSTANDING OF THE PROBLEM

Let’s look at the basic absorption case first. Let’s say the atom of mass m is moving with a velocity v, and the frequency of the photon used is ω in lab’s frame and in atom’s frame frequency Doppler shifts to ωR which is same as the resonant frequency of the atom.

So that, ωR = ω*c/(c-v). (Non-relativistic limit)……………..(1)

• Atoms’ Frame

Atom’s frame refers to the frame in which the Atom is at rest initially.

Initial total energy of the system EiA = 0 (PE) + 0(KE) + hωR (Ep)………………………….(2)

PE is the potential energy of the atom in the ground state,
KE is the kinetic energy of atom.,
Ep is the energy of the photon.

The final total energy of the system EfA = hωR (PE) + ½ mv­2(KE) + 0(Ep)……………………..(3)

v is the velocity of the atom with respect to the same frame. After absorption, the atom is no longer in rest with the respect that frame, it got impacted with some momentum. The potential energy of the atom increases because of the photon absorbed. From 2 and 3 we can see that clearly, energy is not conserved. Where does this excess energy come from?

The initial total momentum of the system PiA = 0(Pa) — hωR/c i^ (Pp)………………….(4)

Pa is the momentum of the atom,
Pp is the momentum of the photon,
i^ resembles the direction of the momentum.

Final total momentum of the system P­fA = mv i^ (Pa) + 0 (Pp)……………..(5)

Since the momentum is imparted to the atom we see the appearance of v in equation 5. We can equate PiA and P­fA and bring out the value of v. substituting this in 2 and 3 still shows the non-conservation of the energy.

• Lab Frame

The initial total momentum of the system
PiL= mu i^ (Pa) — hω/c i^ (Pp)………………(6)

The final total momentum of the system
PfL = [mu — hω/c] i^ (Pa) + 0 (Pp)…………..(7)

The atoms momentum is reduced by an amount equal to the momentum of the photon we hit it with. We can see momentum is conserved in this frame.

Initial total energy of the system EiL = 0(PE) + ½ mu2(KE) + hω(Ep) = ½ mu2 + hω………….(8)

Final total energy of the system EfL = hωR (PE) + ½ [mu-hω/c]2/m (KE) + 0(Ep)
= ½ mu2 + hωc/[c-u] + ½ h2ω2/mc2 — hωu/c………..(9)

The potential energy increases by the same amount in both frames since this value is equal to the transition energy value of the atom which is same in every frame. The change in kinetic energy depends on the amount of momentum change. Clearly from 8 and 9, both energies are not the same. Thus energy is not conserved in this frame too.

In this section, we have seen that energy is not conserved in both frames although momentum is conserved. We can see that the final energy is more than the initial energy. This is a rough approach to the understanding of the absorption process, as I used the relativistic limit when started. But we will see that the problem still arises after considering relativistic cases. It shows that there is something wrong in our understanding of the absorption.

ASSUMPTIONS IN OUR APPROACH

In the above section, we see some excess energies popping up. It means either there is a mistake in the equations or our assumptions are wrong. The relativistic limit isn’t the only assumption made in the above section.

Some of the main assumptions are:

• Mass of atom is taken to be same in every frame
• The energy gap from the ground state to excited is taken to be same in every frame
• Relativistic effects on relative velocities are not taken into account
• Absorbed photon energy is only used in jumping from the lower quantum level to higher i.e. the photon energy is either same or higher than the energy gap

These are some first-hand assumptions we can note. The last assumption is similar to the second assumption mentioned by Carlo in his paper­ ­[1]­. His assumption is that absorption takes place only when the energy of the photon is either higher or equal to the energy gap.

Let’s consider the second assumption here, energy gap from the ground state to the excited state. From Bohr’s quantum model for atoms, we know that the energy gap is dependent on the effectively reduced mass of the nucleus-electron system and the quantum levels. Since in our discussion the quantum levels are the same in every frame, we must only take care of the reduced mass in our further analysis. We know that when looking from different frames mass of a body is not constant but change according to the relativistic motion. Thus the energy gap also changes by the same ratio as the Mass value.

Let’s say we have a body of mass ‘m’ in a frame at rest. Consider a frame in which the mass is moving with velocity v. In that second frame, we know that the mass will change to ‘γm’, where y is Lorentz factor and dependant on v. Since we know that energy gap is proportional to mass, the energy E in the first frame here is seen as γE in the second frame.

In my further analysis, I try to consider relativistic masses, velocities and also the altered quantum gap energies and all the relativistic limit are not taken into consideration.

I end this first article stating the assumptions in my approach. I will finish the report in the second article and try to explain what the report in the third article. Keep reading!

If anyone wants to read the whole report in a single PDF, you can find it here.