2: How deep are our assumptions?
Science as we know now started its progress almost as long as civilizations began. Some of our theories started on the assumptions made long long ago, in Newton’s words it goes as “Standing on the shoulders of Giants” and we kind of take them for granted. It is only when we meet with some paradox we question the assumptions we started on.
Similarly in my previous, I met some paradox in obvious physics. And I started to note down all the assumptions that are clear to me. In this article, I try to rectify the errors in assumptions and re-evaluate the paradox. But what if we meet if the paradox again? What will that mean? I show in my report what it means which is presented below.
CASE OF ABSORPTION
Let’s consider the regular absorption process in this section from both Atom’s frame and Laboratory frame.
Atom’s frame
I am considering an atom of mass M’ in this frame and a photon of frequency ‘f’ w.r.t the present frame is moving towards it. Let’s say the energy gap between the required quantum levels in this frame is E and also that
E = hf…………….(10)
h is Planck’s constant.
By the above equation, I mean that this photon is perfectly fit to excite the atom in this frame.
• The initial energy of the atom and photon system in this frame is given by
Ei = 0 + 0 + hf = hf = E…………………….(11)
First 0 corresponds to the kinetic energy of the atom. It is zero since atom is at rest in this frame. Second 0 corresponds to the potential energy of the atom, which I am considering to be at zero in this frame. ‘hf’ is the energy of the photon same as E from (10).
• Final energy of the atom and the photon system in this frame is given by
Ef = ½ γ1 M Vf2 + (γ1E + γ1EL — EL) + 0………………….(12)
Vf is the velocity of the atom after absorption. Since atom was imparted with some momentum, it is no longer at rest w.r.t the atom’s frame. One must not confuse atom’s frame as the frame which changes with the atom. ‘γ1’ is the Lorentz factor acting on atom from atom’s frame since the atom is in motion now.
½ γ1.M.Vf2 is the kinetic energy of the atom. (γ1E + [γ1EL — EL]) is the increased potential energy of the atom, where the first part is due to transition and second part [γ1EL — EL] shows that from the second frame due to relativity the lower energy level of the atom is also raised. As we have concluded in the previous section the energy gap multiples when we look from a frame of relative motion by corresponding Lorentz factor. And 0 corresponds to the absorbed photon.
Before we analyse whether the energy is conserved or not, let’s go through momentum cases.
• Initial momentum of the atom and the photon system in this frame is given by
Pi = 0 — hf/c …………………(13)
0 corresponds to the momentum of the atom, which is at rest initially. hf/c is the momentum of the photon. It is negative because it is moving towards negative x-direction. Since all the motions here we consider are in the x-direction, I won’t use the directional unit vectors.
• Final momentum of the atom and the photon system in this frame is given by
Pf = — γ1 M Vf …………………(14)
Since the momentum imparted is in negative x-direction, it is negative.
From 13 and 14 we get momentum conservation which gives
— γ1 M Vf = — hf/c……………..(15)
For the energy to be conserved Ei must be equal to Ef. But from equation 11, 12 and 15 we see that’s not the case. Which means energy is not conserved in the atom’s frame.
We get from 11, 12 and 15, that
Ef = E ( ½ Vf/c + γ1) + γ1EL = Ei ( ½ Vf/c + γ1) + γ1EL ……(16)
Ef is clearly greater than Ei and this implies that initial energy is not enough to finish the absorption process.
Limiting case of this when Vf << c also shows Ef >Ei.
Lab’s Frame
Let’s say atom was moving with some velocity Vi in the Lab frame. And let’s continue all the above assumptions. And the frequency of the photon in the lab’s frame will be ‘өf’ where ө is given by sqrt[(1-β)/(1+β)], where β is given by Vi/c. This is because of the relativistic Doppler Effect.
• The initial energy of the atom and photon system in this frame is given by
Ei = ½ γ2 M Vi2 + 0 + өhf…………………….(17)
γ2 is the new Lorentz factor which is a function of Vi since the atom is moving with velocity Vi in this frame. And the energy of the photon is dependent on observed frequency. Potential energy is not 0 if we are at the same ground level as in the previous case. But we can move the reference level by a bit so that the problem is fixed.
• Final energy of the atom and photon system in this frame is given by
Ef = ½ γ3 M Vf22 + (γ3 E + γ3EL –γ2EL) + 0…………………….(18)
γ3 because after the momentum impartation from the photon the atom now has a changed velocity which is Vf2. And similarly, the increase in potential energy in this frame is again dependent on the new relative velocity and the changed value in the lower energy level.
Let’s look at the momentum equations now.
• The initial momentum of the atom and the photon system in this frame is given by
Pi = γ2 M Vi — өhf/c …………………(19)
They are opposite in sign because the atom and photon are moving in the opposite direction in the laser cooling case.
• final momentum of the atom and the photon system in this frame is given by
Pf = γ3 M Vf2 …………………(20)
Again considering the conservation of momentum we get,
γ3 M Vf2 = γ2 M Vi — өhf/c…………….(21)
And from 19, 20 and 21 we end with again non-conservation of energy. In the limiting case, we get Ef < Ei. Which would mean that initial energy is more than the final energy. That means absorption might be taking place but some energy is lost somehow.
CASE OF EMISSION
Here we will have an atom in the excited state and is releasing energy in the form of Photon.
Atom’s frame
• The initial energy of the atom and photon system in this frame is given by
Ei = 0 + 0 + 0 …………………….(22)
Kinetic energy is zero because the atom is at rest. There is no photon released, so photons energy is also 0. And I am considering the potential energy value at 0 in the excited state.
• Final energy of the atom and photon system in this frame is given by
Ef = ½ γ1 M Vf2 — γ1 E + γ1EL — EL + E …………………….(23)
The decrease in potential energy is not E since the atom is in relative motion to the frame. But the emitted photon is still of energy E since at the instant of emission there is no motion and there is the additional raising of energy level (+γ1EL -EL) due to relative motion.
• the initial momentum of the atom and the photon system in this frame is given by
Pi = 0 + 0 …………………(24)
Everything is at rest initially before emission.
• final momentum of the atom and the photon system in this frame is given by
Pf = -γ1.M.Vf + E/c …………………(25)
The momentum of the atom is opposite to the momentum of photon since this is a recoil action.
Assuming the conservation of momentum, we get from 24 and 25
γ1.M.Vf = E/c …………………..(26)
From 26, 22 and 23 we can see that Ef is not same as Ei. And also that Ef>Ei, which implies that there is some excess energy after emission which is not present initially. This means that initial energy is not enough to reach the emission state. Emission should not take place according to the result.
Lab Frame
• The initial energy of the atom and photon system in this frame is given by
Ei = ½ γ2.M.Vi2 + 0 + 0 …………………….(27)
Since the atom is moving with velocity Vi and everything else is at zero level.
• Final energy of the atom and photon system in this frame is given by
Ef = ½ γ3.M.Vf32 + (- γ3E+ γ3EL –γ2EL)+ γ2E…………………….(28)
The energy of a photon depends on γ2 because at the instant of emission there is no relative motion.
• the initial momentum of the atom and the photon system in this frame is given by
Pi = γ2.M.Vi …………………(29)
• final momentum of the atom and the photon system in this frame is given by
Pf = γ3.M.Vf3 +/- γ2 E/c …………………(30)
I wrote +/- in front of the momentum of a photon because the relative direction of motion of the photon and atom affects the results differently.
From 29 and 30, if we assume conservation of momentum, we will arrive at
γ2.M.Vi = γ3.M.Vf3 +/- γ2 E/c………………(31)
From 31, 27 and 28 we can see that again energy is not conserved. Emission case is easy to analyse than Absorption case in the lab frame part since the Doppler Effect need not be considered. In the limiting case, we see that energy tends to conserve but not quite.
ANALYSIS AND CONCLUSION
In all the cases, both Absorption and Emission, in both lab and atom’s frame, we see that energy is clearly not conserved. This means that in absorption case, there is no enough energy for the absorption to take place. Which clearly contradicts with experiment as we know.
All these results would imply that there is something in our assumptions. All the four assumptions mentioned in section ‘ASSUMPTIONS IN OUR APPROACH’ are re-evaluated and corrected. It would imply that either our re-evaluated assumptions had some mistake or that there is some-other assumption we took for granted and didn’t correct all along.
Important assumptions or theories on which we built this analysis are:
a. The special theory of relativity
b. Bohr’s atomic model
We know from rigorous experimental verifications that the theory of relativity is right. That leaves us with the second option, Bohr’s model. This model is an approximated one, and not the accurate one. So Bohr’s model is the one which gave rise to this paradox. This paradox teaches us how theories of even little approximations can lead us to big loops in the understanding of the basic phenomena. Carlo came to the same conclusion in his paper.
In the next article, I present a brief account of all the report in this article and the previous one and conclude what my point.
If anyone wants to read the whole (complicated) report in a single PDF, you find it here.
