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Probability of Quantum Tunnelling in Nucleation (natural units)
My thesis talks about the theoretical possibility of reaching a new stable state of matter composed by strange matter, starting from hadronic matter. This phase transition would happen through the nucleation process.
The system i'm studying has a potential U(r)=-a*r^3+b*r^2+c*r^5 where a,b,c are parameters which i'm changing during my calculations; this potential is expressed in MeV.
This potential is basically a double well with a barrier in the middle and through Wolfram Mathematica i have to calculate the probability of quantum tunnelling through this barrier.
I'm gonna explain briefly the formulas i'm using for this type of calculation:
I have to calculate the eigenvalue of the ground state using the Bohr quantization condition I(E0)=3/2*pi*hbar where I(E)=integral from 0 to r- Sqrt(2*M(r)(E-U))dr where r- is the inferior classical turning point (i'm using the WKB approximation), so i'm giving a value to E0, calculating the integral then finding r- and i do this until i find the correct values for r- and E0 that satisfy Bohr condition.
Once i've done this, i have to calculate another action integral A(E)=integral from r- to r+ Sqrt(2*M(r)(U-E))dr where r+ is the superior classical turning point, i have to calculate then A(E0) (with Mathematica i just put E0 in the int.) and the final formula for the tunnelling probability is p0=exp(-a/hbar).
Getting to the point, the probability that i find is incredibly small. The problem i'm facing is about natural units, i'm forgetting some hbar or c somewhere and for almost 2 weeks i can't find the answer.
M(r) is the mass of the droplet and it's in MeV. So in the integral I(E), dimensionalment speaking, i have sqrt(MeV*MeV)*fm where fm comes from the dr. Thus the integral is MeV*fm. By the way, an action is an energy times a time, so i would have MeV*fm/c. At this point i thought that the factor 1/c comes from the mass of the droplet since the mass is MeV/c^2 and, under square root, it comes out 1/c.
But if i use the mass M(r) in MeV and not MeV/c^2, for the Bohr condition, since my integral is in MeV*fm, i have to multiply the second member (3/2*pi*hbar) times c, so i'm getting 3/2*pi*hbar*c=3/2*pi*200 Mev*fm
Is this right?
Last but not least problem, since i think it's the bigger one, i have to calculate A(E0). Same argument, A(E0) is in MeV*fm, so when i put it in p0=exp(-A/hbar) i have to consider the factor 1/c, so if i use A in Mev*fm, which i do, i have 1/hbar*1/c which is 1/200 Mev*fm and in the end p0=exp(-A[MeV*fm]/200 MeV*fm) and everything seems beautiful and fine.
Sadly this is not Candy Land, and numerically the probability is incredibly small (order of 10^-30 average), but i cannot understand where i'm wrong with units of measurements. This is so frustrating, i have no more ideas left, i tried everything. Does anybody notice an error in my reasoning? You will make my day.
Thank you for your attention, and have a good day.