[math-fun] Alleged exact solution of Dirac eqn in a certain localized magnetic field
Alonso Contreras-Astorga, David J. Fernandez C., Javier Negro: Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators http://arxiv.org/abs/1210.7416 They claim to exactly solve the Dirac equation associated to a charged particle of spin 1/2 immersed in a magnetic field with cylindrical symmetry generated by the vector potential [Ax,Ay,Az] = [0, 0, c*k / (e*r)] where c=lightspeed, e=electron charge, r=distance to symmetry axis, and k is an arbitrary field-strength parameter. This magnetic field wraps round the z-axis and has strength proportional to r^(-2). The energy levels E are (middle of page 8): E = +- sqrt( (m*c/hbar)^2 + d0^2 + n*(2a+n)*b^2 / (a^2 * (a+n)^2) ) where m = electron mass hbar = (planck const)/(2pi) pz = momentum in z-direction d0 = pz * L / (hbar*sqrt(L^2+k^2)) a = sqrt(L^2+k^2) / hbar b = pz * k / hbar^2 n = 0,1,2,3,... L = angular momentum quantum number You may object that this is obviously dimensionally incorrect. True. It's always amazing how physicists publish papers like that. Perhaps they have some magic secret undisclosed units that repair that. You may also object that there is no spin parameter. Also true. Perhaps it is absorbed inside their L parameter somehow. Anyhow, assuming they are not completely full of shit, the important thing to observe is that E when n=0 (and pz and L are nonzero and fixed) is a DECREASING function of the field strength parameter k. Therefore, we here have an alleged exact Dirac solution, where the energy of a state is smaller in higher magnetic field, so that, as I wanted, pair creation should occur if the magnetic field is strong enough and |pz| is large enough. Frankly, though, I have reached the point where I feel one has to assume that every physicist that ever publishes about this topic, is publishing a result which you should assume as your default assumption, is utterly false garbage. That is because that definitely has been the case in over 50% of the papers I've looked at. Under that assumption, life is difficult since you have to do everything yourself from the ground up.
The quoted formula, once the dimensions are corrected, gives E > mc^2. Spontaneous pair creation requires E < -2 mc^2. Taking the minus sign on the square root doesn't count. That's just the Dirac sea of negative energy states that are occupied in the ground state, from which the absence of an electron is a positive energy positron. The vector potential in this calculation arises from a current carrying line. There is something annoying about solving the Schrödinger or Dirac equation in a constant field. The physical situation is invariant under translation, yet quantum theory insists that you use the potential, not the field, and then invariance under translation is lost. That's how you end up with a messy confluent hypergeometric or whatever. -- Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, June 5, 2015 5:03 PM Subject: [math-fun] Alleged exact solution of Dirac eqn in a certain localized magnetic field Alonso Contreras-Astorga, David J. Fernandez C., Javier Negro: Solutions of the Dirac Equation in a Magnetic Field and Intertwining Operators http://arxiv.org/abs/1210.7416 They claim to exactly solve the Dirac equation associated to a charged particle of spin 1/2 immersed in a magnetic field with cylindrical symmetry generated by the vector potential [Ax,Ay,Az] = [0, 0, c*k / (e*r)] where c=lightspeed, e=electron charge, r=distance to symmetry axis, and k is an arbitrary field-strength parameter. This magnetic field wraps round the z-axis and has strength proportional to r^(-2). The energy levels E are (middle of page 8): E = +- sqrt( (m*c/hbar)^2 + d0^2 + n*(2a+n)*b^2 / (a^2 * (a+n)^2) ) where m = electron mass hbar = (planck const)/(2pi) pz = momentum in z-direction d0 = pz * L / (hbar*sqrt(L^2+k^2)) a = sqrt(L^2+k^2) / hbar b = pz * k / hbar^2 n = 0,1,2,3,... L = angular momentum quantum number You may object that this is obviously dimensionally incorrect. True. It's always amazing how physicists publish papers like that. Perhaps they have some magic secret undisclosed units that repair that. You may also object that there is no spin parameter. Also true. Perhaps it is absorbed inside their L parameter somehow. Anyhow, assuming they are not completely full of shit, the important thing to observe is that E when n=0 (and pz and L are nonzero and fixed) is a DECREASING function of the field strength parameter k. Therefore, we here have an alleged exact Dirac solution, where the energy of a state is smaller in higher magnetic field, so that, as I wanted, pair creation should occur if the magnetic field is strong enough and |pz| is large enough. Frankly, though, I have reached the point where I feel one has to assume that every physicist that ever publishes about this topic, is publishing a result which you should assume as your default assumption, is utterly false garbage. That is because that definitely has been the case in over 50% of the papers I've looked at. Under that assumption, life is difficult since you have to do everything yourself from the ground up.
Correcting my mistake: I said "The quoted formula, once the dimensions are corrected, gives E > mc^2. Spontaneous pair creation requires E < -2 mc^2." In this context, spontaneous pair creation only requires E < 0, provided that rest energy is included in E. I was confusing this with a different situation. If a nucleus is sufficiently charged that the binding energy of an electron in its lowest level exceeds 2 mc^2, then it is energetically favorable to create a pair, keep the electron, and eject the positron. This requires Z = 137 for a point nucleus, Z = 170 for a real nucleus. Positron production in collisions of heavy nuclei has been observed. -- Gene
participants (2)
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Eugene Salamin -
Warren D Smith