S8_Heisenberg_Shells - My notes

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1 REGARDING HEISENBERG SHELLS Dr Spyridon Koutandos [email protected] [email protected] Abstract: In this article the author examines the possibility of some hidden variables existing behind quantum mechanics formulation, a phrase first used by Albert Einstein. This time we derive new methods of explaining integrations used in quantum mechanics. In further calculations we will assume for ease as is the case usually that the angle part of the wave function is already normalized and divided by 4 pi. For the behavior of polarization regarding the signs the reader might want to refresh his knowledge with the book of reference [9]. The result of the author [1] about the surface polarization charge sigma is that it is possibility times the radius and the polarization charge contained within equal psi surfaces is: r N e 2 ψ σ = (1) enclosed sphere p V N r e r N r e S d r e N r S d P q π 4 ) ( 3 ) ( | ) ( ) ( ) ( 2 3 2 2 2 = = = Ω = = r r r r (2) We should calculate the mean magnetic pressure on the surfaces of constant probability. Coming back to the problem of calculating expression 14, we may notice that , since the gradient of phase changes only along the equipotentials of psi squared and the magnetic pressure is proportional to this gradient times the vector potential A which again is the magnetic induction times the vertical projection of the radius used in expression 14 we have for a radial integration (where psi(r) is of course constant):
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2 ( ) B m r mN e d rB r m mN e d r A mN e r P P B M l l m magnetic frontiers 2 2 2 0 2 2 ) ( sin ) ( 2 sin ) ( ) ( 2 ψ θ π φ h h r h r r = = = = = = (3) We assume as is the case that the anfle parts of the wave function are already normalized to 4 pi, so we have: m magnetic l sphere m E N r E N r B m r mN e P 2 2 2 ) ( ) ( ) ( = = = h (4) At this point the reader should be reminded that after psi comes the last letter of the Greek alphabet, Omega=-PV which is the big thermodynamic potential used by Gibbs[7] along with some other symbols now used in quantum mechanics. At conditions of equilibrium and no particle generations, therefore of zero chemical potential, the free energy F is equal to Omega. The author has included pressure that arises from the validity of the virial theorem. The polarization pressure is to be associated with surface polarization charge and the normal pressure with the potential. A part of the omega should include the magnetic pressure. m p l m enclosed m m E e q V N B m m e V P V P 3 4 2 = = = = Ω h (5) In equation 15d stands the Bohr magneton. We have made some assumptions such as constant magnetic field but in general magnetic fields depend on angles and this rule should apply. Specifically the magnetic field is formed by closed loops and may be expressed as a gradient of the solid angle at which we look at it:
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This note was uploaded on 12/02/2009 for the course NONE None taught by Professor Spyridonkoutandos during the Spring '09 term at 東京国際大学.

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S8_Heisenberg_Shells - My notes

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