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6-4 Spin Orbit Interaction

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6-4 Spin Orbit Interaction Spin-orbit interaction Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Due Date: March 14, 2015) In quantum physics, the spin–orbit interaction is an interaction of a particle's spin with its motion. The first and best known example of this is that spin–orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus. ((Llewellyn Hilleth Thomas)) http://www.aip.org/history/acap/biographies/bio.jsp?thomasl Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician. He is best known for his contributions to atomic physics, in particular: Thomas precession, a correction to the spin-orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom. The Thomas–Fermi model, a statistical model of the atom subsequently developed by Dirac and Weizsäcker, which later formed the basis of density functional theory. Thomas collapse - effect in few-body physics, which corresponds to infinite value of the three body binding energy for zero-range potentials. http://en.wikipedia.org/wiki/Llewellyn_Thomas 1. Biot-Savart law The electron has an orbital motion around the nucleus. This also implies that the nucleus has an orbital motion around the electron. The motion of nucleus produces an orbital current. From the Biot-Savart’s law, it generates a magnetic field on the electron. The current I due to the movement of nucleus (charge Ze, e>0) is given by Idl ZevN , dl where v is the velocity of the nucleus and v . Note that N dt N q dl Idl dl q Zev . t dt N The effective magnetic field at the electron at the origin is I dl r B 1 , v ve eff c 3 N r1 where v is the velocity of the electron. Then we have Ze v r Ze ve r B N 1 1 . eff c 3 c 3 r1 r1 Since r1 r , Beff can be rewritten as Ze ve r Zev r e Zev e Zem v B z e e eff c 3 c 3 c 2 2 z r r r mec r where me is the mass of electron. The Coulomb potential energy is given by Ze 2 dV (r) Ze 2 V (r) , c . c r dr r2 Thus we have 2 2 Ze rv 1 Ze 1 dVc (r) Beff 3 ez 2 mervez Lzez , cer mecer r mecer dr or 1 1 dVc (r) Beff Lzez , mece r dr where Lz is the z-component of the orbital angular momentum, Lz mevr . 2. Derivation of the expression for the spin-orbit interaction Fig. Electron in the proton frame, and proton in the electron frame. The direction of magnetic field B produced by the proton is the same as that of the orbital angular momentum L of the electron (the z axis in this figure). We consider the circular motion of the nucleus (e) around an electron at the center. The nucleus rotates around the electron at the uniform velocity v. Note that the velocity of the proton in the electron frame is the same as the electron in the proton frame. The magnetic field (in cgs) at the center of the circle along with the current I flow. 2I B . cr according to the Biot-Savart law. The current I is given by e e ev I , T 2r 2r v where T is the period. Then the magnetic field B at the site of the electron (the proton frame) can be expressed as emevN r eL B 3 ez 3 , mecr mecr where L is the orbital angular momentum of the electron (the electron frame). Note that the direction of B is the same as that of L. The spin magnetic moment is given by 2B μs S , where S is the spin angular momentum and B is the Bohr magneton, e B . 2mec Then the Zeeman energy of the electron is given by 1 1 2 eL e2 H μ B ( B S) L S , s 3 2 2 3 2 2 mecr 2me c r where the factor (1/2) is the Thomas correction. In quantum mechanics we use the notation 2 ˆ e 1 ˆ ˆ H so 2 2 3 L S . 2me c r av 3. Thomas correction The spin magnetic moment is defined by 2B μs S , where S is the spin angular momentum. Then the Zeeman energy is given by 1 1 2 1 1 dV (r) B c H so μs Beff S L 2 2 mece r dr 1 1 dVc (r) 2 2 S L 2me c r dr (S L) where the factor 1/2 is the Thomas correction and the Bohr magneton B is given by e B . 2mec Thomas factor 1/2, which represents an additional relativistic effect due to the acceleration of the electron. The electron spin, magnetic moment, and spin-orbit interaction can be derived directly from the Dirac relativistic electron theory. The Thomas factor is built in the expression. H so S L , with 2 1 1 dV (r) 1 e 1 c . 2 2 3 2me c r dr 2 mec r av When we use the formula 1 r 3 , 3 3 n aB l(l 1/ 2)(l 1) the spin-orbit interaction constant is described by e2 m e8 e 2 2 3 3 2 3 6 2me c n aB l(l 1/ 2)(l 1) 2c n l(l 1/ 2)(l 1) where 2 aB 2 = 0.53 Å (Bohr radius). mee The energy level (negative) is given by 2 e 0 En 2 2 , 2n aB n where 4 2 mee e 0 2 . 2 2aB 2 The ratio / En is 2 4 2 e 2 2 2 2 En c n l(l 1/ 2)(l 1) n l(l 1/ 2)(l 1) with e2 1 c 137.037 ((Note)) For l = 0 the spin-orbit interaction vanishes and therefore = 0 in this case. ((Summary)) The spin-orbit interaction serves to remove the l degeneracy of the eigenenergies of hydrogen atom. If the spin-orbit interaction is neglected, energies are dependent only on n (principal quantum number). In the presence of spin-orbit interaction (n, l, s = 1/2; j, m) are good quantum numbers. Energies are dependent only on (n, l, j). 4. Commutation relations We introduce a new Hamiltonian given by ˆ ˆ ˆ H H0 H so , The total angular momentum J is the addition of the orbital angular momentum and the spin angular momentum, Jˆ Lˆ Sˆ , The spin-orbit interaction is defined by 1 Hˆ Lˆ Sˆ (Jˆ 2 Lˆ2 Sˆ 2 ) . so 2 where ˆ ˆ ˆ ˆ ˆ ˆ L L iL , S S iS , and ˆ ˆ ˆ ˆ ˆ ˆ [Lx ,S x ] 0 , [Lx ,S y ] 0 , [Lx ,Sz ] 0 , ˆ ˆ ˆ ˆ ˆ ˆ [Ly ,S x ] 0 , [Ly ,S y ] 0, [Ly ,Sz ] 0, ˆ ˆ ˆ ˆ ˆ ˆ [Lz ,S x ] 0 , [Lz ,S y ] 0, [Lz ,Sz ] 0 . ˆ (a) The unperturbed Hamiltonian H 0 ˆ ˆ ˆ The unperturbed Hamiltonian H0 commutes with all the components of L and S . ˆ ˆ2 ˆ ˆ 2 ˆ ˆ [H0 , L ] [H0 , Lz ] [H0 , Lz ] 0 , ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ [H0 ,S ] [H0 ,Sz ] [H0 ,Sz ] 0 , and ˆ ˆ ˆ ˆ ˆ [H0, J z ] [H0, Lz Sz ] 0 . We note that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [H 0 , L S] [H 0 ,(Lx S x Ly S y Lz Sz )] 0 . Then we have ˆ ˆ 2 ˆ2 ˆ 2 [H0 ,J L S ] 0 , or ˆ ˆ 2 [H 0 ,J ] 0 . We also note that [Jˆ 2 , Lˆ2 ] [Lˆ2 Sˆ 2 2Lˆ Sˆ, Lˆ2 ] 2[Lˆ Sˆ, Lˆ2 ] 0 , [Jˆ 2 , Sˆ 2 ] [Lˆ2 Sˆ 2 2Lˆ Sˆ,Sˆ 2 ] 2[Lˆ Sˆ,Sˆ 2 ] 0 , ˆ 2 ˆ ˆ2 ˆ 2 ˆ ˆ ˆ ˆ [J , J z ] [L S 2L S,Lz Sz ] ˆ ˆ ˆ ˆ 2[L S,Lz Sz ] 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2[Lz ,Lx ]S x 2[Ly ,Lz ]S y 2Lx [Sz ,S x ] 2Ly [S y ,Sz ] ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2iLy S x 2iLx S y 2iLx S y 2iLy S x 0 Thus we conclude that 0 is the simultaneous eigenket of the mutually commuting ˆ ˆ2 ˆ 2 ˆ 2 ˆ observables { H0 , L , S , J , and J z }. 0 j,m;l, s , ˆ (0) H0 0 En 0 , ˆ2 2 L 0 l(l 1) 0 , ˆ 2 2 S 0 s(s 1) 0 , ˆ 2 2 J 0 j( j 1) 0 , ˆ J z 0 m 0 . ˆ (b) The perturbed Hamiltonian H so ˆ ˆ ˆ ˆ Here we note that L S does not commute with Lz or Sz . ˆ ˆ ˆ [L S,Lz ] 0 ˆ ˆ ˆ ˆ ˆ ˆ [Lz ,Lx ]Sx [Ly ,Lz ]S y ˆ ˆ ˆ ˆ iLy S x iLx S y ˆ ˆ ˆ [L S,Sz ] 0 ˆ ˆ ˆ ˆ ˆ ˆ Lx[Sz ,Sx ] Ly[S y ,Sz ] ˆ ˆ ˆ ˆ iLxS y iLy Sx and ˆ ˆ ˆ [L S, J z ] 0 These commutation relations can be also derived from the invariant of the scalar product Lˆ Sˆ, under the rotation.
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