Lecture8-9

53 ground state for two electrons sym 100 r1 100

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Unformatted text preview: f the spin- orbit interac2on and the other fine- structure correc2ons, are not eigenstates of Sz or Lz . u༇  They will be eigenstates of the total angular momentum (J2) and of it’s z component. o  eigenstates of E, L2, J2, Jz and e always spin ½ o  not Lz or Sz 40 j states have 2 terms (FYI) u༇  One with spin up and the other with spin down 41 Spectrascopic Nota2on u༇  We name the states with their principal quantum number and a leOer for l. o  1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s,… o  s, p, d, f, g, h,… o  these names were made up early and have stuck u༇  Now we know the fine structure splits these states and the new quantum number is j. o  for example 2p1/2 and 2p3/2 u༇  We will not bother with the labels for mul2electron states, but they are used in atoms. 42 Hydrogen Fine Structure u༇ Energy Eigenstates are also Eigenstates of Total Angular Momentum u༇ It is reasonable that the total angular momentum will be conserved in a spherically symmetric problem, o  not the individual parts of the angular momentum o  with interac2on between the parts. u༇ So the energy states all split into two, o  except for l=0. o  adding 0 plus ½ just gives ½ 43 Fine Structure Energy Diagram 44 Iden2cal Par2cles One electron is indis2nguishable from another u༇  Therefore there is a new Interchange symmetry to the problem u༇  Consider the interchange operator P12 u༇  o  Energy states must be eigenstates of P12 o  We prove the possible eigenvalues are ±1 Therefore, some states may be even under interchange and some may be odd. u༇  But the eigenvalue is sta2onary, created odd, it must stay odd. u༇  The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Electrons are always odd under interchange u༇  Photons are always even. u༇  45 Spin: Fermions and Bosons u༇ Internal angular momentum u༇ Electron has spin ½: Fermion o  an2symmetric under interchange o  implies no two in same state o  only created as par2cle an2par2cle pair o  maOer par2cle u༇ Photon has spin 1: Boson o  symmetric (under interchange) o  implies as many par2cles in same state (LASER) o  can be create...
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