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(normal) Zeeman Effect with Spin In strong external magnetic fields B, orbital and spin magnetic moments are both coupled to B (and not to each other): In this case, z-components of the moments are additive. No magnetic field With magnetic field, no spin With magnetic field and spin l=1, ml=+1, ms=+1/2 l=1, ml=0, ms=+1/2 l=1, ml=+1, ms=-1/2 and l=1, ml=-1, ms=+1/2 l=1, ml=0, ms=-1/2 n=2, l=1 l=1, ml=-1, ms=-1/2 l=0, ml=0, ms=+1/2 n=1, l=0 l=0, ml=0, ms=-1/2 (note: no spin-orbit coupling in this diagram. We will later learn that some transitions are forbidden) Spin-Orbit Interaction In the absence of an external magnetic field (or a weak field), orbital and spin magnetic moments interact. From the perspective of the electron, the (positively charged, Ze+) nucleus orbits around it. This creates a magnetic moment at the site of the electron. The electron energy is higher (less stable), if spin S is aligned with B. The electron energy is lower (more stable), if spin S is aligned opposite to B. Spin-Orbit Vector Addition When spin-orbit coupling is in effect, neither L nor S are angular momentum conserved; only the total angular momentum J=L+S is conserved. This replaces the set of quantum numbers (l, ml, s, ms) with a new set. (l, s, j, mj) l=2, s=1/2 ! j=5/2 and j=3/2 Total angular momentum vector: l=2, j=5/2, mj=3/2 Quantum numbers: Spectroscopic Notation (Term Symbols) General Form: l : orbital angular momentum quantum number. s: spin quantum number. j: total angular momentum quantum number. 2s+1 : “Spin Multiplicity” " total number of spin states [e.g. 2s+1=2 (up and down) for s=1/2]. Example: hydrogen atom (lowest states): (note: In this example, spin multiplicity is always 2 because hydrogen has only one electron. Total angular momentum quantum number j is either l+1/2 or l-1/2). Dirac Relativistic Hydrogen Atom The Bohr/Schrödinger result for hydrogen Bohr/Schrödinger Dirac is based on non-relativistic (Newtonian) mechanics. The (special-) relativistic equivalent to the Schrödinger equation is the Dirac equation. The Dirac equation, naturally explains spin (and spin-orbit coupling), giving the following energy equation of hydrogen. (Fine structure constant) Shift of Dirac levels relative to Bohr/Schrödinger is exaggerated by a factor of (1/#2).