Quantum Ch'4 continued Physical Systems, 27'Feb'2003 EJZ

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Quantum Ch.4 - continuedPhysical Systems,
27.Feb.2003 EJZ
Recall solution to Schrödinger eqn in spherical
coordinates with Coulomb potential (H atom) Work
on HW help sheet (linked to Help page) Probs.1
and 10. Angular Momentum - Minilecture by Don
Verbeke (Do Prob 4.18, and 4.20 p.150 as you did
Prob.1 above) Spin - Minilecture by Andy Syltebo
Do the example on p.157, try problem 4.28
together
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Schrödinger eqn. in spherical coords with
Coulomb potential
The time-independent SE
has solutions where and Rnl(r) Plm
associated Legendre functions of argument (cosq)
and LLaguerre polynomials
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Quantization of l and m
In solving the angular equation, we use the
Rodrigues formula to generate the Legendre
functions Notice that l must be a
non-negative integer for this to make any
sense moreover, if mgtl, then this says that
Plm0. For any given l, then there are (2l1)
possible values of m (Griffiths p.127)
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Solving the Radial equation
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finish solving the Radial equation
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Hydrogen atom a few wave functions
Radial wavefunctions depend on n and l, where l
0, 1, 2, , n-1
Angular wavefunctions depend on l and m, where m
-l, , 0, , l
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Angular momentum L review from Modern physics
Quantization of angular momentum direction for
l2
Magnetic field splits l level in (2l1) values of
ml 0, 1, 2, l
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Angular momentum L from Classical physics to QM
L r x p Calculate Lx, Ly, Lz and their
commutators Uncertainty relations Each
component does commute with L2 Eigenvalues
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Spin - review

• Hydrogen atom so far 3D spherical solution to
  Schrödinger equation yields 3 new quantum
  numbers
• l orbital quantum number
• ml magnetic quantum number 0, 1, 2, , l
• ms spin 1/2
• Next step toward refining the H-atom model
• Spin with
• Total angular momentum JLs
• with jls, ls-1, , l-s

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Spin - new

• Commutation relations are just like those for L
• Can measure S and Sz simultaneously, but not Sx
  and Sy.
• Spinors spin eigenvectors
• An electron (for example) can have spin up or
  spin down
• NEW operate on these with Pauli spin matrices

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Total angular momentum

• Multi-electron atoms have total J SL where
• S vector sum of spins,
• L vector sum of angular momenta
• Allowed transitions (emitting or absorbing a
  photon of spin 1)
• ?J 0, 1 (not J0 to J0) ?L 0, 1
  ?S 0
• ?mj 0, 1 (not 0 to 0 if ?J0)
• ?l 1 because transition emits or absorbs a
  photon of spin1
• ?ml 0, 1 derived from wavefunctions and
  raising/lowering ops

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Review applications of Spin

• Bohr magneton
• Stern Gerlach measures me 2 m B
• Diracs QM prediction 2Bohrs semi-classical
  prediction
• Zeeman effect is due to an external magnetic
  field.
• Fine-structure splitting is due to spin-orbit
  coupling (and a small relativistic correction).
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• Hyperfine splitting is due to interaction of
  melectron with mproton.
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• Very strong external B, or normal Zeeman
  effect, decouples L and S, so geffmL2mS.