QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ

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QM in 3DQuantum Ch.4, Physical Systems,
24.Feb.2003 EJZ
Schrödinger eqn in spherical coordinates Separatio
n of variables (Prob.4.2 p.124) Angular equation
(Prob.4.3 p.128 or 4.23 p.153) Hydrogen atom
(Prob 4.10 p.140) Angular Momentum (Prob 4.20
p.150) Spin
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Schrödinger eqn. in spherical coords.
The time-dependent SE in 3D
has solutions of form where Yn(r,t)
solves Recall how to solve this using
separation of variables
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Separation of variables
To solve Let Then the 3D diffeq becomes two
diffeqs (one 1D, one 2D) Radial
equation Angular equation
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Solving the Angular equation
To solve Let Y(q,f) Q(q) F(f) and separate
variables The f equation has solutions F(f)
eimf (by inspection) and the q equation has
solutions Q(q) C Plm(cosq) where Plm
associated Legendre functions of argument
(cosq). The angular solution spherical
harmonics Y(q,f) C Plm(cosq) eimf where C
normalization constant
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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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Solutions to 3D spherical Schrödinger eqn
Radial equation solutions for V Coulomb
potential depend on n and l (LLaguerre
polynomials, a Bohr radius)
Rnl(r) Angular solutions Spherical
harmonics As we showed earlier, Energy Bohr
energy with nnl.
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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
• Next time, 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.