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QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ Schrödinger eqn in spherical coordinates Separation 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 n (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( ) = and separate variables: The equation has solutions = e im (by inspection) and the equation has solutions = C P l m (cos ) where P l m = associated Legendre functions of argument (cos ). The angular solution = spherical harmonics: Y( )= C P l m (cos ) e im 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 |m|>l, then this says that P l m =0. For any given l, then there are (2l+1) 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 (L=Laguerre polynomials, a = Bohr radius) R nl (r)= Angular solutions = Spherical harmonics As we showed earlier, Energy = Bohr energy with n’=n+l.

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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 l=2 Magnetic field splits l level in (2l+1) values of m l = 0, ±1, ± 2, … ± l

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Angular momentum L: from Classical physics to QM L = r x p Calculate L x, L y, L z and their commutators: Uncertainty relations: Each component does commute with L 2 : 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 m l = magnetic quantum number = 0, ±1, ±2, …, ±l m s = spin = ±1/2 Next step toward refining the H-atom model: Spin with Total angular momentum J=L+s with j=l+s, l+s-1, …, |l-s|

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Spin - new Commutation relations are just like those for L: Can measure S and S z simultaneously, but not S x and S y. 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 = S+L where S = vector sum of spins, L = vector sum of angular momenta Allowed transitions (emitting or absorbing a photon of spin 1) ΔJ = 0, ±1 (not J=0 to J=0) ΔL = 0, ±1 ΔS = 0 Δm j =0, ±1 (not 0 to 0 if ΔJ=0) Δl = ±1 because transition emits or absorbs a photon of spin=1 Δm l = 0, ±1 derived from wavefunctions and raising/lowering ops

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Review applications of Spin Bohr magneton Stern Gerlach measures e = 2 B : Dirac’s QM prediction = 2*Bohr’s 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). Hyperfine splitting is due to interaction of electron with proton. Very strong external B, or “normal” Zeeman effect, decouples L and S, so g eff =m L +2m S.

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