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P460 - 3D S.E.1 3D Schrodinger Equation Simply substitute momentum operator do particle in box and H atom added dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity. (We skipped when doing perturbations). Quantization of angular momentum split between E+R Chaps 7+8 Solve by separating variables

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P460 - 3D S.E.2 If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z). Looking at second one: LHS depends on x,y RHS depends on z S = separation constant. Repeat for x and y

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P460 - 3D S.E.3 Example: 2D/3D particle in a Square Box solve 2 differential equations and get symmetry as square. “broken” if rectangle

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P460 - 3D S.E.4 2D gives 2 quantum numbers. Level nx ny Energy 1-1 1 1 2E0 1-2 1 2 5E0 2-1 2 1 5E0 2-2 2 2 8E0 for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….) this still satisfies S.E. with E=5E0

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P460 - 3D S.E.5 Spherical Coordinates Can solve S.E. if V(r) function only of radial coordinate volume element is solve by separation of variables multiply each side by

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P460 - 3D S.E.6 Spherical Coordinates-Phi Look at phi equation first constant (knowing answer allows form) must be single valued the theta equation will add a constraint on the m quantum number

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P460 - 3D S.E.7 Spherical Coordinates-Theta Take phi equation, plug into (theta,r) and rearrange knowing answer gives form of constant. Gives theta equation which depends on 2 quantum numbers. Associated Legendre equation. Can use either analytical (calculus) or algebraic (group theory) to solve. Do analytical. Start with Legendre equation

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P460 - 3D S.E.8 Spherical Coordinates-Theta Get associated Legendre functions by taking the derivative of the Legendre function. Prove by substitution into Legendre equation Note that power of P determines how many derivatives one can do. Sets value on m quantum number Solve Legendre equation by series solution

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P460 - 3D S.E.9 Solving Legendre Equation Plug series terms into Legendre equation let k-1=j+2 in first part and k=j in second (think of it as having two independent sums). Combine all terms with same power gives recursion relationship series ends if a value equals 0 end up with odd/even (Parity) series

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P460 - 3D S.E.10 Solving Legendre Equation Can start making Legendre polynomials. Be in ascending power order can now form associated Legendre polynomials. Can only have l derivatives of each Legendre polynomial. Gives constraint on m (theta solution constrains phi solution)

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P460 - 3D S.E.11 Spherical Harmonics The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M. They hold whenever V is function of only r. They are related to angular momentum (discuss later)

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