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3D Schrodinger Equation

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. Solve by separating variables.

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3D Schrodinger Equation

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  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. • Solve by separating variables 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 P460 - 3D S.E.

  3. Example: 2D (~same as 3D) particle in a Square Box • solve 2 differential equations and get • symmetry as square. “broken” if rectangle 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 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 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 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 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. • Solve Legendre equation by series solution P460 - 3D S.E.

  9. Solving Legendre Equation • Plug series terms into Legendre equation • let k=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  L=j=integer • end up with odd/even (Parity) series 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) P460 - 3D S.E.

  11. Spherical Harmonics • The product of the theta and phi terms are called Spherical Harmonics. Also occur in E&M. See Table on page 127 in book • They hold whenever V is function of only r. Seen related to angular momentum P460 - 3D S.E.

  12. 3D Schr. Eqn.-Radial Eqn. • For V function of radius only. Look at radial equation. L comes in from theta equation (separation constant) • can be rewritten as (usually much better...) • and then have probability P460 - 3D S.E.

  13. 3D Schr. Eqn.-Radial Eqn. • note L(L+1) term. Angular momentum. Acts like repulsive potential and goes to infinity at r=0 (ala classical mechanics) • energy eigenvalues typically depend on 2 quantum numbers (n and L). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=L+s. P460 - 3D S.E.

  14. Particle in spherical box • Good first model for nuclei • plug into radial equation. Can guess solutions • look first at l=0 P460 - 3D S.E.

  15. Particle in spherical box • l=0 • boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 and • note plane wave solution. Supplement 8-B discusses scattering, phase shifts. General terms are P460 - 3D S.E.

  16. Particle in spherical box • ForLl>0 solutions are Bessel functions. Often arises in scattering off spherically symmetric potentials (like nuclei…..). Can guess shape (also can guess finite well) • energy will depend on both quantum numbers • and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on details • gives what nuclei (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms: • in nuclei (with j subshells) P460 - 3D S.E.

  17. H Atom Radial Function • For V =a/r get (use reduced mass) • Laguerre equation. Solutions are Laguerre polynomials. Solve using series solution (after pulling out an exponential factor), get recursion relation, get eigenvalues by having the series end……n is any integer > 0 and L<n. Energy doesn’t depend on L quantum number. • Where fine structure constant alpha = 1/137 used. Same as Bohr model energy P460 - 3D S.E.

  18. H Atom Radial Function • Energy doesn’t depend on L quantum number but range of L restricted by n quantum number. l<n  n=1 only l=0 1S n=2 l=0,1 2S 2P n=3 l=0,1,2 3S 3P 3D • eigenfunctions depend on both n,L quantum numbers. First few: P460 - 3D S.E.

  19. H Atom Wave Functions P460 - 3D S.E.

  20. H Atom Degeneracy Energy n l m D -13.6 eV 1 0(S) 0 1 -3.4 eV 2 0 0 1 1(P) -1,0,1 3 • As energy only depends on n, more than one state with same energy for n>1 (only first order) • ignore spin for now -1.5 eV 3 0 0 1 1 -1,0,1 3 2(D) -2,-1,0,1,2 5 1 Ground State 4 First excited states 9 second excited states P460 - 3D S.E.

  21. Probability Density • P is radial probability density • small r naturally suppressed by phase space (no volume) • can get average, most probable radius, and width (in r) from P(r). (Supplement 8-A) P460 - 3D S.E.

  22. Most probable radius • For 1S state • Bohr radius (scaled for different levels) is a good approximation of the average or most probable value---depends on n and L • but electron probability “spread out” with width about the same size P460 - 3D S.E.

  23. Radial Probability Density P460 - 3D S.E.

  24. Radial Probability Density note # nodes P460 - 3D S.E.

  25. Angular Probabilities • no phi dependence. If (arbitrarily) have phi be angle around z-axis, this means no x,y dependence to wave function. We’ll see in angular momentum quantization • L=0 states are spherically symmetric. For L>0, individual states are “squished” but in arbitrary direction (unless broken by an external field) • Add up probabilities for all m subshells for a given L get a spherically symmetric probability distribution P460 - 3D S.E.

  26. Orthogonality • each individual eigenfunction is also orthogonal. • Many relationships between spherical harmonics. Important in, e.g., matrix element calculations. Or use raising and lowering operators • example P460 - 3D S.E.

  27. Wave functions • build up wavefunctions from eigenfunctions. • example • what are the expectation values for the energy and the total and z-components of the angular momentum? • have wavefunction in eigenfunction components P460 - 3D S.E.

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