Lecture8-9

B given spherical symmetry the radial wave func2ons

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Unformatted text preview: entum will always yield quan2zed results. 12 Hydrogen u༇ Spherically symmetric poten2al. u༇ We can label the Energy eigenfunc2ons with the eigenvalues of L2 and Lz. u༇ And this will simplify the differen2al equa2on to a 1D equa2on in r. u༇ note the 1/r2 behavior of the psuedopoten2al due to L2 u༇ no dependence on m 13 Hydrogen: Basic Structure u༇ The simple Schrödinger equa2on gives a very accurate descrip2on of the atom’s proper2es. u༇ When we require that the probability can be normalized, we get quan2zed energy states as observed using diffrac2on gra2ngs. u༇ The Hydrogen wavefunc2ons are in three dimensions and it turns out we will need 3 quantum numbers to label each spa2al state. Solu2on to Hydrogen Radial Eq. u༇  Solving the Hydrogen Radial Eq. u༇  As a result of the normaliza2on requirement, we get the Rydberg formula for the energies. u༇  the principle quantum number n is really a sum: n=nr+l+1 o  nr is the number of nodes in the radial wave func2on, not at 0 or infinity. o  radial excita2ons and angular excita2ons happen to have the same energy in the Coulomb poten2al u༇  We also get a recursion rela2on for the radial wave func2ons. 15 Solu2ons to Radial Equa2on Rnl u༇  The radial probability density [email protected], P(r), is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr Clicker: Spherical Symmetry u༇  Which of the following statements is false? A.  We can use the Spherical Harmonics to reduce any 3D bound state problem with spherical symmetry to one equa2on in the radial coordinate. B.  Given spherical symmetry, the radial wave func2ons do not depend on the angular momentum quantum numbers. C.  Spherical symmetry implies conserva2on of angular momentum. D.  Spherical symmetry implies the energy eigenstates will also be eigenstates of two angular momentum operators. E.  none of the above 17 The 3D Wavefunc2on u༇  The Radial wfn is specific to the Coulomb poten2al but it only depends on the radius. 18 Example: Expecta2on of 1/r u༇  plug in u༇  Ylm orthonormal u༇  Integrate by parts u༇  Finite term 0 u༇  Integrate exp u༇  answer u༇  general integral for Hydrogen from int by parts 19 Example: Expecta2on of r u༇  We use the integral fo...
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This note was uploaded on 02/24/2014 for the course PHYS 2D taught by Professor Hirsch during the Winter '08 term at UCSD.

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