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Unformatted text preview: CHEM 356, Lecture 22, Fall 2009 1 The Hydrogen Atom and Hydrogenic Ions. II. We saw in the previous lecture that the Schr¨ odinger equation for the relative motion of the electron about the nucleus in a hydrogen atom or a hydrogenic ion could be separated if we work in terms of spherical polar coordinates: the separated version of the equations that we obtained there was − ℏ 2 ? 2 ( ? ) ( 1 ? ? ?? ? ) 2 ( ? ) − 2 ?? 2 [ − ( ? )] + 1 ?? ( , ) ℒ 2 ?? ( , ) = 0 The separation constant ? can be evaluated explicitly from our knowledge that the spherical harmonics are eigenfunctions of the orbital angular momentum operator ℒ 2 , with eigenvalue ℏ 2 ℓ ( ℓ + 1), to give − ℏ 2 2 ?? 2 ? ?? ( ? 2 ? ?? ) ( ? ) + [ ( ? ) − ] ( ? ) + ℓ ( ℓ + 1) ℏ 2 2 ?? 2 ( ? ) = 0 , as the differential equation governing the radial dependence of ( r ). ∙ The commutator [ ℒ 2 , r ] − = 0 implies that [ ℒ 2 , ( ? )] − = 0; also, we know that [ ℒ 2 , ∇ 2 ] − = 0, so that [ ℒ 2 , ℋ ] − = 0. This verifies that the angular eigenfunctions ℓ,? ( , ) of ℒ 2 , ℒ provide the angle dependence of the energy eigenfunctions of ℋ itself. ∙ Because the quantum number ℓ appears in the radial equation, the solutions ( ? ) will depend upon it. We may also expect the solution procedure to introduce a new quantum number (in addition to ℓ ), and indeed, it does: it is called the radial quantum number, and is designated by ′ (we shall see why ′ is used shortly). The radial equation is solved traditionally by the series solution method. The solution is somewhat lengthier (and messier!) than those we have seen so far in this course, and for this reason, we shall leave its details to a later course. CHEM 356, Lecture 22, Fall 2009 2 If we now use the explicit functional form for the Coulombic interaction represented by ( ? ), which is ( ? ) = − ?...
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 Fall '09
 Prof.Iaskjd
 Electron, Nucleus, Angular Momentum, Laguerre polynomials, Schr¨dinger equation, Hydrogenic Ions

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