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lecture15_umn

lecture15_umn - Lecture 15 Hydrogenic Orbitals Atomic Units...

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Lecture 15 October 14, 2009 Hydrogenic Orbitals Atomic Units Hydrogenic One-Electron Wave Functions and their Energies

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Hydrogenic Orbitals Quantum restrictions on the energy levels of one-electron atoms Final step: putting final quantum numbers into various differential equations to pull out the eigenfunctions themselves . In a completely general way (arbitrary quantum numbers n and l ): one can write a general wave function in terms of the associated Laguerre polynomials . Consider some specific functions associated with smaller quantum numbers . What if we pick the principle quantum number n = 1 ? From eq. 14-30: l and m both =0, and β = n = 1. l + m + 1 " # = 0
Principle Quantum Number n = 1 Requirement l =0: the spherical harmonic is Y 0,0 . From the table of Lecture 12 (15-1) very simple! As for the radial function, since m = 0, the only term in Σ ( ρ ) that is non-zero is the zeroth order term a 0 . Y 0, 0 = 1 4 " # \$ % & ( 1/ 2

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Case n=1, l=0, ml=0 This means (15-2) ρ was defined as the dimensionless quantity 2 α r . If we plug back in all of the constants we get (15-3) " # ( ) = a 0 S # ( ) = # 0 a 0 = a 0 \$ # ( ) = a 0 e % # / 2 " = # 2 μ E h 2 = 2 μ 2 Z 2 e 4 32 \$ 2 % 0 2 n 2 h 4 = μ Ze 2 4 \$% 0 n h 2
Hydrogenic Orbitals So, the full wave function becomes (15-4) To solve for a 0 , we simply have to normalize Ψ 100 . Subscripts: principle quantum number n azimuthal quantum number l z component of the angular momentum quantum number m l . " 100 r , # , \$ ( ) = % 1 & ( ) Y 0,0 # , \$ ( ) = 1 4 a 0 e ( μ Ze 2 r / 4 ’) 0 n h 2

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Atomic Units A new system of units: " atomic units " (a.u.) In a.u.: the magnitude of various quantities is defined to be 1 . one atomic unit of action. Dielectric permittivity of the vacuum ε 0 =1. Charge on the electron e = 1. Mass of the electron (which is very close to μ ) =1. In atomic units, eq. 15-4 simplifies to (15-5) To convert from atomic units to other units: use of the proper conversion factors " 100 r , # , \$ ( ) = 1 4 % a 0 e & Zr / n h
Useful Quantities in Atomic and Other Units Physical quantity (unit name) Symbol Value in a.u.

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lecture15_umn - Lecture 15 Hydrogenic Orbitals Atomic Units...

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