Lecture 19 Highlights Now, back to the He atom. The ground state of He must put both of the electrons in to the lowest energy single-particle state100ψ (promoting one of the two electrons to the next available state is too energetically costly). This will put the two electrons into a symmetric space wavefunction. To make the overall wavefunction anti-symmetric under exchange of all the coordinates of the two identical fermions, we must have an anti-symmetric spin wavefunction. The spin singlet wavefunction will do the trick; 00)2()1()2,1(100100ψψ=ΨGSHe The unperturbed eigenstate has a ground state energy of 8.108−=TEeV. Now if we include the Coulomb repulsion perturbing Hamiltonian, the first-order correction to the energy is given by; GSHeGSHeGSE ΨΗΨ= '1 00)2()1(400)2()1(1001002102**2313100100ψψπεψψrrexdxdrr−=∫∫∫∫∫∫ This integral can be done analytically (see Griffiths, page 300), and yields eV. The first-order corrected ground state energy of He is predicted to be eV. The measured ground state energy of He is 78.98 eV, within 5% of our estimate. 341+=GSE8.741−=+=GSTGSEEE What is the first excited state of He? Clearly one of the electrons must be promoted from the n=1 state to the n=2 state. But now there are two choices, (code-letter s) or (code-letter p). The short-hand for these two states is 1s2s and 1s2p. The 1s2s configuration has a lower energy than the 1s2p for the following reason. The first electron (1s) resides in a Hydrogenic state with strong binding energy and small average radius. The second electron goes into an n=2 state with less binding energy and a larger average radius. The n=2 electron will experience a partially screened nucleus, since the 1s electron “wraps around” the nucleus and reduces its effective charge from +2e to something closer to +e. This weakens the attraction that the n=2 electron experiences with the nucleus, and is called ‘screening’. Now the orbital angular momentum comes in to play. An electron in an 0=l 1=l0=lstate will spend more time traveling through the nucleus and penetrating the inner 1s screening cloud (at least classically), compared to a 2p electron that will be in more of a traditional high angular momentum classical “orbit” about the nucleus. This allows the 1s electron to enjoy a stronger attraction to the nucleus, without paying too high a price in terms of Coulomb repulsion from the 1s electron. This same general idea explains many un-expected features of the periodic table. There are now many ways to write down the excited state wavefunctions (super-scripts “A” and “S” stand for Antisymmetric and Symmetric, respectively): {}00)1()2()2()1(21)2,1(212121 ssssSAssψψψψ+=Ψ or {}mssssASss1)1()2()2()1(21)2,1(212121ψψψψ−=Ψ , with 1,0,1 −+=m 1and {}00)1()2()2()1(21)2,1(212121 pspsSApsψψψψ+=Ψ or {}mpspsASps1)1()2()2()1(21)2,1(212121ψψψψ−=Ψ , with 1,0,1 −+=m The anti-symmetric space wavefunctions will have lower energy than their symmetric counterparts because of the “exchange splitting” discussed in the last lecture. E1s21s2s1s2p'ΗExchange1P13P1S03S11S0Spin-OrbitNotation:2S+1LJNot to scale!He Atom Low-Lying States0H These energy level orderings agree with the experimental data on He, as posted on the NIST Atomic Spectra Database web site.
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