O lled shells are 2ghtly bound and spherically

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Unformatted text preview: spin state 1 χ anti = (↑↓ − ↓↑) 2 1 χ sym,0 = (↑↓ + ↓↑) 2 χ sym,1 = (↑↑) χ sym,−1 = (↓↓) implies an2symmetric spa2al state. 53 Ground State for Two Electrons ψ sym = ψ100 (r1 )ψ100 (r2 ) + ψ100 (r2 )ψ100 (r1 ) ψ anti = ψ100 (r1 )ψ100 (r2 ) − ψ100 (r2 )ψ100 (r1 ) = 0 1 ψ gs = (ψ100 (r1 ) ↑ ψ100 (r2 ) ↓ −ψ100 (r1 ) ↓ ψ100 (r2 ) ↑) 2 u༇ For both par2cles in the same spa2al state, there is no an2symmetric spa2al state of 2e. u༇ So the He ground state uses the an2symmetric (singlet) spin state and the symmetric space state. u༇ This will not be true in l>0 states. 54 Excited State for Two Electrons 1 ψ sym = (ψ100 (r1 )ψ200 (r2 ) + ψ100 (r2 )ψ200 (r1 )) χ s 2 1 ψ anti = (ψ100 (r1 )ψ200 (r2 ) − ψ100 (r2 )ψ200 (r1 )) χ t 2 For different spa2al states, there are both an2symmetric and symmetric spa2al states of 2e. u༇  So the He excited state could be either of these two states. u༇  The an2symmetric state will have much lower energy due to the Coulomb repulsion. u༇  o  note that for two electrons in the same place, the an2symmetric state is zero while the symmetric state tends to put them in the same place. u༇  This is an important general result: The an2symmetric spa2al state has lower energy, forcing us to use the spin triplet state. o  including when space states differ only in m 55 Helium 1st Excited States with V All states increase in energy due to the Coulomb repulsion of the electrons. u༇  Before the perturba2on, the first excited state is degenerate. u༇  AGer the perturba2on, the singlet and triplet spin states split significantly due to the symmetry of the spa2al part of the wavefunc2on. u༇  The Fine Structure effects can be neglected on the scale of the Coulomb correc2ons. u༇  But the Coulomb repulsion causes a large difference between the singlet and triplet spin states. u༇  56 Hydrogen- like Energy Levels Principle Total Quantum Angular name Number momentum n = 1: n=2: n = 3: n=4: =0 =0 =1 =0 =1 =2 =0 =1 =2 =3 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f z angular momentum m=0 m=0 m = −1, 0,1 m=0 m = −1, 0,1 m = −2, −1, 0,1, 2 m=0 m = −1, 0,1 m = −2, −1, 0,1, 2 m = −3, −2, −1, 0,1, 2, 3 Electrons In shell 2 electrons 2 electrons 6 electrons 2 electrons 6 electrons 10 electrons 2 electrons 6 electrons 10 electrons 14 electrons 57 The Atomic States u༇ The Hydrogen- like states fill up as we increase Z and add more electrons. u༇ The lowest energy states fill first. u༇ Inside an orbital (3d f...
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