Lecture8-9

# O lled shells are 2ghtly bound and spherically

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern