(1) We want to consider an atomic system containing two electrons.For now, we will consider that the electrons do not interact witheach other through the repulsive Coulomb interaction and we willneglect all interactions explicitly involving spin, such as thespin-orbit interaction. We can then write the one-electron-atomeigenfunctions as a product of a spatial function and a spinfunction in the form,
?nlmlms =?nlmlsms,
where the sigma function is the spin eigenfunction, the exactrepresentation of which you will not need. For an electron labeledby 1, applying our formalism in class, this can be expressed in theform,
?a(1)=?a(1)smsa (1), where a represents both the spatial and spinquantum numbers and a represents only the
spatial quantum numbers. Two-electron product eigenfunctions can bewritten in the form,
?aà = ?a (1)?à (2)= ?a (1)?b (2)smsa (1)smsb (2), and, finally, inthe more compact notation that we adapted in class,
?aÃ=?absms ms . ab
Of course, this is not an acceptable total eigenfunction for ourtwo-electron system since it is not antisymmetric with respect toan interchange of the electron labels.
(a) Write down both symmetric and antisymmetric forms for thespatial, two-electron eigenfunctions.
(b) Do the same for the spin, two-electron eigenfunctions,remembering that ms can assume the two values + 1/2 and â1/2.
(c) Now write down all of the antisymmetric total, two-electroneigenfunctions formed by the products. Label the eigenfunctionsthat are spatially symmetric with the letter s and those that arespatially antisymmetric with the letter t. These two-electron,total eigenfunctions are degenerate under our initial assumptions.
(1) We want to consider an atomic system containing two electrons.For now, we will consider that the electrons do not interact witheach other through the repulsive Coulomb interaction and we willneglect all interactions explicitly involving spin, such as thespin-orbit interaction. We can then write the one-electron-atomeigenfunctions as a product of a spatial function and a spinfunction in the form,
?nlmlms =?nlmlsms,
where the sigma function is the spin eigenfunction, the exactrepresentation of which you will not need. For an electron labeledby 1, applying our formalism in class, this can be expressed in theform,
?a(1)=?a(1)smsa (1), where a represents both the spatial and spinquantum numbers and a represents only the
spatial quantum numbers. Two-electron product eigenfunctions can bewritten in the form,
?aà = ?a (1)?à (2)= ?a (1)?b (2)smsa (1)smsb (2), and, finally, inthe more compact notation that we adapted in class,
?aÃ=?absms ms . ab
Of course, this is not an acceptable total eigenfunction for ourtwo-electron system since it is not antisymmetric with respect toan interchange of the electron labels.
(a) Write down both symmetric and antisymmetric forms for thespatial, two-electron eigenfunctions.
(b) Do the same for the spin, two-electron eigenfunctions,remembering that ms can assume the two values + 1/2 and â1/2.
(c) Now write down all of the antisymmetric total, two-electroneigenfunctions formed by the products. Label the eigenfunctionsthat are spatially symmetric with the letter s and those that arespatially antisymmetric with the letter t. These two-electron,total eigenfunctions are degenerate under our initial assumptions.
