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Unformatted text preview: We have seen that the wavefunctions of electrons are completely anti- symmetric. By this statement we mean that under the exchange of any two particles’ coordinates and spins the wavefunction changes sign. Inclusion of the spin degree of freedom complicates the matters a little bit, but the dis- cussion is fairly simple for two electrons. When the total spin is conserved eigenfunctions of the Hamiltonian can also be chosen as the eigenfunctions of the total spin. More precisely, we can choose the energy eigenfunctions as the common eigenstates of H , S 2 and S z . In that case the eigenfunctions will be Ψ(1 , 2) = φ ( r 1 , r 2 ) χ spin where φ is the part of the wavefunction describing positions of the two elec- trons and χ spin is the part that describe the “spin motion” of the electrons. Since that last part is chosen as an eigenstate of S 2 and S z , it can be written as χ spin = | S = 1 ,m s i = |↑↑i m s = 1 , 1 √ 2 ( |↑↓i + |↓↑i ) m s = 0 , |↓↓i m s =- 1 , and χ spin = | S = 0 ,m s = 0 i = 1 √ 2 ( |↑↓i- |↓↑i ) . Since for S = 1 case (triplet) χ spin is symmetric under the exchange of par- ticles’ spin, the space part of the wavefunction should be antisymmetric: φ triplet ( r 1 , r 2 ) =- φ triplet ( r 2 , r 1 ) . For the S = 0 case (singlet), spin part is antisymmetric and as a result the space part should be symmetric: φ singlet ( r 1 , r 2 ) = φ singlet ( r 2 , r 1 ) . We should add that this simplification occurs only if the total spin, S = S 1 + S 2 , is a conserved quantity (i.e., it commutes with the Hamiltonian). In real atoms and molecules, this is not the case due to the spin-orbit coupling. Only the total angular momentum is conserved and not the total spin. As a result, true eigenstates of realistic Hamiltonians will be a mixture of singlet and triplet states, i.e., a very complicated wavefunction where spin and space are “correlated”. 1 For the case of three or more particles, the breakdown of the wavefunc- tion into a space and spin part will be quite complicated, even when the total spin is conserved. You may try to see this yourself for the case of three electrons with total spin quantum number S = 1 / 2. If the complete an- tisymmetry of the whole wavefunction is imposed, no wavefunction can be written as a product of a space part and a spin part. This is a result of the non-commutativeness of the different exchange operators for three or more particles. As a result in some cases (e.g., when S = 1 / 2 restriction is im- posed) you cannot find common eigenfunctions of all the exchange operators. 1 Slater Determinants Therefore a general method is needed to construct electron wavefunctions which are completely antisymmetric in all cases. For this purpose, we can use the complete antisymmetry property of determinants. Since interchang- ing any two rows (or columns) of a determinant changes the sign of the determinant, completely antisymmetric wavefunctions can be expressed in...
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
- Spring '11