4.9 Magnetic properties of the two electron system: singlet and triplet state

For each electron the spin can be +1/2 or 1/2, leading to four possible states

 |↑↑,|↑↓,|↓↑,|↓↓.(4.66)

According to the following table we can combine these states to get several states which differ in the values for the complete spin S and the component Sz:

State SSz name picture
12(|↑↓|↓↑) 0 0 singlet PIC
|↑↑ 1 1 triplet PIC
12(|↑↓+|↓↑) 1 0 triplet PIC
|↓↓ 1 -1 triplet PIC


HINT:


Spin add like vectors. In addition to the norm of the spin only one additional component (mostly referred to as the z-component) can be chosen. This may be interpreted as a rotation e.g. around the z-axis as illustrated in the right figure: two spins add to the complete spin leading to no effective z-component.


PIC


Since quantum mechanical particles are not distinguishable we already recognized that the symmetry for changing two indices is very important; exchanging the indices for a singlet state we get an additional minus sign which not occurs for a triplet state. Thus the singlet state is antisymmetric and the triplet state is symmetric when exchanging the indices. Always the complete wave function of the two electron state has to be antisymmetric; consequently the ”local” wave function of a singlet state must be symmetric and for a triplet state it has to be antisymmetric. Therefore we always find a magnetic momentum if the local wave function is symmetric in the local space when exchanging two indices and otherwise no magnetic momentum.
Following these considerations we will now solve the Schrödinger equation (4.64). Since we have a sum of two independent one electron Hamiltonians we can construct the complete solution as a product of solutions of

 hψ(r)=ϵψ(r)(4.67)

Let ψ0(r) and ψ1(r) be the solutions for the two lowest energy levels ϵ0<ϵ1 of Eq. (4.67); the symmetric solution of the two electron state with the lowest energy is

 ψs(r1,r2)=ψ0(r1)ψ0(r2),Es=2ϵ0(4.68)

and the antisymmetric solution with the lowest energy is

 ψt(r1,r2)=ψ0(r1)ψ1(r2)ψ0(r2)ψ1(r1),Et=ϵ0+ϵ1(4.69)

This leads to an energy gap between the singlet and triplet state of

 EsEt=ϵ0ϵ1(4.70)

The singlet state will always be the ground state if the Coulomb coupling is negligible.
This singlet ground state corresponds perfectly to what we calculated in the last sections as the band structure of a solid:

  1. Neglect the electron-electron coupling

  2. Calculate single electron states

  3. Fill all states with electrons, starting with the lowest energy level,

  4. with two electrons of opposite spin

  5. We will never find a magnetic momentum

Now we apply the Ritz’ variational method to get approximations for ψ0 and ψ1. As a test function we chose a linear combination of the atomic wave functions with the lowest energies of both atoms

 ψ=α1Φ1+α2Φ2(4.71)

For the expectation value for the energy we find

 ϵ=ψhψdVψψdV=α12H11+α22H22+2α1α2H12α12+α22+2α1α2S(4.72)

with

 S=Φ1Φ2dV,H11=Φ1hΦ1dV=H22andH12=Φ1hΦ2dV.(4.73)

Minimizing Eq. (4.72) for αi we get a system of linear equations

 α1(H11ϵ)+α2(H12ϵS)=0α2(H12ϵS)+α2(H11ϵ)=0(4.74)
This system has non trivial solutions only if the coefficient determinant vanishes, i.e.

 (H11ϵ)2(H12ϵS)2=0(4.75)

We find energy values

 ϵ±=H11±H121±S(4.76)

and Eigenvectors α1=α2 and α1=α2, leading to

 ψ0=12(Φ1+Φ2)(4.77)

and

 ψ1=12(Φ1Φ2)(4.78)

So according to Eq. (4.68) the complete symmetric wave function is calculated as

 ψs(r1,r2)=12{Φ1(r1)Φ2(r2)+Φ2(r1)Φ1(r2)+Φ1(r1)Φ1(r2)+Φ2(r1)Φ2(r2)}(4.79)

and the antisymmetric one:

 ψt(r1,r2)=12{Φ2(r1)Φ1(r2)Φ1(r1)Φ2(r2)}(4.80)

Eq. (4.79) is an excellent solution of the Schrödinger equation (4.64) for negligible Coulomb interaction.
Not taking into account the lattice periodicity and the large number of atoms in a solid we repeated the LCAO-calculation:

We will check now, if the singlet state is as well a good approximation in the case of strong Coulomb interaction as it is included in the Schrödinger equation (4.62):

4.9.1 Mean-Field-Approximation for the description of the Coulomb-Coupling


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© J. Carstensen (Quantum Mech.)