- #1

lelouch_v1

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- TL;DR Summary
- Couldn't find the 8 eigenvalues of the problem

I was reading a paper on Radical-Pair mechanism (2 atoms with 1 valence electron each) and the author used the hyperfine hamiltonian $$H_{B}=-B(s_{D_z}+s_{A_z})+As_{D_x}I_x+As_{D_y}I_y+as_{D_z}I_z$$ and found the following eigenvalues: a/4 (doubly degenerate) , a/4±B , (-a-2B±2√(A^2+B^2)) , (-a+2B±2√(A^2+B^2))

What i used was the eigenvalue equation $$H{\ket{\psi}}=E{\ket{\psi}}$$ where ψ is a singlet and a triplet state, i.e. $$\ket{S}={\frac{1}{\sqrt{2}}}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$$ $$\ket{T_1}=\ket{\uparrow\uparrow}$$ $$\ket{T_{-1}}=\ket{\downarrow\downarrow}$$ $$\ket{T_{0}}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})$$ Using these states, i was able to get the first four eigenvalues. Since there are eight eigenvalues, should i right the terms in the hamiltonian as tensor products? Also, since I is the nuclear spin operator, do i take it always as 1/2 when in the z-direction, since i suppose that the nucleus is a hydrogen atom with one proton? Can i write the term $$As_{D_x}I_x+As_{D_y}I_y$$ in terms of the annihilation and creation operators, i.e. $$As_{+}I_{-}+As_{-}I_{+}$$ or is it wrong?

P.S.: In the hamiltonian, the subscript D is used for Donor and the subscript A for Acceptor.

What i used was the eigenvalue equation $$H{\ket{\psi}}=E{\ket{\psi}}$$ where ψ is a singlet and a triplet state, i.e. $$\ket{S}={\frac{1}{\sqrt{2}}}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$$ $$\ket{T_1}=\ket{\uparrow\uparrow}$$ $$\ket{T_{-1}}=\ket{\downarrow\downarrow}$$ $$\ket{T_{0}}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})$$ Using these states, i was able to get the first four eigenvalues. Since there are eight eigenvalues, should i right the terms in the hamiltonian as tensor products? Also, since I is the nuclear spin operator, do i take it always as 1/2 when in the z-direction, since i suppose that the nucleus is a hydrogen atom with one proton? Can i write the term $$As_{D_x}I_x+As_{D_y}I_y$$ in terms of the annihilation and creation operators, i.e. $$As_{+}I_{-}+As_{-}I_{+}$$ or is it wrong?

P.S.: In the hamiltonian, the subscript D is used for Donor and the subscript A for Acceptor.