Eigenvalues of Hyperfine Hamiltonian

In summary, the author of the paper used the hyperfine hamiltonian to find the eigenvalues of a radical-pair mechanism involving two atoms with one valence electron each. The eigenvalues were obtained by solving the eigenvalue equation using singlet and triplet states. The dipolar term in the hamiltonian, which accounts for the dipole-dipole interaction between the electron spin and nuclear spin, is usually neglected due to its small contribution compared to the Zeeman terms.
  • #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.
 
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  • #2
A:Let me try to answer some of your questions.The hyperfine Hamiltonian is usually written in terms of the spin operators, so you don't need to write it in terms of creation and annihilation operators. The $\hat{I}$ operator is usually written in terms of Pauli matrices $\hat{\sigma}$, so if $I=1/2$ then $\hat{I}_z = \sigma_z/2$. The terms $As_{D_x}I_x+As_{D_y}I_y$ are usually called the 'dipolar' term, and are a result of the dipole-dipole interaction between the electron spin and the nuclear spin. This term is often neglected because its contribution is much smaller than that of the Zeeman terms. To get all eight eigenvalues, you need to solve the full matrix equation $\hat{H}\psi = E \psi$. As you mentioned, the states $\psi$ can be chosen as the spin singlet and triplet states.
 

FAQ: Eigenvalues of Hyperfine Hamiltonian

1. What are eigenvalues of a hyperfine Hamiltonian?

Eigenvalues of a hyperfine Hamiltonian refer to the possible energy values that a system can have when subjected to the hyperfine interaction. These values are determined by solving the Schrödinger equation for the system.

2. How are eigenvalues of a hyperfine Hamiltonian calculated?

The eigenvalues of a hyperfine Hamiltonian are calculated by solving the Schrödinger equation for the system. This involves finding the eigenvalues of the Hamiltonian matrix, which represents the energy operator of the system.

3. Why are eigenvalues of a hyperfine Hamiltonian important?

Eigenvalues of a hyperfine Hamiltonian are important because they provide information about the possible energy states of a system. This can be used to understand the behavior of atoms, molecules, and other quantum systems, and can also be used in applications such as nuclear magnetic resonance.

4. How do eigenvalues of a hyperfine Hamiltonian affect the behavior of a system?

The eigenvalues of a hyperfine Hamiltonian determine the energy levels of a system, which in turn affect the behavior of the system. These energy levels can determine the stability, reactivity, and other properties of the system.

5. Can the eigenvalues of a hyperfine Hamiltonian change?

Yes, the eigenvalues of a hyperfine Hamiltonian can change if there is a change in the system's parameters or if it interacts with other systems. This can result in a different energy spectrum and can affect the behavior of the system.

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