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## Homework Statement

A spin-1/2 electron in a magnetic field can be regarded as a qubit with Hamiltonian

$$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$. This matrix can be written in the form of a qubit matrix

$$

\begin{pmatrix}

\frac{1}{2}\epsilon & t\\

t^* & -\frac{1}{2}\epsilon

\end{pmatrix}$$

Where it fulfils the eigenvalue equation ##\hat{H}|\psi\rangle = E|\psi\rangle## in the basis of ##|1\rangle## and ##|0\rangle##.

Let us choose ##|0\rangle## and ##|1\rangle## to be the eigenstates of ##\sigma_x## with eigenvalues ##\pm 1## and find ##\epsilon## and ##t## in terms of ##\textbf{B}## if we write the Hamiltonian in this basis.

## Homework Equations

The eigenstates of ##\sigma_x## are

$$\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix}, \quad \text{and} \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}$$

with eigenvalues 1 and -1 respectively.

The qubit Hamiltonian can be written as

$$

\hat{H} = \frac{\epsilon}{2}\{|1\rangle\langle1|-|0\rangle\langle0|\} + t|1\rangle\langle0|+t^*|0\rangle\langle1|$$

And the magnetic field Hamiltonian can be written as $$

\hat{H} = \frac{1}{2}g\mu_B\begin{pmatrix}B_z & B_x-iB_y\\

B_x+iB_y & -B_z\end{pmatrix}$$

## The Attempt at a Solution

I actually do not see the connection how to write this in the basis spanned by the eigenstates of ##\sigma_x##. The solution for eigenvalues ##E## and eigenstates of the qubit Hamiltonian are given later in the book and I thought about using these, but I do not think they are necessary since these are explained after this question.

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