Magnetic field Hamiltonian in different basis

[IanBerkman]

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.

 

[vela]

To express the Hamiltonian in matrix form, you have to first choose a basis. The matrix you wrote down (in terms of B) is written with respect to the basis consisting of eigenstates of $\sigma_z$. The problem seems to be asking you to express $\hat H$ in terms of the eigenstates of $\sigma_x$ instead, and then compare it to the qubit matrix and identify the corresponding values of $\varepsilon$ and $t$.

 

[IanBerkman]

Yes I see. Can someone verify if the following is correct?

I use $\langle x |\hat{H}| z\rangle$ where $x$ and $z$ denote the eigenstates of the corresponding Pauli matrices.
This can be written as $T_x^\dagger \hat{H}T_z$ (I think this is wrong but I do not see why).
Where $T_x$ is the matrix given by the eigenstates of $\sigma_x$:
$$T_x = \frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 1\\
1 & -1\end{pmatrix}$$

From this we get:
$$
\hat{H}_x = \frac{g \mu_B}{2\sqrt{2}}
\begin{pmatrix}
B_x+B_z+iB_y & B_x-B_z-iB_y\\
-B_x+B_z-iB_y & B_x+B_z-iB_y
\end{pmatrix}$$
Which I think is not correct since this Hamiltonian is not Hermitian.

 

[vela]

Yes I see. Can someone verify if the following is correct?

I use $\langle x |\hat{H}| z\rangle$ where $x$ and $z$ denote the eigenstates of the corresponding Pauli matrices.

What's your reasoning for doing this?

 

[IanBerkman]

I see, I thought completely wrong and was confused about notations. I found the answer in another thread, thank you.