# Magnetic field Hamiltonian in different basis

## 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.

Last edited:

vela
Staff Emeritus
Homework Helper
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##.

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
Staff Emeritus