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Magnetic field Hamiltonian in different basis

  1. Jul 27, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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 $$
    \hat{H} = \frac{1}{2}g\mu_B\begin{pmatrix}B_z & B_x-iB_y\\
    B_x+iB_y & -B_z\end{pmatrix}$$

    3. 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: Jul 27, 2016
  2. jcsd
  3. Jul 27, 2016 #2

    vela

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    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##.
     
  4. Jul 28, 2016 #3
    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.
     
  5. Jul 28, 2016 #4

    vela

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    What's your reasoning for doing this?
     
  6. Jul 29, 2016 #5
    I see, I thought completely wrong and was confused about notations. I found the answer in another thread, thank you.
     
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