Magnetic field Hamiltonian in different basis

  • Thread starter IanBerkman
  • Start date
  • #1
54
1

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:

Answers and Replies

  • #2
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,093
1,672
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##.
 
  • #3
54
1
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.
 
  • #4
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,093
1,672
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?
 
  • #5
54
1
I see, I thought completely wrong and was confused about notations. I found the answer in another thread, thank you.
 

Related Threads on Magnetic field Hamiltonian in different basis

Replies
12
Views
7K
  • Last Post
Replies
2
Views
732
  • Last Post
Replies
2
Views
2K
Replies
1
Views
590
Replies
17
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
2K
Replies
0
Views
2K
Replies
5
Views
581
Top