# 1.6: Example of matrix representation method and choice of basis

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In practical quantum problems, we almost always describe the state of the system in terms of some basis set. Consider a simple spin 1/2 system, choosing as basis states $$S_z = \pm \frac{1}{2}$$. Consider this system in a magnetic field pointing in the $$x$$ direction, the operator corresponding to this is $$\mu B \hat{S}_x$$. We wish to find the eigenstates and eigenenergies.

Evaluating the required matrix elements such as $$\langle S_z = \frac{1}{2} |\mu B\hat{S}_x | S_z = \frac{1}{2} \rangle$$ (see QP3) gives a matrix:

$\begin{pmatrix} 0 & \mu B/2 \\ \mu B/2 & 0 \end{pmatrix} \nonumber$

The normalised eigenvectors of this matrix are $$(\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{2}})$$ and $$(\sqrt{\frac{1}{2}}, -\sqrt{\frac{1}{2}})$$ with eigenvalues $$(\mu B/2)$$ and $$(-\mu B/2)$$. Of course these represent the eigenstates $$|S_x = \pm \frac{1}{2} \rangle$$ in the basis of $$|S_z = \pm \frac{1}{2} \rangle$$:

$|S_x = \pm \frac{1}{2} \rangle = \left[|S_z = \frac{1}{2} \rangle \pm |S_z = -\frac{1}{2} \rangle \right] / \sqrt{2} \nonumber$

Had we chosen $$|S_y = \pm \frac{1}{2} \rangle$$ as our basis set, then the matrix would have been:

$\begin{pmatrix} 0 & -i\mu B/2 \\ i\mu B/2 & 0 \end{pmatrix} \nonumber$

Once again, the eigenvalues of this matrix are $$(\mu B/2)$$ and $$(-\mu B/2)$$, as they must be since these are the measurable quantities. Coincidently, the eigenvectors in this basis set are also $$(\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{2}})$$ and $$(\sqrt{\frac{1}{2}}, -\sqrt{\frac{1}{2}})$$.

Had we chosen $$|S_x = \pm \frac{1}{2} \rangle$$ as our basis set in the first place, the problem would have been much simplified. The matrix would then be:

$\begin{pmatrix} \mu B/2 & 0 \\ 0 & -\mu B/2 \end{pmatrix} \nonumber$

Once again, the eigenvalues of this matrix are $$(\mu B/2)$$ and $$(-\mu B/2)$$, and now the eigenvectors are (1,0) and (0,1): i.e. the eigenstates are simply the basis states.

This page titled 1.6: Example of matrix representation method and choice of basis is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.