Take a spin-1/2 particle of mass ##m## and charge ##e## and place it in a magnetic field in the ##z## direction so that ##\mathbf B=B\mathbf e_z##. The corresponding Hamiltonian is(adsbygoogle = window.adsbygoogle || []).push({});

$$\hat H=\frac{eB}{mc}\hat S_z.$$

This must have units of joules overall, and since the eigenvalues of ##\hat S_z## are proportional to ##\hbar## with units ##\text{J s}##, the prefactor ##eB/mc## should have units ##\text s^{-1}##, i.e. it is an angular frequency - specifically the Larmor frequency - and is denoted ##\omega##.

But if we work out the units of ##\omega=eB/mc##, with

\begin{align*}

[e]&=\text C\\

[\mathbf B]&=\text T=\text{kg C}^{-1}\text{ s}^{-1}\\

[m]&=\text{kg}\\

[c]&=\text{ m s}^{-1}

\end{align*}

we get ##\text m^{-1}## overall and not ##\text s^{-1}##.

What am I doing wrong?

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# I Hamiltonian for spin-1/2 particle in B-field: units issue

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