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

## Main Question or Discussion Point

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
$$\hat H=\frac{eB}{mc}\hat S_z.$$
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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Demystifier
Gold Member
There should be no ##c## in the first formula, i.e. the correct one is
$$\hat H=\frac{eB}{m}\hat S_z$$

Last edited:
dextercioby
Homework Helper
Well, the formula is derivable in non specially-relativistic physics (Levy-Leblond's work in the 1960s), therefore couldn't possibly have a "c" in it, IF ONE USES SI UNITS.

Last edited:
dextercioby
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