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