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

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1. Apr 27, 2017

### BrokenPhysics

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?

2. Apr 27, 2017

### Demystifier

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: Apr 27, 2017
3. Apr 27, 2017

### dextercioby

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: Apr 27, 2017
4. Apr 27, 2017

### BrokenPhysics

5. Apr 27, 2017

### dextercioby

This has to be SI vs. CGS issue.

6. Apr 27, 2017