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

  1. Apr 27, 2017 #1
    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. jcsd
  3. Apr 27, 2017 #2

    Demystifier

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    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
  4. Apr 27, 2017 #3

    dextercioby

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    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
  5. Apr 27, 2017 #4
  6. Apr 27, 2017 #5

    dextercioby

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    This has to be SI vs. CGS issue.
     
  7. Apr 27, 2017 #6

    jtbell

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    Staff: Mentor

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