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In classical mechanics and solid-state physics texts, you often see the Hamiltonian $H$ of a system of harmonic oscillators written in normal mode coordinates:

$$ H = \frac{1}{2}\sum_n\dot{X_n}^2 + \frac{1}{2}\sum_n\omega_n^2X_n^2$$

where $X_n$ and $\dot{X_n}$ are the amplitudes velocities of mode $n$. In the potential energy term, $\omega_n$ is the angular frequency of mode $n$, because $\omega_n$ comes from the eigenvalue equation for normal modes. Let's look at the units of this Hamiltonian.

Consider SI units:

  • $\omega_n = rad/s$
  • $X_n = \sqrt{kg}m$
  • $\dot{X_n} = \sqrt{kg}m/s$
  • $\dot{X_n}^2 = J$
  • $\omega_n^2X_n^2 = rad^2J$

Shouldn't the potential energy have units of $J$ instead of $rad^2J$? Do we need to divide by $(2\pi)^2$?

In my experience of calculating mode energies in molecular dynamics simulations, the use of angular frequency gives the correct energy, but the units don't make sense!

Tobias Fünke
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Thermodynamix
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  • 'The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.' from https://en.wikipedia.org/wiki/Radian . Does this answer your question? – Tobias Fünke Feb 06 '21 at 16:27
  • Consider that this problem (or non-problem) appears in much simpler situations. if you have a harmonic motion with $x(t) = A\sin(\omega t)$, where $\dim(x)= \dim(A) = \mathsf{L}$, then the velocity is $\dot{x}(t) = \omega A \cos(\omega t)$, where the amplitude now has units, say, $\mathrm{rad}\ \mathrm{m}/s$. But the dimension is $\dim(\dot{x}) = \dim(\omega A) = \mathsf{L}\ \mathsf{T}^{-1}$, because angles have dimension $\textsf{1}$. So it's important here to distinguish between units and dimensions, as Jakob remarked: angles in radians are ratios of two lengths, so they are dimensionless. – pglpm Feb 06 '21 at 16:42
  • ...That said, I personally also feel somewhat unsatisfied with this situation, and I've heard that there are people at ISO who work on possibly different points of view and different ways of handling this. – pglpm Feb 06 '21 at 16:45
  • @pglpm, thanks for that example. I'm concerned about the factors of $2\pi$ to get the proper final value in a practical calculation. If I want to calculate the velocity you mentioned in $m/s$, should I use angular frequency in $rad/s$ or normal frequency in Hz? – Thermodynamix Feb 06 '21 at 17:05
  • @Jakob Maybe I used the wrong language in my title/question, I don't want to get caught up in the semantics of what's a unit or not, and I agree $rad$ is dimensionless. I'm wondering whether or not in a practical calculation if I should use angular frequency or frequency. I think the answer is angular frequency, because that's what the math says, and I shouldn't concern myself with the radians. – Thermodynamix Feb 06 '21 at 17:10
  • Check out the ISO standard and the discussion there: https://www.iso.org/obp/ui#iso:std:iso:80000:-1:ed-1:v1:en – pglpm Feb 06 '21 at 17:54
  • There's also a summary by the NIST: https://physics.nist.gov/cuu/pdf/sp811.pdf Regarding Hz, that usually indicates cycles/second, so I personally think that Hz is not appropriate for an angular frequency, and would also use rad/s. But when we multiply $\omega$ and $A$ the "rad" disappears, because an angle in radians multiplied by a length gives a length. As already said I also feel that all this should be made more consistent. But in practice it works without difficulties. – pglpm Feb 06 '21 at 18:04
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    See this answer of mine, it's virtually the same thing. – Massimo Ortolano Feb 06 '21 at 20:15

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