Conversion from natural units to SI

Question

I got questions about converting units from natural system of units to SI. To be exact, I'm solving the problem in Heisenber interpretation of quantum mechanics, and I'm using Heisenberg equation of motion

$$ \frac{d}{dt}A(t)=i[H,A(t)] $$

instead of

$$ \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]. $$

And, of course every operator that I'm using in my calculation i.e. (momentum) is used in this convention $\hbar = 1$. Finally, in my calculations I have got the wanted expression for some expectation value for some variable $t$, but the problem now is how to convert that expression in SI, where $\hbar$ is not $1$. Is there some rule of thumb, or dimensional analysis is the right way to go?

Answer 1

Yes, dimensional analysis is enough to reconstruct all the powers of $\hbar$ (and/or $c$ and other constants you may want to set equal to one) in all such formulae. For example, the first form of the Heisenberg equation is schematically $A/t = HA$. It's enough to analyze the powers of one kilogram in the units: $1/t$ has none while $H$ has units of one joule so it contains the first power of a kilogram. Obviously, this has to be cancelled on the right hand side and because $\hbar$ has units of joule-seconds which also has the first power of a kilogram, you must add $1/\hbar$. If you set $\hbar=c=1$ or many more constants equal to one, you would have to add general powers of all these constants. Dimensional analysis would lead to a set of linear equations for the exponents that you could solve.