# What are the units of time when planck's constant is equal to 1?

If I express a Hamiltonian $$H$$ in units of Hz by dividing the energy terms in the Hamiltonian by hbar $$\tilde{H}=\dfrac{H}{\hbar}$$ which means you set $$\hbar =1$$. Then what are the units of time? Also let's say you wanted to calculate the evolution of a system with $$U=e^{i/\hbar H t}$$ between $$t = 1$$ to $$t=10$$ s. What numbers do you have to use for $$t$$ when $$\tilde{U}=e^{i \tilde{H} t}$$. Do you have to multiply $$t$$ with $$\hbar$$ to obtain the same results?

It depends what else, if anything, you set to $$1$$. If nothing, time has energy dimension $$-1$$. If $$c=1$$, we can also say time has mass dimension $$-1$$. (You could even say the momentum dimension is $$-1$$, but no-one does.) In the convention $$c=\hbar=1$$, known as natural units, mass/energy dimensions are often just called dimensions, so time has dimension $$-1$$. If we also take $$G=1$$ as in Planck units, time can be nondimensionalised, with the Planck time given by $$\sqrt{G\hbar/c^5}$$. (Well, in our $$4$$-dimensional spacetime, anyway; the analogous result in $$n$$-dimensional spacetime, with one time dimension, is $$(G\hbar/c^{n+1})^{1/(n-2)}$$ provided $$n\ne 2$$.)
• What's "energy dimension -1"? Do you mean $(energy \, dimension)^{-1}$? – badjohn Nov 9 '18 at 16:03
• @badjohn The dimension $E^{-1}$ can just be called dimension $-1$ if everything has to be of the form $E^p$, but the prefix energy clarifies that it's powers of energy we're using. – J.G. Nov 9 '18 at 16:33
$$E=h\nu$$, so the units of frequency and energy become identical if you set h=1. The answer is therefore the inverse of whatever energy unit you have. You still have a lot of freedom to fix this.