My quantum mechanics book says that $ħ$ is the Planck's constant. The book uses ħ throughout and not one single use of $h$.
My statistical mechanics book says that $h$ is the Planck's constant and doesn't use $ħ$ at all.
Now I know that one of the constant is the other scaled by $2\pi$. But one of them is the Planck's constant and the other is not. Which one of them is true Planck's constant?
In the usual terminology we have \begin{align} h &&&\text{Planck's constant} \\ \hbar &= \frac{h}{2\pi} &&\text{reduced Planck's constant} \end{align}
The significance of $2\pi$ here is the ratio between a full circle and a radian, because the energy of a photon is $$ E = hf = \hbar \omega \;,$$ where $f$ is the cyclic frequency of the light and $\omega = 2 \pi f$ is its angular frequency. Both are common because—by long tradition—the frequency and wavelength of waves are generally measured with respect to a full cycle, but mathematical expressions involving waves may be written down more compactly in terms of angular (radian-based) quantities such as the angular frequency and the wavenumber ($k = 2\pi/\lambda$).
It is $h$. $\hbar$ is $\frac{h}{2π}$.
Planck constant $h$ reduced constant $\hbar = \frac{h}{2\pi}$
Have a look at the original: 10.1002/andp.19013090310. Planck uses $h$ has it is about the relation of frequency and energy.
