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We all know Schrödinger's equation $$\mathrm{i}\hbar \partial_t |\Psi\rangle = H|\Psi\rangle$$ I'm trying to figure out why we multiply by $\hbar$ instead of e.g. $h$. What is causing us to specifically use $\hbar$? Any ideas are appreciated.

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If we consider, e.g., the stationary solutions: $$ H|\psi_n\rangle = E_n|\psi_n\rangle,\\ |\Psi_n(t)\rangle = e^{-\frac{iE_n t}{\hbar}}|\psi_n\rangle, $$ we see that the time phase is: $$ \frac{E_n t}{\hbar} = \omega_n t, $$ which is consistent with the DeBroglie relation $$ E=\hbar\omega=2\pi \hbar\nu $$

In other words, if we expect the oscillations with frequency $\nu$, they will appear in a complex exponent with an additional factor of $2\pi$: $$e^{i2\pi \nu t}=e^{i\omega t} $$

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