# Quantum mechanics with multiple values of $\hbar$

Having different Planck constants for different particles violates energy-momentum conservation unless the different types of particles do not interact with each other. This can be seen by following the arguments in "New Test of Quantum Mechanics: Is Planck's Constant Unique?", E. Fischbach, G.L. Greene, R.J. Hughes, Physical Review Letters 66 (1991) 256-259.

Consider a simple non-relativistic one-dimensional system of two spinless particles with the same mass $m$ but different Planck constants $h_A$ and $h_B$, interacting through a potential $V$. Their Hamiltonian is $$H=\frac{p^2_A}{2m}+\frac{p^2_B}{2m}+V(x_A-x_B) =\frac{P^2}{2M}+\frac{k^2}{m}+V(r)$$ where $r=x_A-x_B$, $k=(p_A-p_B)/2$, $M=2m$, and $P=p_A+p_B$. The Planck constants for each particle relate their momentum and position through the commutation relations $$[x_A,p_A]=i\hbar_A,\quad[x_B,p_B]=i\hbar_B, \qquad\textrm{with }[x_A,x_B]=[p_A,p_B]=0$$ A quantity is conserved if it commutes with the Hamiltonian, but we find that $$[H,P]=[V(r),P]=i(h_A-h_B)\frac{\partial V}{\partial r}$$ which is not zero unless either $h_A=h_B$ or $V(r)$ is independent of $r$ (i.e. there is no force between the particles). So for momentum to be conserved, the two particles must have the same Planck's constant or they must not interact.

Experimental constraints on differences in Planck's constant are set by how well theories such as quantum electrodynamics work. If different types of charged particles had different $h$, each would also have their own value for the fine structure constant $\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c}$. The extremely good agreement between measurements of $\alpha$ in systems involving different types of particles means that any differences in Planck's constant between those particles must be tiny. Fischbach, Greene, and Hughes set limits on fractional differences in the Planck's constants of electrons, photons, and neutrons at $< 10^{-7}$ in 1991, and newer measurements set even stronger constraints.

You may also want to look at the answer to the similar questions Why is Planck's constant the same for all particles? and Universality of Planck's Constant.