In non-relativistic Quantum Mechanics how does one write the momentum and position operator in a non-inertial reference frame? How is the Schrodinger Wave equation modified to account for non-zero acceleration? Does the commutation relation $px-xp= - i \hbar$ hold?
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Sometimes one can reason as follows:
We can write the action as $S=\int dS,$ where $dS=pdq-Hdt$, and the integral is along a curve in $P\times \mathbb{R}$, where $P$ is phase space and $\mathbb{R}$ is an extra time axis.
Now if one works in a non-inertial frame, one will be given non-inertial coordinates $\tilde q(q,t).$ If you manage to define additional coordinates $\tilde p(q,p,t)$ such that $dS=\tilde p d\tilde q-\tilde H dt$ for some other function $\tilde H$, then $\tilde p$ and $\tilde q$ will be a canonical pair at equal times and the Hamiltonian $\tilde H$ correctly takes into account any pseudo-forces arising due to the non-inertial frame.
The system can now be quantised in the normal manner by putting $\tilde q \mapsto x,$ $\tilde p \mapsto -id/dx$ and $\tilde H$ becomes the Hamiltonian operator.
Basically, a system where you can play the above trick with $dS$ is one where the pseudo forces allow for a Hamiltonian formulation. It should also be noted that the definition of $\tilde p$ can be quite complicated, and its physical interpretation can be quite complicated.
Anton Quelle
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