# Quantum mechanical origin of pseudo forces

I am thinking about this from quite some time but could not come up with any satisfactory explanation. In a nutshell, how would one explain the pseudo forces felt by non-inertial observers given that the fundamental laws of physics are quantum mechanical?

Since in quantum mechanics one always talks about potentials instead of forces, I cannot think of anything that I can relate to the acceleration. In other words, given an electron for example, can we say that in the frame of an electron there is exists a pseudo force? I think no because of couple of reasons. First, it doesn't have a precise position to attach a reference frame to and second, even if we could do that, there is no way we would be able to talk about its acceleration because it doesn't exist in quantum mechanics. Which should also generalize to any macroscopic object that is simply made up of trillion quantum particles. Thus, I see no reason for pseudo force to exist for any observer in the first place! I am extremely confused about this. Any ideas?

• -1. Not useful, IMO. Your problem seems to be the "existence" of pseudo-forces. Trying to bring Quantum Mechanics into it does not make them any easier to understand. Try understanding pseudo-forces in Classical Mechanics first - eg Does centrifugal force exist? - and then perhaps the origin of real forces in QM. – sammy gerbil Jan 23 '17 at 19:07
• @sammygerbil You might want to add -2,-3 etc. "Not useful" : to whom? If you are not interested, leave it. "Try to understand Classical mechanics" : Thanks for the suggestion. I have doctorate in that, no worries. That simply ignores the real question. This is like asking a guy trying to work on quantum gravity to try to understand atoms first. – Peaceful Jan 24 '17 at 3:31
• I can only vote once, and it is my view that the question is not useful to the community of users. There is no indication in the question that you hold a doctorate. It seems to me that your arguments about pseudo-forces apply equally to real forces. ... Have you done any research on this topic? (Down-votes can also indicate lack of research effort.) Googling your title returns this paper at position #4. – sammy gerbil Jan 24 '17 at 4:13

In classical Lagrangian mechanics for a single particle moving in a non-inertial reference frame, pseudoforces can be described in terms of so-called generalized potentials. There is a Lagrangian $$L(t,{\bf x}, \dot{\bf x}) =m \dot{\bf x}^2/2 + {\bf A}(t, {\bf x}) \cdot \dot{\bf x} + u(t,{\bf x})$$ where $A$ an $u$ are respectively a vector field and a scalar field constructed out of the known motion of the reference frame with respect a given inertial reference frame. Euler-Lagrange equations produce the usual law of motion including centrifugal, Coriolis force, and so on if you want to describe the dynamics of the particle in Newtonian fashion. However this approach only relies upon the Lagrangian/Hamiltonian framework.
N.B.: The case of the electromagnetic interaction is treated exactly with the same formalism and, up to factors including the value of the charge, ${\bf A}$ and $u$ are the given vector and scalar potentials.
The afore-mentioned Lagrangian defines a corresponding Hamiltonian via the standard Legendre transformation. $$H(t,{\bf x}, {\bf p}) = \frac{({\bf p}-{\bf A}(t,{\bf x}))^2}{2m} - u(t,{\bf x})$$ Replacing canonical coordinates for the corresponding operators, you have the quantum Hamiltonian operator.