(For simplicity I work in 1+1 dimensions)
In this post I show how one can define the wavefunction of a particle from its trajectory and how it's consistent with the orthodox definitions of quantum mechanical operator.
Let
ψ:=x−i
Then dψdt=∂ψ∂t=−ix−i−1dxdt=−ix−ixv=−iψvx,
but from the above definition vx=ω,
so finally
∂ψ∂t=−iψω,
which is
ˆω=i∂∂t,
when expressed as an operator/eigenvalue problem.
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