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basically, as far as I know we can derive 1/2mv

∫F⋅ds=1/2mv

for wave equation we use Hamiltonian H=P

However, I wonder how we can say that P

∫F⋅ds=∫(dp/dt)⋅ds=1/2mv

Then, we can say that P

Then KE is dependent on both s and p, and this seems a bit ambiguous to me because in Hamiltonian KE is dependent on momentum and potential energy is dependent on position. Dimension-wise, it would agree with energy but I wonder how we took (p

^{2}from∫F⋅ds=1/2mv

^{2}=(p^{2})/2mfor wave equation we use Hamiltonian H=P

^{2}/2m+V where P and V are both operatorsHowever, I wonder how we can say that P

^{2}/2m is the term for kinetic energy because∫F⋅ds=∫(dp/dt)⋅ds=1/2mv

^{2}is saying that knowing F and path s, we can determine (p^{2})/2m. and given with an appropriate boundary condition, vice versa.Then, we can say that P

^{2}/2m is a functional F[p,s]=∫(dp/dt)⋅ds in which we have to determine p and s separately?Then KE is dependent on both s and p, and this seems a bit ambiguous to me because in Hamiltonian KE is dependent on momentum and potential energy is dependent on position. Dimension-wise, it would agree with energy but I wonder how we took (p

^{2})/2m from ∫F⋅ds=∫(dp/dt)⋅ds=1/2mv^{2}
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