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let be a Hamiltonian system $ H= H(x,p) $

for this system there is a conserved quantity namely $ C=xp $ so the total system is invariant under rotation

if we 'quantizy' this function $ Cf(x)=\lambda _{n}f(x) $ the result is known and

$ f(x)= Ax^{{1/2}+i\lambda _{n}} $ for some constant 'A'

then how is this function $ f(x) $ related to the quantization of the Hamiltonian

$ H\Psi (x,t)= E_{n}\Psi (x,t) $

for example if the system has a SYMMETRY so there is a conserved quantity (there can be more constants of motion of course) how does this affect to the quantization of the TOTAL hamiltonian ??

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Which symmetry gives rise to that conserved quantity? The operator corresponding to that symmetry's generator will commute with the quantized Hamiltonian. – Chay Paterson Jul 20 '13 at 17:39

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