# quantization with constraints

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