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amplitude approximation associated assumed beam behavior calculate canonical Chapter classical mechanics commutation relations component Consequently considered constant coordinates corresponding cross section defined degeneracy density matrix diagonal discussed effect eigen eigenfunctions eigenvalue equation energy levels ensemble equal equations of motion example expectation value expression flux formalism frequency given by Eq gives Hamiltonian Hermitian Hermitian adjoint hydrogen atom incident integral interaction kinetic energy ladder operator linear lowest energy magnetic field matrix elements measurement momenta nucleus obtained one-dimensional operator Q orthogonal parity perturbation photon physical plane wave Poisson bracket polarization position possible potential energy probability problem quantization quantum mechanics quantum numbers radiation region representation represents result rotating scattering Schrodinger equation seen Show simple harmonic oscillator singlet solution space spherical spin angular momentum square statistics symmetric theory tion triplet unitary transformation vanish variables vector velocity wave function wave packet wavelength written z-component z-direction zero