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In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
I guess mathematics wouldn't work and you would not be able to reduce those operators into entities with commutation operators leading to simple algebraic relations. Plus this harmonic potential corresponds to free fields, and this "freeness" i guess is the physical reason why things are simple and tractable and meaningful.In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.