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I want to solve Schrödinger's equation with the potential $$V(x)=\frac{1}{2}mx^2+\lambda x$$ algebraically? Is there any way to construct ladder operators that are similar to the one for the harmonic oscillator in order to get the full Hamiltonian written as $H=k_1(b^*b+k_2)$?
A good reference may be suitable as well.
This is just the potential of a standard harmonic oscillator. The presence of the linear term is just due to the fact that you're using a coordinate system where the minimum of the potential is not at $x=0$. You can re-write it as $$V(x)=\frac{1}{2}mx'^2+c$$ where $x'=x+\frac{\lambda}{m}$ is the coordinate centered at the minimum of the potential and $c=-\frac{\lambda^2}{2m}$ is an overall constant.
