The shifted harmonic oscillator

Problems involving harmonic oscillators that are shifted in their equilibrium positions relative to some preset origin are ubiquitous in simple models of quantum dynamical processes. We consider a few examples in this section. 

1 Harmonic oscillator under an additional constant force 

Consider a particle of charge q moving in one dimension (along the x-axis, say) in a harmonic potential. The Hamiltonian describing its motion is 

Let us switch on an external uniform electrostatic field along the same direction. The Hamiltonian becomes 

It is easy to find the eigenfimctions and eigenvalues of the Hamiltonian (2.169) given the corresponding eigenfunctions and eigenvalues of (2.168). Making the 

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