Quantum Harmonic Oscillator with Additional Potential

Mr_Allod
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Homework Statement
Apply and additional potential ##V'(x) = \alpha x## to the standard Hamiltonian of a harmonic oscillator. Find the solutions to the Schrodinger equation.
Relevant Equations
Harmonic Oscillator Hamiltonian: ##H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac {m\omega^2 x^2}{2}##
Hello there, I am trying to solve the above and I'm thinking that the solutions will be Hermite polynomials multiplied by a decaying exponential, much like the standard harmonic oscillator problem. The new Hamiltonian would be like so:

$$H = - \frac \hbar {2m} \frac {d^2}{dx^2}\psi + \frac {m\omega^2 x^2}{2} + \alpha x$$

Normally one would make the substitutions:

$$y = \sqrt{\frac {m\omega}{\hbar}}$$
$$\epsilon = \frac {2E}{\hbar\omega}$$

This would produce a solvable dimensionless differential equation:
$$\frac {d^2}{dy^2}\psi + (\epsilon-y^2)\psi = 0$$

Now I'm having trouble finding the correct substitution to make to reduce the new problem to a dimensionless one like above. I would appreciate it if someone could give me some suggestions, thank you!
 
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Rather think about the Hamiltonian a bit first.

Hint: Think about the classical case. What's the change of the potential due to the additional linear piece? Equivalently you can think about, what it changes for the force acting on the particle and how does it affect the solutions of the classical equations of motion?
 
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