- #1

Mr_Allod

- 42

- 16

- 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!

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

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!