Mr_Allod
- 39
- 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!