The standard Damped one-dimensional Harmonic Oscillator with sinusoidal driving force has equation
$$\frac{d^2}{dt^2}x(t)+2\zeta\omega_0\frac{d}{dt}x(t)+\omega_0^2x(t)=\frac{1}{m}F_0\sin(\omega t).$$
Here is $\zeta>0$ for damping.
The solutions of this system are linear combinations of a transient function (that goes to 0 with time) and a steady-state solution that is periodic with frequency $\omega$ (see Wikipedia).
I would like to quantize this to a Quantum Harmonic Oscillator. In the case without damping and driving force we get
$$i\hbar \frac{\partial}{\partial t}\psi = \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac12
m\omega_0^2x^2\right)\psi$$
To add the driving force, we can create a potential energy field $U(x,t)$ such that the driving force is the gradient of this. Then add $U$ to the quantum hamiltonian:
$$i\hbar \frac{\partial}{\partial t}\psi = \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac12
m\omega_0^2x^2+F_0\sin(\omega t)x\right)\psi$$
Now we have to add the damping term. But I am not sure how to do that. My question is then: Are the steady-state solutions of this system also periodic with frequency $\omega$? Is there an easy way to prove that?
Best Answer
It is usually convenient to write problems about a quantum harmonic oscillator in terms of the creation-annihilation operators $a^\dagger$, $a$. The Schrödinger equation now looks like $$ i\hbar\frac{\partial}{\partial t}|\psi, t\rangle = \left(\hbar\omega_0\ a^\dagger a + \alpha\sin(\omega t)(a+a^\dagger)\right)|\psi,t\rangle $$ Let's look for a solution to this equation in the form of a coherent state: $$ |\psi,t\rangle = A(t)\exp(\phi(t)\ a^\dagger)|0\rangle $$ The functions $A(t)$, $\phi(t)$ must satisfy the following equations $$ i\hbar\dot{A} = A\alpha\sin(\omega t)\phi,\quad i\hbar\dot{\phi} = \hbar\omega_0\phi + \alpha\sin(\omega t) $$ These equations posses periodic solution of the form $$ \phi(t) = -\frac{\alpha}{2i\hbar}\left(\frac{e^{i\omega t}}{\omega+\omega_0} + \frac{e^{-i\omega t}}{\omega-\omega_0}\right), $$ $$ A(t) = A(0)\exp\left(-i\frac{\alpha^2}{2\hbar^2}\frac{\omega_0}{\omega^2-\omega_0^2}t \right)\exp\left(-\frac{\alpha^2}{8\hbar^2\omega}\left( \frac{e^{i2\omega t}}{\omega+\omega_0} + \frac{e^{-i2\omega t}}{\omega-\omega_0} \right)\right) $$ Due to the phase factor in $A(t)$, the vector $|\psi,t\rangle$ is not strictly periodic with frequency $\omega$. However, the average value of the coordinate, for example, demonstrates the expected time dependence $$ \langle x\rangle \sim \langle a + a^\dagger\rangle = \frac{\alpha}{\hbar} \frac{2\omega_0}{\omega^2-\omega_0^2}\sin(\omega t) $$ I suppose there may be other solutions with properties similar to those of an explicitly constructed solution.