## Posts

### Harmonic oscillator with a time-dependent frequency

Consider a quantum particle in a harmonic trap with a time-dependent frequency $\omega = \omega(t) \,.$ In atomic units ($$\hbar = m = 1$$), the Schrödinger equation describing this system reads $i \dot{\psi} = -\frac{1}{2} \psi'' + \frac{1}{2} \omega^2 x^2 \psi \,,$ where $$\psi = \psi(x,t)$$ is the particle's wave function, and $$\; '$$ and $$\dot{}$$ denote $$\frac{\partial}{\partial x}$$ and $$\frac{\partial}{\partial t}$$, respectively. A general treatment of this problem can be found, for instance, in Propagator for the general time-dependent harmonic oscillator with application to an ion trap . Here we only construct an example solution to the Schrödinger equation above. Let us look for solutions $$\psi$$ in the form of a Gaussian wave packet centered at the origin: $\psi = \left( \frac{2 \alpha}{\pi} \right)^{1/4} e^{-(\alpha + i \beta) x^2 + i \gamma} \,,$ where $$\alpha = \alpha(t)$$, $$\beta = \beta(t)$$, and $$\gamma = \gamma(t)$$ are ye