Below are basic properties of quantum Gaussian packets describing the motion of a free particle on a line (taken to be the $$x$$ axis). Position representation A time-dependent Gaussian wave function, in its most general form, is given by $\psi(x,t) = \left( \frac{2 \operatorname{Re} \alpha_t}{\pi} \right)^{1/4} \exp \left( -\alpha_t (x - x_t)^2 + \frac{i}{\hbar} p_0 (x - x_t) + i \frac{p_0^2 t}{2 \hbar m} + \frac{i}{2} \arg \frac{\alpha_t}{\alpha_0} \right) \,,$ where $x_t = x_0 + \frac{p_0 t}{m} \,, \qquad \alpha_t = \frac{\alpha_0}{1 + i \frac{2 \hbar t}{m} \alpha_0} \,.$ Here, $$m$$ is the particle's mass, $$x_0$$ and $$p_0$$ are real-valued parameters, and $$\alpha_0$$ is a complex-valued parameter, such that $$\operatorname{Re} \alpha_0 > 0$$. The wave function $$\psi$$ is normalized, $\int_{-\infty}^{+\infty} dx \, |\psi(x,t)|^2 = 1 \,,$ and satisfies the free-particle Schrödinger equation, $i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{ ### 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
Consider a quantum particle of mass $$m$$ moving along a line, taken to be the $$x$$ axis, under the action of a spatially uniform time-dependent force $$F(t)$$. The time-evolution of the particle's wave function $$\Psi(x,t)$$ is governed by the Schrödinger equation, $i \hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi(x,t)}{\partial x^2} - F(t) x \Psi(x,t) \,.$ The solution to this equation can be written as $\Psi(x,t) = \exp \left[ i \frac{p(t) x - s(t)}{\hbar} \right] \Psi_0 \big( x-q(t), t \big) \,,$ where $$\Psi_0$$ is the free-particle wave function that initially coincides with $$\Psi$$, $\Psi_0(x,0) = \Psi(x,0) \,,$ and the functions $$p(t)$$, $$q(t)$$, and $$s(t)$$ are defined as $p(t) = \int_0^t d\tau \, F(\tau) \,,$ $q(t) = \frac{1}{m} \int_0^{t} d\tau \, p(\tau) = \frac{1}{m} \int_0^{t} d\tau \int_0^{\tau} d\tau' \, F(\tau') \,,$ \[s(t) = \frac{1}{2 m} \int_0^t d\tau \, p^2(\tau) = \frac{1}{2 m} \int_