Skip to main content

Posts

Showing posts with the label Approximations

Free particle: Long time evolution

Consider a free quantum particle of mass \(m\) moving in an \(n\)-dimensional space. Let \(\Psi(\boldsymbol{r}, t)\) be the particle's wave function, with \(\boldsymbol{r} \in \mathbb{R}^n\) and \(t \ge 0\) denoting position and time, respectively. Suppose that initially, at \(t = 0\), the wave function \(\Psi(\boldsymbol{r},0)\) is localized around \(\boldsymbol{r} = 0\). Then, at long times (\(t \to \infty\)), the wave function is approximately given by \[\Psi(\boldsymbol{r},t) \simeq \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}|^2 \right) \Phi \left( \frac{m \boldsymbol{r}}{t} \right) \,,\] where \[\Phi(\boldsymbol{p}) = \int_{\mathbb{R}^n} d^n\boldsymbol{r} \, \Psi(\boldsymbol{r},0) e^{-i \boldsymbol{p} \cdot \boldsymbol{r} / \hbar}\] is the initial wave function in momentum space. Proof: Using the free-particle propagator \[K(\boldsymbol{x},t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{