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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 \[K(\boldsymbol{x},t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{x}|^2 \right) \,,\] the wave function \(\Psi(\boldsymbol{r},t>0)\) can be written as \[\begin{align} \Psi(\boldsymbol{r},t) &= \int_{\mathbb{R}^n} d^n\boldsymbol{r}' \, K(\boldsymbol{r}-\boldsymbol{r}',t) \Psi(\boldsymbol{r}',0) \\[0.2cm] &= \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}|^2 \right) \int_{\mathbb{R}^n} d^n\boldsymbol{r}' \, \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}'|^2 - \frac{i m}{\hbar t} \boldsymbol{r} \cdot \boldsymbol{r}' \right) \Psi(\boldsymbol{r}',0) \,. \end{align}\] Suppose that \(\Psi(\boldsymbol{r}',0)\) is non-negligible only if \(|\boldsymbol{r}'| < \sigma\). (That is, the initial wave function is assumed to be localized within a distance \(\sigma\) from the origin.) Then, for \(t \gg m \sigma^2 / \hbar\), we have \[\exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}'|^2 \right) = 1 + \mathcal{O} \left( \frac{m \sigma^2}{\hbar t} \right) \,.\] Hence, up to terms of order \(m \sigma^2 / \hbar t\), \[\Psi(\boldsymbol{r},t) \simeq \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}|^2 \right) \underbrace{ \int_{\mathbb{R}^n} d^n\boldsymbol{r}' \, \exp \left(-\frac{i}{\hbar} \frac{m \boldsymbol{r}}{t} \cdot \boldsymbol{r}' \right) \Psi(\boldsymbol{r}',0) }_{\Phi(m \boldsymbol{r} / \hbar)} \,.\]