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 $$m \sigma^2 / \hbar t$$, $\Psi(\boldsymbol{r},t) \simeq \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}|^2 \right) \underbrace{ \int_{\mathbb{R}^n} d^n\boldsymbol{r}' \, \exp \left(-\frac{i}{\hbar} \frac{m \boldsymbol{r}}{t} \cdot \boldsymbol{r}' \right) \Psi(\boldsymbol{r}',0) }_{\Phi(m \boldsymbol{r} / \hbar)} \,.$