这是一份uwa西澳大学PHYS3005的成功案例

The sudden removal of the shutter marks the beginning of a “quantum race” where the particles run along the positive $x$-axis. In order to elucidate the spreading of the signal all one has to do is to calculate $\psi(x, t)$ starting with
$$
\psi_{0}(x)=\psi(x, t=0)=\Theta(-x) \mathrm{e}^{\mathrm{i} / x}
$$
Using one finds
$$
\langle x \mid \psi(t \geq 0)\rangle=M(x ; k ; h t / m)
$$
where the Moshinsky function $\mathrm{M}$ is defined in terms of the complementary error function ,
$$
M(x ; k ; \tau)=\frac{1}{2} \exp \left(\mathrm{i} k x-\mathrm{i} k^{2} \tau / 2\right) \operatorname{erfc}\left[\frac{x-k \tau}{(2 \mathrm{i} \tau)^{1 / 2}}\right]
$$
with $\mathrm{i}^{1 / 2}=\exp (\mathrm{i} \pi / 4)$.
An interesting property of is revealed when we evaluate the particle number probability. Introducing $u=(h k t / m-x) /(\pi h t / m)^{1 / 2}$, we obtain

PHYS3005 COURSE NOTES ：

$$
\nabla \cdot \mathbf{j}(\mathbf{r})=-\frac{2}{h} \mathfrak{S}\left[\boldsymbol{\sigma}(\mathbf{r})^{} \psi_{\mathrm{sc}}(\mathbf{r})\right] $$ where $\mathfrak{S}[x]$ stands for the imaginary part of $x$. Thus, the inhomogeneity $\sigma(\mathbf{r})$ acts as a source for the particle current $j(r)$. By integration over the source volume, and inserting we obtain a bilinear expression for the total particle current $J(E)$, i. e., the total scattering rate: $$ J(E)=-\frac{2}{h} \mathfrak{J}\left[\int \mathrm{d}^{3} r \int \mathrm{d}^{3} r^{\prime} \sigma(\mathbf{r})^{} G\left(\mathbf{r}, \mathbf{r}^{\prime} ; E\right) \sigma\left(\mathbf{r}^{\prime}\right)\right]
$$
Some important identities concerning the total current $J(E)$ are most easily recognized in a formal Dirac bra-ket representation. In view we may express $J(E)$ by
$$
J(E)=-\frac{2}{h} S[\langle\sigma|G| \sigma\rangle]=\frac{2 \pi}{\hbar}\langle\sigma|\delta(E-H)| \sigma\rangle,
$$