# 量子现象 Physics 4- Quantum Phenomena PHYS231101

To calculate $R$ and $T$ we need to evaluate the current $J$ for $z<0$ and for $z>0$. Using the wavefunction from we have
$$J(z<0)=-\frac{|C|^{2} g \hbar k_{1}}{m^{}}\left(1-|r|^{2}\right)=J_{i}-J_{\mathrm{r}}$$ Similarly, using the wavefunction from eq. $(1.39)$, $$J(z>0)=-\frac{|C|^{2} q \Phi k_{2}}{m^{}}|t|^{2}=J_{1}$$
Hence from
\begin{aligned} &R=J_{1} / J_{1}=|r|^{2} \ &T=J_{3} / J_{1}=|r|^{2} k_{2} / k_{1} \end{aligned}

## PHYS231101 COURSE NOTES ：

Continuity Equation: Consider a single electron whose probability density is given
$$n(\mathbf{r}, t)=\Psi *(\mathbf{r}, t) \Psi(\mathbf{r}, t)$$
and whose probability current density is given
$$\mathbf{J}(\mathbf{r}, t)=-\frac{i q h}{2 m *}[(\nabla \Psi r) * \Psi-\Psi *(\nabla \Psi)]$$