91影视

A particle is defined by the wave function: $\mathit{B}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}$ for $\mathit{x}\mathbf{<}\mathbf{0}$and$\mathit{C}{\mathit{e}}^{\mathbf{4}\mathbf{x}}$ for$\mathit{x}\mathbf{>}\mathbf{0}$ . For the given wave function to becontinuous, at$\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{B}\mathbf{=}\mathit{C}$

To left of the step $x<0$as:

$$\begin{gathered} 蠄={蠄}_{inc}+{蠄}_{ref} \\ 蠄=e^{ikx}+B{e}^{-ikx} \end{gathered}$$

To the right of the step $x>0$as:

$蠄=C{e}^{-伪x}$

Her, $蠄$must be continuous at $x=0$.

$$\begin{gathered} e^0+B{e}^0=C{e}^0 \\ 1+B=C \end{gathered}$$

Here, $\frac{d蠄}{dx}$ must be continuous at $x=0$.

$$\begin{gathered} ik{e}^0-ikB{e}^0=-伪C{e}^0 \\ ik\left(1-B\right)=-伪C \end{gathered}$$

From first and second condition:

$$\begin{gathered} ik\left(1-B\right)=伪\left(1+B\right) \\ \frac{ik+伪}{ik-伪}=\frac{i\frac{\sqrt{2mE}}{h}+\frac{\sqrt{2m\left(\frac{5}{4}E-E\right)}}{h}}{i\frac{\sqrt{2mE}}{h}-\frac{\sqrt{2m\left(\frac{5}{4}E-E\right)}}{h}} \end{gathered}$$

Divide by 鈭�2mE/$\mathrm{鈩�}$everywhere:

$$\begin{gathered} B=\frac{i+\sqrt{\left(\frac{5}{4}-1\right)}}{i-\sqrt{\left(\frac{5}{4}-1\right)}}=\frac{i+\frac{1}{2}}{i-\frac{1}{2}} \\ B=\frac{3}{5}-i\frac{4}{5} \end{gathered}$$

Substitute this in C, the equation obtained is:

$C=\frac{8}{5}-i\frac{4}{5}$

If,${蠄}_{refl}=B{e}^{-伪x}$

By putting value of B from Step 3 in the above-mentioned equation and solve:

${蠄}_{refl}=\left(\frac{3}{5}-i\frac{4}{5}\right)e^{-伪x}$

If, ${蠄}_{x>0}=C{e}^{-伪x}$

By putting value of C from Step 3 in the above-mentioned equation solve as:

${蠄}_{x>0}=\left(\frac{8}{5}-i\frac{4}{5}\right)e^{-伪x}$

Hence, the reflected wave is given by, ${蠄}_{refl}=\left(\frac{3}{5}-i\frac{4}{5}\right)e^{-伪x}$ and the wave inside the step can be written as, ${蠄}_{x>0}=\left(\frac{8}{5}-i\frac{4}{5}\right)e^{-伪x}$

The ratio of reflected and incident probability can be calculated by the following equation,

$$\begin{gathered} B^{*}B=\left(\frac{3}{5}+i\frac{4}{5}\right)\left(\frac{3}{5}-i\frac{4}{5}\right) \\ =1 \end{gathered}$$