91Ó°ÊÓ

Probability density is a function whose value at any given sample in the sample space gives the likelihood of that random value would be close to that sample.

Consider the given function

${\left|{\mathrm{ψ}}_{\mathrm{x}<0}\right|}^2={\left|{\mathrm{Ae}}^{\mathrm{ikx}}+{\mathrm{Be}}^{-\mathrm{ikx}}\right|}^2$

$\mathrm{B}=-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}\mathrm{A}$ ….. (1)

Here, data-custom-editor="chemistry" $\mathrm{ψ}$is the wave function, data-custom-editor="chemistry" $\mathrm{A},\mathrm{B},\mathrm{Î\pm }$is the Arbitrary constants$\mathrm{k}=\frac{\sqrt{2\mathrm{mE}}}{\mathrm{h}}$

${\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|{\mathrm{Ae}}^{\mathrm{ikx}}+{\mathrm{Be}}^{-\mathrm{ikx}}\right|}^2$

$\mathrm{B}=-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}\mathrm{A}$ …..(2)

After using equation (1) in equation (2), and taking common from the Right-Hand-Side, you get,

role="math" localid="1660032485327" ${\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2{\left|{\mathrm{e}}^{\mathrm{ikx}}-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}{\mathrm{e}}^{-\mathrm{ikx}}\right|}^2$

After splitting the squared expression and further solving it as:

$$\begin{gathered} {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left({\mathrm{e}}^{\mathrm{ikx}}-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}{\mathrm{e}}^{-\mathrm{ikx}}\right)\left({\mathrm{e}}^{-\mathrm{ikx}}-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}{\mathrm{e}}^{\mathrm{ikx}}\right) \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(1-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}{\mathrm{e}}^{-2\mathrm{ikx}}-\frac{\mathrm{Î\pm }+\mathrm{ik}}{\mathrm{Î\pm }-\mathrm{ik}}{\mathrm{e}}^{2\mathrm{ikx}}+1\right) \end{gathered}$$

Using${\mathrm{Î\pm }}^2+{\mathrm{k}}^2$as the common denominator solve as:

$$\begin{gathered} {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(\frac{2\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)-{\left(\mathrm{Î\pm }+\mathrm{ik}\right)}^2{\mathrm{e}}^{-2\mathrm{ikx}}-{\left(\mathrm{Î\pm }-\mathrm{ik}\right)}^2{\mathrm{e}}^{2\mathrm{ikx}}}{\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)}\right) \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(\frac{2\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)-\left({\mathrm{Î\pm }}^2-{\mathrm{k}}^2+2\mathrm{¾\pm °ìÎ\pm }\right){\mathrm{e}}^{-2\mathrm{ikx}}-\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2-2\mathrm{¾\pm °ìÎ\pm }\right){\mathrm{e}}^{2\mathrm{ikx}}}{\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)}\right) \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(\frac{2\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)-\left({\mathrm{Î\pm }}^2-{\mathrm{k}}^2\right)\left({\mathrm{e}}^{-2\mathrm{ikx}}+{\mathrm{e}}^{2\mathrm{ikx}}\right)+2\mathrm{¾\pm °ìÎ\pm }\left({\mathrm{e}}^{-2\mathrm{ikx}}-{\mathrm{e}}^{2\mathrm{ikx}}\right)}{\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)}\right) \end{gathered}$$

Now, after using Euler’s equation:${\mathbf{e}}^{\mathbf{¾\pm θ}}\mathbf{=}\mathbf{³¦´Ç²õθ}\mathbf{+}\mathbf{¾\pm ²õ¾\pm ²Ôθ}$ and further solving it as:

$$\begin{gathered} {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(\frac{2\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)-\left({\mathrm{Î\pm }}^2-{\mathrm{k}}^2\right)2\cos \left(2\mathrm{kx}\right)+4\mathrm{°ìÎ\pm ²õ¾\pm ²Ô}\left(2\mathrm{kx}\right)}{\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)}\right) \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2={\left|\mathrm{A}\right|}^2\left(\frac{2\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)-\left({\mathrm{Î\pm }}^2-{\mathrm{k}}^2\right)\left({\cos }^2\mathrm{kx}-{\sin }^2\mathrm{kx}\right)+4\mathrm{°ìÎ\pm }\left(2\mathrm{sinkxcoskx}\right)}{\left({\mathrm{Î\pm }}^2+{\mathrm{k}}^2\right)}\right) \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2=4{\left|\mathrm{A}\right|}^2{\left(\frac{\mathrm{Î\pm }}{\sqrt{{\mathrm{Î\pm }}^2+{\mathrm{k}}^2}}\mathrm{sinkx}-\frac{\mathrm{k}}{\sqrt{{\mathrm{Î\pm }}^2+{\mathrm{k}}^2}}\mathrm{coskx}\right)}^2 \\ {\left|{\mathrm{ψ}}_{\mathrm{I}}\right|}^2=4{\left|\mathrm{A}\right|}^2{\left(\mathrm{³¦´Ç²õθsinkx}-\mathrm{²õ¾\pm ²Ô賦´Ç²õ°ì³\ae }\right)}^2 \end{gathered}$$

Hence, the wave function is obtained as ${\left|\mathrm{ψ}\right|}^2=4{\left|\mathrm{A}\right|}^2{\sin }^2\left(\mathrm{kx}-\mathrm{θ}\right)$

Here,$\mathrm{θ}={\tan }^{-1}\left(\frac{\mathrm{k}}{\mathrm{Î\pm }}\right)$

If k=0, $\mathrm{θ}=0^{\mathrm{o}}$and D=0. The wave is zero at the step. The step is relatively so high that the wave doesn’t penetrate it. If $\mathbf{Î\pm }\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{θ}\mathbf{=}{\mathbf{90}}^{\mathbf{o}}\mathbf{}\mathbf{and}\mathbf{}\mathbf{D}\mathbf{=}\mathbf{2}\mathbf{A}$. The wave is maximum at the step and there is much penetration.

Hence, the required parameters are obtained as $\mathbf{θ}\mathbf{=}{\mathbf{90}}^{\mathbf{o}}\mathbf{}\mathbf{and}\mathbf{}\mathbf{D}\mathbf{=}\mathbf{2}\mathbf{A}$.