A particle of mass m and energy E moving in a region where there is initially no potential energy encounters a potential dip of width L and depth $\mathit{U}\mathbf{=}\mathbf{-}{\mathit{U}}_{\mathbf{o}}$.

$$\begin{gathered} \mathit{U}\left(x\right)\mathbf{=}\left\{0x⩽0 \\ -U_{o}0<x<L \\ 0x⩾L\right\} \end{gathered}$$

Show that the reflection probability is given by

$\mathit{R}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\left[\sqrt{2m\left(E+U_o\right)}L/\hslash \right]}{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\left[\sqrt{2m\left(E+U_o\right)}L/\hslash \right]\mathbf{+}\mathbf{4}\left(E/U_o\right)\left[\left(E/U_o\right)+1\right]}$

(Hint: All that is needed is an appropriate substitution in a known probability.)

In a potential step, the expression for the reflection probability is

$R=\frac{{\sin }^2[\sqrt{2\mathrm{m}(\mathrm{E}+{\mathrm{U}}_{\mathrm{o}})}\mathrm{L}/\mathrm{\hslash }]}{{\sin }^2[\sqrt{2\mathrm{m}(\mathrm{E}+{\mathrm{U}}_{\mathrm{o}})}\mathrm{L}/\mathrm{\hslash }]+4(\mathrm{E}/{\mathrm{U}}_{\mathrm{o}})[(\mathrm{E}/{\mathrm{U}}_{\mathrm{o}})+1]}$

where,

R is the reflection probability

E is the energy of the particle.

$U_o$is the least potential energy in the potential dip.

m is the mass of the particle.

It is often assumed that the particle gets reflected only off the potential step because the potential energy increases. But the particle can be reflected off a potential drop/dip. Finding the reflection probability in this case is similar to finding reflection probability in the potential step. Replacing $U_{o}by-U_o$in the above expression will lead to reflection probability in potential drop.

$$\begin{gathered} R=\frac{{\sin }^2\left[\sqrt{2m\left(E-\left(-U_o\right)\right)}L/\hslash \right]}{{\sin }^2\left[\sqrt{2m\left(E-\left(-U_o\right)\right)}L/\hslash \right]+4\left(E/\left(-U_o\right)\right)\left[\left(E/\left(-U_o\right)\right)-1\right]} \\ =\frac{{\sin }^2\left[\sqrt{2m\left(E+U_o\right)}L/\hslash \right]}{{\sin }^2\left[\sqrt{2m\left(E+U_o\right)}L/\hslash \right]+4\left(E/\left(U_o\right)\right)\left[\left(E/U_o\right)+1\right]} \end{gathered}$$

Thus, the expression for reflection probability is proved by replacing $U_{o}by-U_o$.