Question

For a particle in a finite potential well of width L and depth${\mathrm{U}}_0$ , what is the ratio of the probability Prob ( in $\mathrm{\delta x}$at $\mathrm{x}=\mathrm{L}+\mathrm{\eta }$) to the probability Prob ( in $\mathrm{\delta x}$ at$\mathrm{x}=\mathrm{L}$)?

Short answer

The probability ratio will be$0.135.$

Step-by-step solution

Given information

We have given,

Length of potential well =L

Depth = ${\mathrm{U}}_0$

We have to find the ratio of probability of the system.

Simplify

We know that the probability of any system is describe by

$\mathrm{P}=\int {\left|\mathrm{\phi }\left(\mathrm{x}\right)\right|}^2\mathrm{\delta x}$

Where $\mathrm{\phi }\left(\mathrm{x}\right)$is eigenstate of the system.

Then the ration will be found out as

$$\begin{gathered} \mathrm{ratio}=\frac{\mathrm{Probability}\mathrm{at}\mathrm{x}=\mathrm{L}+\mathrm{\eta }}{\mathrm{Probability}\mathrm{at}\mathrm{x}=\mathrm{L}} \\ \mathrm{ratio}=\frac{\int {\left|\mathrm{\phi }(\mathrm{L}+\mathrm{\eta })\right|}^2\mathrm{\delta x}}{\int {\left|\mathrm{\phi }\left(\mathrm{L}\right)\right|}^2\mathrm{\delta x}} \\ \mathrm{ratio}=\frac{{\left|\mathrm{\phi }(\mathrm{L}+\mathrm{\eta })\right|}^2}{{\left|\mathrm{\phi }\left(\mathrm{L}\right)\right|}^2} \end{gathered}$$

Since, we can use here the probability of parodic wave function. Then the wave function at the $\mathrm{L}+\mathrm{\eta }$will be

$\mathrm{\phi }(\mathrm{L}+\mathrm{\eta })=\mathrm{\phi }\left(\mathrm{L}\right){\mathrm{e}}^{-(\mathrm{x}-\mathrm{L})/\mathrm{\eta }}$

then,

$\mathrm{ratio}={\mathrm{e}}^{-2}=0.135$