An Electron in a Finite Well — Question 5

Question 5

Conditional penetration depth follows the probability decay rather than amplitude decay

A finite-well bound state has a right-hand tail $\psi(x)=Be^{-\kappa(x-a)}$ for $x>a$, with $\kappa=4.00\,\mathrm{nm}^{-1}$. Condition on the electron being detected to the right of the well, and set $y=x-a\ge0$.

1. Normalize the conditional density of y and find its mean.

2. Find the conditional probability of penetrating more than $.300\,\mathrm{nm}$.

Question 5 - Solution

Conditional penetration depth follows the probability decay rather than amplitude decay

(a) Method and calculation

$$f(y\mid x>a)=\frac{|B|^2e^{-2\kappa y}}{|B|^2/(2\kappa)}=\boxed{2\kappa e^{-2\kappa y}}.$$$$\boxed{\langle y\rangle=\int_0^\infty y(2\kappa)e^{-2\kappa y}dy=\frac1{2\kappa}=.125\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$P(y>d\mid x>a)=\int_d^\infty2\kappa e^{-2\kappa y}dy=e^{-2\kappa d}.$$$$\boxed{P(y>.300\,\mathrm{nm}\mid x>a)=e^{-2.4}=.0907180}.$$

Physical check

This is conditional on being outside on the right. The unconditional probability also includes the probability of reaching that side of the tail.