Schrödinger's Equation — Question 4

Question 4

Amplitude decay and probability decay have different characteristic lengths

In a semi-infinite region with constant $V_0>E$, a stationary electron has $\psi(x)=Ce^{-\kappa x}$ for $x\ge0$ and $\kappa>0$. The potential exceeds the energy by 1.00 eV. Use $\hbar^2/(2m_e)=.0381\,\mathrm{eV\,nm^2}$. Use $-(\hbar^2/2m)\psi\prime\prime+V\psi=E\psi$.

1. Find $\kappa$ and the amplitude e-folding length.

2. Find the distance over which the density decreases to one tenth of its boundary value.

Question 4 - Solution

Amplitude decay and probability decay have different characteristic lengths

(a) Method and calculation

$$\kappa^2=(V_0-E)/(\hbar^2/2m_e)\quad\Rightarrow\quad\boxed{\kappa=5.12316\,\mathrm{nm}^{-1}}.$$$$\boxed{\ell_{\mathrm{amp}}=1/\kappa=.195192\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\rho(x)/\rho(0)=e^{-2\kappa x}=.100\quad\Rightarrow\quad x=\ln10/(2\kappa).$$$$\boxed{x=.224723\,\mathrm{nm}}.$$

Physical check

The density e-folding length is half the amplitude e-folding length. This real decaying solution alone carries zero probability current.