= Solution
In the time-independent <Schrödinger equation>, $-\hbar^2\psi''/(2m)$ is the kinetic-energy operator acting on the state, $V\psi$ its potential-energy contribution and $E\psi$ the energy-eigenvalue term. The complex <wave function> is a probability amplitude: the <Born rule> assigns density $|\psi|^2$ for a normalized single-particle state. A plane-wave beam uses an incident-amplitude or flux normalization instead of a unit-integral wavefunction.
For a free plane wave, $\psi''=-k^2\psi$, so $\boxed{E=\hbar^2k^2/(2m)}$ and its time factor obeys $\hbar\omega=E$. Above the upward <potential step>, the bounded subthreshold solution has decay constant
$$
\boxed{\alpha=\frac{\sqrt{2m(V_0-E)}}\hbar.}
$$
For the PDF's unit incident amplitude, write $\psi=e^{ikx}+Re^{-ikx}$ on the left and $\psi=Te^{-\alpha x}$ on the right, omitting their common time phase. Continuity of both $\psi$ and $\psi'$ at this finite step gives
$$
1+R=T,\qquad ik(1-R)=-\alpha T,\qquad
R=\frac{k-i\alpha}{k+i\alpha},\quad T=\frac{2k}{k+i\alpha}.
$$
Consequently
$$
\boxed{|R|^2=1,\qquad |T|^2=\frac{4k^2}{k^2+\alpha^2}=\frac{4E}{V_0}.}
$$
The <evanescent amplitude versus transmitted current> distinction matters: the second quantity is a squared amplitude, \b[not a transmitted-flux probability]. The decaying tail has zero <probability current>, despite nonzero density and the possible value $|T|^2>1$.
Using $\hbar=1.05457\times10^{-34}\,\mathrm{J\,s}$ and the electron mass and elementary charge specified above, the energy difference1eV gives $\alpha=5.12\,\mathrm{nm^{-1}}$. Thus at the stated position
$$
\boxed{|\psi_t(1\,\mathrm{nm})|^2=\frac83e^{-2(5.123)(1)}\simeq9.46\times10^{-5}.}
$$
This is relative to the unit incident density in the source figure, not an absolute normalized probability per metre. <CODATA recommended values of the fundamental physical constants: 2022 by National Institute of Standards and Technology>[Reduced Planck constant, electron mass and electron volt conversion]{ref}
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1-ia2011-step-density.png]
{title=Evanescent step density}
{source=paper-1-ia2011-step-density.py}
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