= Solution
Inside the <rectangular potential barrier>, the electron's <wavefunction> is evanescent, with decay constant
$$
\kappa=\frac{\sqrt{2m_e(V_0-E)}}\hbar
=\frac{\sqrt{2m_e(2\,\mathrm{eV})}}\hbar
\simeq7.25\,\mathrm{nm}^{-1}.
$$
For width $a=1\,\mathrm{nm}$, the squared exponential suppression in the <quantum tunnelling> <probability> is
$$
\boxed{T\sim e^{-2\kappa a}=e^{-14.49}\simeq5.1\times10^{-7}.}
$$
This is the usual leading-exponential estimate. The <rectangular-barrier tunnelling probability>, retaining the interface-matching prefactor for equal potentials on the two sides, is
$$
T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}
\simeq\frac{16E(V_0-E)}{V_0^2}e^{-2\kappa a}.
$$
Here that prefactor is $16(8)(2)/10^2=2.56$, so the more accurate result is $\boxed{T\simeq1.30\times10^{-6}}$. Both estimates show that transmission is of order one part in a million. A classical particle with this <kinetic energy> cannot cross the barrier at all.
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