Question of the Day: Can a particle be detected beyond a barrier even when classical mechanics says it should not cross?
Learning objectives
Notebook 03 established quantized stationary states in an infinite well. We now ask whether its perfect exclusion is a special limit.
Notebook 03 used an infinite hard wall: the wave function was forced to zero at the boundary and no probability existed beyond it. This notebook replaces that idealization by a finite rectangular barrier of height $V_0$ and width $a$. The question is whether a particle with energy $E<V_0$ can still be detected on the far side.
A classically forbidden region should be impenetrable, yet quantum mechanics predicts that a stationary wave can still have non-zero amplitude beyond a finite barrier. The experiment asks how that transmitted signal depends on barrier properties and why a tiny change in width or distance can change the signal by orders of magnitude — the physics behind tunnelling devices and STM.
Prediction. For $E<V_0$, do you expect the transmitted signal to change by a small additive amount or by a multiplicative factor when the barrier becomes slightly wider? What would classical mechanics predict for the same change?
We send electrons of energy $E$ toward a one-dimensional rectangular barrier of height $V_0$ and width $a$. The barrier is finite, so the wave can have different character in the incident, barrier, and transmitted regions.
What the student changes: incident energy $E$ (below the barrier) and barrier width $a$ (barrier height and particle mass remain fixed at $V_0=4.0$ eV for an electron).
What the detector reports: the fraction of incident flux that is reflected and the fraction that is transmitted.
Before interpreting, note whether a non-zero transmitted signal appears at all for $E<V_0$, and how that signal responds when you make the barrier slightly wider or bring the incident energy closer to the barrier height.
Mathematical Window — evanescent wave in the barrier
For the rectangular barrier with $0 < E < V_0$:
$$ k = \frac{\sqrt{2 m E}}{\hbar} $$
$$ \kappa = \frac{\sqrt{2 m (V_0 - E)}}{\hbar} $$
The barrier region supports the evanescent basis
$$ \psi_{\mathrm{II}}(x) = C \exp(\kappa x) + D \exp(-\kappa x) $$
and the stationary matching requires $\psi$ and $d\psi/dx$ continuous at $x = 0$ and $x = a$.
Interpretation: $\kappa$ is real for $E < V_0$; $1/\kappa$ is the amplitude decay length. The full solution uses both $\exp(\pm\kappa x)$ terms. Matching to the far-side propagating wave permits a non-zero transmitted flux even though the interior wave shows evanescent attenuation through the finite barrier.
Question. How much can a fraction of a nanometre change the transmission?
Prediction. If barrier width increases by the same small amount each time, do you expect transmission to fall by the same absolute amount, the same factor, or neither?
Vary the barrier width $a$ at fixed electron energy $E = 2.0$ eV and compare three barrier heights: $V_0 = 3$, $4$, and $5$ eV. The curves predict the exact transmitted flux fraction $T$, rather than individual detector events. The ideal calculation assumes one-dimensional, stationary scattering at a lossless rectangular barrier with the same asymptotic potential on both sides.
The exact transmission $T$ falls by orders of magnitude as the barrier width $a$ increases. On the logarithmic transmission axis the thick-barrier behavior approaches a near-linear trend. Larger $V_0 - E$ suppresses transmission more strongly at every width. The three curves are exact finite-barrier results.
Mathematical Window — exact and opaque transmission
Exact for the finite rectangular barrier with $0 < E < V_0$:
$$ T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1} $$
$$ R = 1 - T $$
Opaque-barrier approximation for $\kappa a \gg 1$:
$$ T \propto \exp(-2\kappa a) $$
The first relation is exact for this model; the last is an approximation for the opaque barrier when $\kappa a \gg 1$ and explains the observed exponential sensitivity — it does not replace the exact calculation used for F2.
Here $T = J_{\mathrm{trans}}/J_{\mathrm{inc}}$ and $R = |J_{\mathrm{refl}}|/J_{\mathrm{inc}}$ with $R + T = 1$ for the symmetric lossless barrier. $R$ and $T$ are flux fractions, not normalized spatial probabilities.
Because $\kappa \propto \sqrt{m}$, increasing particle mass suppresses tunnelling more strongly under otherwise identical conditions. Mass is not varied in this experiment.
Question. Why can moving an STM tip by only a small fraction of a nanometre change the measured current so strongly?
Prediction. If the tip moves only $0.10$ nm closer to the sample, would you expect the relative current to change slightly, by a fixed additive amount, or by a large multiplicative factor?
We use a representative effective barrier $\Phi_{\mathrm{eff}} = 4.0$ eV for an electron. The experiment varies the tip-sample separation $d$ while the reference distance is fixed at $d_{\mathrm{ref}} = 0.50$ nm.
A fraction-of-a-nanometre change in $d$ causes a large multiplicative change in $I/I_{\mathrm{ref}}$; the logarithmic graph makes the exponential sensitivity visible. Current increases when the tip moves closer and decreases when it moves farther — the simplified physical origin of STM's extreme vertical sensitivity.
Mathematical Window — STM relative current
Approximate effective barrier:
$$ \kappa_{\mathrm{STM}} \approx \frac{\sqrt{2 m_e \Phi_{\mathrm{eff}}}}{\hbar} $$
Relative current:
$$ \frac{I(d_2)}{I(d_1)} = \exp[-2\kappa_{\mathrm{STM}}(d_2-d_1)] $$
Interpretation: $\Phi_{\mathrm{eff}}$ is a representative effective barrier; the relation predicts relative distance sensitivity in this simplified model; if $d_2<d_1$ the ratio $>1$, if $d_2>d_1$ ratio $<1$; no absolute-current formula is used.
The simple rectangular-barrier model explains exponential distance sensitivity, not the full quantitative current of a real STM. It omits: 1) detailed tip shape; 2) applied bias dependence; 3) electronic density of states; 4) different tip and sample work functions; 5) non-rectangular barrier shape; 6) constant-current feedback; 7) surface chemistry/contamination.
Therefore an STM image is not simply a geometric height map — electronic structure and feedback also affect contrast.
STM can probe suitable conductive surfaces at nanoscale/atomic-scale resolution and is relevant to molecular/nanoscale surface science, but it is not a general live-cell or soft-tissue imaging method.
Notebook 03's infinite wall enforced exact exclusion beyond the wall. A finite barrier changes the answer: the stationary wave has evanescent behavior inside the classically forbidden region, and a finite-width barrier can still produce nonzero transmitted flux.
Across the three investigations, the controls and model quantities had different roles. In the barrier explorer, $E$ and $a$ were controls while $R$ and $T$ were incident-flux fractions and $\kappa$ described evanescent decay. The transmission sweep used the exact rectangular-barrier result, while the opaque-barrier approximation explained why the width sensitivity becomes approximately exponential. In the STM model, $d$ was the control and $I/I_{\mathrm{ref}}$ was the relative detector-like reading; the decay constant and current-distance relation remained model quantities.
The same exponential sensitivity that suppresses transmission through a barrier makes STM extraordinarily sensitive to nanoscale separation. A real STM also depends on electronic structure, tip geometry, bias, feedback, and surface effects, so its image contrast is not simply geometric height.
This answers the Question of the Day: for a finite barrier, the quantum model predicts nonzero transmitted flux even when $E<V_0$. Classically forbidden does not mean exactly impossible.
So far the course has moved from rigid confinement to a finite barrier that a quantum particle can tunnel through. Notebook 05 replaces hard or piecewise-constant boundaries with a smooth restoring potential, where quantization appears in a different form.
What changes when a particle is bound not by rigid walls, but by a smooth restoring force?