Quantum Experiments · III · Control and reconstruction · Chapter 12
How can transmission occur when the particle energy is below a barrier?
When a particle of total energy \(E\) approaches a one-dimensional rectangular potential barrier of height \(V_0>E\) and thickness \(a\), classical mechanics requires the particle to reflect at the front interface because kinetic energy \(E-V_0\) cannot be negative. In quantum mechanics, the stationary wavefunction \(\psi(x)\) and its spatial derivative \(d\psi/dx\) must remain continuous across both barrier boundaries \(x=0\) and \(x=a\). Because the evanescent wavefunction decays exponentially rather than vanishing instantly inside \(0<x<a\), a finite barrier thickness \(a\) leaves a nonzero transmitted wave for \(x>a\).
In a scanning tunnelling microscope (STM), an atomically sharp conducting tip is held a few angstroms (\(a\sim 0.5\text{ nm}\)) above a conducting surface under a small bias voltage. Electrons tunnel across the vacuum work-function barrier between tip and sample, and because the transmission probability varies exponentially with the gap width \(a\), changing the tip–surface height by just \(0.1\text{ nm}\) typically alters the tunnelling current by an order of magnitude—enabling atomic-scale topographic and local density-of-states imaging.
In the apparatus schematic below, a stationary scattering wave at energy \(E=V_0/2\) impinges on a rectangular barrier of height \(V_0\) and width \(a\). Elastic tunnelling does not involve a particle "borrowing" extra energy \(\Delta E\ge V_0-E\) to hop over the top of the barrier: the transmitted wave emerges on the far side with the exact same energy \(E\) and wavelength as the incident wave, only with reduced amplitude.
Visual animation · Section 12.1
Sub-barrier scattering and scanning tunnelling probes
Inside the classically forbidden region \(0<x<a\) where \(V(x)=V_0>E\), the time-independent Schrödinger equation \(-\frac{\hbar^2}{2m}\psi''(x)+V_0\psi(x)=E\psi(x)\) has real exponential solutions \(e^{\pm\kappa x}\) characterized by the decay constant
\[\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}.\]
When the barrier is thick (\(\kappa a\gg 1\)), the transmission coefficient is dominated by the exponential attenuation factor \(T\propto e^{-2\kappa a}\), explaining why small changes in \(a\), mass \(m\), or barrier excess \(V_0-E\) produce dramatic changes in transmission.
Matching \(\psi(x)\) and \(\psi'(x)\) at \(x=0\) and \(x=a\) for symmetric potentials \(V=0\) outside the barrier gives the exact transmission probability for all \(\kappa a\ge 0\):
\[T(\kappa a) = \left[1 + \frac{V_0^2\,\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}.\]
In our interactive model, the energy ratio is fixed at the midpoint \(E/V_0=1/2\), where \(V_0^2/[4E(V_0-E)]=1\) and the exact formula simplifies cleanly to
\[T(\kappa a) = \frac{1}{1+\sinh^2(\kappa a)} = \frac{1}{\cosh^2(\kappa a)} = \operatorname{sech}^2(\kappa a), \qquad R(\kappa a) = 1 - T(\kappa a) = \tanh^2(\kappa a).\]
Unlike the thick-barrier asymptotic approximation \(e^{-2\kappa a}\), the exact expression \(\operatorname{sech}^2(\kappa a)\) properly satisfies \(T(0)=1\) as the barrier width \(a\to 0\).
Because the Hamiltonian is Hermitian and the potential \(V(x)\) is real, probability current is conserved across the barrier, enforcing unitarity \(T(\kappa a)+R(\kappa a)=1\) at every barrier width.
Visual animation · Section 12.2
Evanescent decay constant and exact rectangular transmission
In solid-state and molecular junctions, observing a small conductance across an insulating layer does not automatically prove coherent single-step elastic tunnelling: thermally activated hopping over the barrier (\(\propto e^{-(V_0-E)/(k_B T)}\)), defect-assisted multistep hopping, and pinhole shorts can also conduct current. Verifying elastic tunnelling requires checking that the conductance decays exponentially with barrier thickness \(a\) while remaining nearly independent of temperature \(T\) at low temperatures.
Because the stationary scattering eigenstates have definite energy \(E\), measuring the static transmission ratio \(T\) alone does not define a unique classical "tunnelling time" spent inside the forbidden region; wave-packet group delay, Larmor clock rotation, and modulated-barrier experiments measure distinct dynamical response functions.
While the one-dimensional rectangular barrier captures the essential \(\kappa a\) scaling, quantitative STM spectroscopy also incorporates the three-dimensional tip curvature, image-charge rounding of the barrier, and the energy-dependent electronic density of states of both electrodes.
Visual animation · Section 12.3
Separating elastic tunnelling from thermal activation
Sweep the dimensionless barrier thickness \(\kappa a\) from \(0\) to \(4\). Compare the exact transmission \(T=\operatorname{sech}^2(\kappa a)\) and reflection \(R=\tanh^2(\kappa a)\), and check both the zero-width limit \(\kappa a=0\) and the thick-barrier decay at \(\kappa a\ge 2\).
Interactive model · Idealized predictions
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
At \(E/V_0=1/2\) and dimensionless barrier width \(\kappa a=1\), compute the exact transmission probability \(T\) and reflection probability \(R\), and compare \(T\) against the naive exponential factor \(e^{-2\kappa a}\).
At \(E/V_0=1/2\) and \(\kappa a=1\), \(\cosh(1)=(e+e^{-1})/2\approx 1.5431\), so the exact transmission is \(T=1/\cosh^2(1)\approx 0.41997\approx 0.420\) and the reflection probability is \(R=1-T\approx 0.580\). By contrast, the bare exponential \(e^{-2(1)}=e^{-2}\approx 0.1353\) omits the prefactor \(4\approx(2/\cosh 1)^2 e^2\) for \(E/V_0=1/2\) and underestimates \(T\) by more than a factor of three at \(\kappa a=1\).
What does the exact transmission formula \(T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}\) give as the barrier width \(a\to 0\)?
Use \(\sinh(0)=0\).
Since \(\sinh(0)=0\), the denominator equals \(1+0=1\), giving \(T(0)=1\) and \(R(0)=0\) as required when the barrier disappears.
In elastic tunnelling through a static rectangular barrier, is the kinetic energy of a transmitted particle on the far side (\(x>a\)) smaller than its incident kinetic energy on the near side (\(x<0\))?
Compare the potential energy \(V(x)=0\) for \(x<0\) and \(x>a\) at fixed total energy \(E\).
No. Because the static barrier exchanges no energy with the particle and \(V(x)=0\) on both sides, a transmitted particle has the exact same kinetic energy \(E\) as an incident particle; the barrier reduces the fraction \(T\) of particles transmitted, not the energy of each transmitted particle.
In the thick-barrier regime (\(\kappa a\gg 1\)), by what factor does the transmission probability \(T\) change when the barrier width \(a\) increases by \(\Delta a=1/\kappa\)?
Evaluate \(e^{-2\kappa(a+1/\kappa)}/e^{-2\kappa a}\).
In the thick-barrier regime where \(\sinh(\kappa a)\approx\frac{1}{2}e^{\kappa a}\), increasing \(a\) by \(1/\kappa\) multiplies \(T\propto e^{-2\kappa a}\) by \(e^{-2}\approx 0.135\).
Quantum tunnelling arises from boundary continuity of \(\psi(x)\) and \(\psi'(x)\) across a finite forbidden region, yielding exact transmission \(T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}\) while conserving energy \(E\) and total probability \(T+R=1\).
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.