How can transmission occur below the barrier energy?
How can transmission occur below the barrier energy?
- Set E=1 eV, barrier 4 eV and width 0.2 nm.
- Increase width in 0.05 nm steps.
- Raise particle energy while remaining below the barrier.
What to measure
Units and parameter definitions are shown beside every laboratory control and live reading.
The idea underneath
For E below a finite rectangular barrier, the stationary wavefunction decays exponentially inside it rather than vanishing. Matching the wavefunction and its derivative at both boundaries produces a nonzero transmission probability. The readout uses the exact one-dimensional rectangular-barrier formula, while the displayed envelope illustrates evanescence. A small width increase can suppress transmission by orders of magnitude. This is not a classical particle borrowing energy: transmitted particles retain the incident energy.
κ=√(2m(V₀−E))/ℏ · T=[1+V₀²sinh²(κa)/(4E(V₀−E))]⁻¹Units and parameter definitions are shown beside every laboratory control and live reading.
Build the argument
- State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
- Normalise the wavefunction so the integral of |ψ|² is one.
- Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.
Worked example 1
At E=1 eV, V₀=4 eV and a=0.2 nm, T≈8.45%.
Worked example 2
The decay constant is approximately 8.87 nm⁻¹ for V₀−E=3 eV.
Applications
Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.
Common misconception
A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.
A further investigation
Increasing barrier width usually makes transmission…
Check your understanding
Increasing barrier width usually makes transmission…
- Smaller
- Larger
- Exactly one
Explanation
The evanescent region grows and transmission is strongly suppressed.
Model limits
Stationary nonrelativistic electron; rectangular barrier and E<V₀; envelope illustration is not a time-dependent scattering solution.