Quantum Mechanics — Wave Mechanics & Scattering
1D Quantum Tunneling
Schrödinger → Scattering → Tunneling → WKB → QEDT ≈ exp(−2κw)
Wave Packet
Energy E1 eV
Kinetic energy of incident particle
Packet width σ1.5 Å
Spatial spread of Gaussian (uncertainty)
Mass1 mₑ
In units of electron mass
Potential
Height |V₀|2 eV
Barrier height (well depth for Well)
Width w2 Å
Barrier thickness
Position25 Å
Distance from left edge
Animation
Speed1 ×
Time evolution speed
Zoom1 ×
Canvas zoom level
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Calculated
Energy E1.00 eV
Momentum p0.512 Å⁻¹
Wavelength λ12.27 Å
Barrier V₀2.00 eV
Width w2.0 Å
Decay κ0.512 Å⁻¹
Theoretical T40.48 %
Measured T0.0 %
Measured R0.0 %
Formulas
κ = √(2m(V₀−E))/ℏ
κ = 0.512 Å⁻¹
———————
T ≈ 16(E/V₀)(1−E/V₀)e^(−2κw)
T ≈ 5.16e-1
Curriculum
WAECNECOIGCSESATJUPEBA-LevelUndergrad
Rectangular Barrier — The Canonical Tunneling Problem
- •When E < V₀, the wavefunction decays exponentially inside the barrier as ψ ∼ e^(−κx), where κ = √(2m(V₀−E))/ℏ. This is called an evanescent wave.
- •The transmission probability depends exponentially on width: T ≈ 16(E/V₀)(1−E/V₀)e^(−2κw). Doubling the width squares T — this extreme sensitivity is why tunneling only matters at nanometer scales.
- •Classically, the particle would bounce back with 100% probability. Quantum mechanically, there is always a non-zero probability of penetration, however small.
- •Watch for the evanescent wave inside the barrier — it carries no probability current, yet it "connects" the incident and transmitted waves, enabling the particle to appear on the far side.
- •When E > V₀, transmission is NOT 100%. The wave partially reflects at each boundary due to impedance mismatch (change in wavelength), creating interference oscillations in T(E).
General Principles
- •Quantum tunneling is the phenomenon where a particle passes through a potential energy barrier that it classically cannot surmount. It arises from the wave nature of matter encoded in the Schrödinger equation.
- •The wavefunction ψ(x,t) is not zero inside the barrier even when E < V₀. It decays exponentially as ψ ∼ exp(−κx), where κ = √(2m(V₀−E))/ℏ. This evanescent wave carries no probability current, yet it connects to a non-zero transmitted wave on the far side.
- •Transmission probability for a thick rectangular barrier: T ≈ 16(E/V₀)(1−E/V₀) exp(−2κw). The exponential dependence on width w and height V₀ makes tunneling extremely sensitive to atomic-scale geometry.
- •Classically, a particle with E < V₀ is always reflected. The classical turning point is where E = V(x). Quantum mechanics permits non-zero |ψ|² beyond this point because the particle does not have a definite position — only a probability amplitude.
- •When E > V₀, transmission is not 100%. The wave partially reflects from the barrier boundaries due to impedance mismatch (change in wavelength), creating interference oscillations in T(E). This is purely a wave phenomenon.
- •The split-step Fourier method used here is unconditionally stable and unitary (conserves total probability exactly). It is spectrally accurate in space, making it far superior to finite-difference methods for teaching the time-dependent Schrödinger equation.