Quantum Tunneling Calculator for a Rectangular Barrier
Calculate exact transmission and reflection for a finite one-dimensional rectangular barrier, then compare the result with the simple WKB approximation. Choose common particles or an effective mass, with energy in eV, keV, or MeV and width in fm, pm, or nm; the model is appropriate for idealized single-barrier teaching and estimates.
Exact barrier transmission
A realistic electron example is loaded. Results recalculate after each valid change; use Calculate to confirm or submit with Enter.
Calculation steps and substitutions
- Enter valid inputs to see each step.
Interactive barrier and logarithmic plots
Rectangular barrier
The energy line, barrier, decay, reflection, and transmission update with the current inputs.
Transmission vs barrier width
Logarithmic transmission plot versus width.
Transmission vs particle energy
Logarithmic transmission plot versus energy.
How to use the calculator
- Choose a particle or select a custom/effective mass, then choose its mass unit.
- Enter barrier height, particle energy, and barrier width in the displayed units. Presets supply reproducible starting points.
- Read exact T first. Compare R, WKB, log₁₀(T), “1 in N,” the diagram, plots, and substitution steps.
Exact formula, WKB approximation, and assumptions
For a barrier V(x)=V₀ from x=0 to x=a, with zero potential on both sides, continuity of the wavefunction and its derivative gives:
m is particle mass, E is incident energy, V₀ is barrier height, a is barrier width, ℏ is the reduced Planck constant, κ is the evanescent decay constant, and q is the wave number inside an above-barrier region. The calculator’s simple WKB comparison is TWKB = exp(−2κa). It captures the dominant exponential when E < V₀ and κa ≫ 1, but omits the interface prefactor; therefore it need not closely match the exact value near the barrier top or for thin barriers.
Worked examples
Electron crossing a nanoscale barrier
Inputs: electron mass 1 mₑ = 9.1093837139 × 10⁻³¹ kg; V₀ = 5 eV = 8.01088317 × 10⁻¹⁹ J; E = 1 eV = 1.602176634 × 10⁻¹⁹ J; a = 0.5 nm = 5 × 10⁻¹⁰ m. Thus V₀−E = 4 eV = 6.408706536 × 10⁻¹⁹ J, κ = 1.02463 × 10¹⁰ m⁻¹, and κa = 5.12317. Substitution in the exact formula gives T = 9.08458 × 10⁻⁵; WKB gives 3.54873 × 10⁻⁵. Interpretation: about 1 in 11,008 incident electrons transmits in this idealized model.
Nuclear-scale alpha-particle illustration
Inputs: alpha mass 4.001506179127 u = 6.6446573450 × 10⁻²⁷ kg; V₀ = 25 MeV = 4.005441585 × 10⁻¹² J; E = 5 MeV = 8.01088317 × 10⁻¹³ J; a = 10 fm = 1 × 10⁻¹⁴ m. Then V₀−E = 20 MeV = 3.204353268 × 10⁻¹² J, κ = 1.95679 × 10¹⁵ m⁻¹, and κa = 19.5679. The rectangular-barrier result is T = 2.58068 × 10⁻¹⁷; WKB gives 1.00808 × 10⁻¹⁷, or roughly 1 transmission in 3.87 × 10¹⁶ encounters. A real alpha-decay calculation integrates through a Coulomb barrier instead.
Parameter sensitivity
| Scenario (one change only) | Exact T | log₁₀(T) |
|---|---|---|
| Electron example, width 0.4 nm | 7.05069 × 10⁻⁴ | −3.15177 |
| Electron example, width 0.5 nm | 9.08458 × 10⁻⁵ | −4.04170 |
| Electron example, width 0.6 nm | 1.17038 × 10⁻⁵ | −4.93167 |
| Electron example, E raised from 1 to 2 eV | 5.37613 × 10⁻⁴ | −3.26953 |
| Alpha illustration, width 8 fm | 6.47196 × 10⁻¹⁴ | −13.1890 |
| Alpha illustration, width 12 fm | 1.02904 × 10⁻²⁰ | −19.9876 |
Applications
Scanning tunneling microscopy: tunneling current changes sharply with tip–sample separation. Semiconductors: thin junctions enable tunnel diodes, memory, and leakage paths. Fusion: nuclei can penetrate Coulomb repulsion at sub-barrier energies. Alpha decay: an alpha cluster escapes through a nuclear Coulomb barrier. The rectangular model explains the trend, while realistic device and nuclear work requires the actual potential profile.
Methodology and verification
The implementation evaluates the closed-form finite rectangular-barrier solution in logarithmic form below the barrier to avoid overflow and underflow. It uses ℏ = 1.054571817 × 10⁻³⁴ J·s, exact 1 eV = 1.602176634 × 10⁻¹⁹ J, and 2022 CODATA particle masses. Last reviewed: .
- NIST Reference on Constants, Units, and Uncertainty (2022 CODATA)
- OpenStax University Physics, §7.6: Quantum Tunneling
- Verification cases: the two reproducible examples and six sensitivity rows above, plus the required limits
a→0 ⇒ T→1,V₀→0 ⇒ T→1, andR+T=1.
Calculation methodology is transparent in the formulas and live substitution steps above. No external credentialed physics reviewer is currently identified; no reviewer credentials or accuracy rating are claimed.
Quantum tunneling FAQ
What is tunneling probability?
Tunneling probability, or transmission T, is the fraction of incident particle flux predicted to emerge beyond a potential barrier. For this lossless model, R + T = 1.
What is the exact rectangular-barrier formula?
For E < V₀, it is T = [1 + V₀²sinh²(κa)/(4E(V₀−E))]⁻¹. The finite limit at E = V₀ and oscillatory sine form for E > V₀ are shown in the formula section.
When is T ≈ exp(−2κa) valid?
It is a useful simple WKB estimate below the barrier when κa is much greater than 1. It gets the exponential sensitivity right but omits an interface prefactor, so compare it with exact T.
Why does mass matter?
κ ∝ √m. A heavier particle’s wavefunction decays faster inside the same sub-barrier region, so its transmission generally falls exponentially.
Can transmission occur above the barrier?
Yes. Transmission is classically allowed for E > V₀, but it is not always 100%: quantum waves can reflect from abrupt changes in potential. The exact result also shows interference oscillations with width and energy.
Why is T sometimes shown as zero?
Software often rounds or underflows an extremely small positive number to zero. This calculator retains log₁₀(T) and displays 10n when ordinary decimal representation can no longer preserve it.
How is tunneling used in STM and semiconductors?
An STM maps a surface using the strong distance dependence of tunneling current. Semiconductor tunnel junctions, flash memory, diodes, and unwanted gate leakage likewise depend on barrier transmission.
How is tunneling related to fusion and alpha decay?
Tunneling permits nuclei to penetrate electrostatic barriers in fusion and alpha particles to escape nuclei. Quantitative nuclear calculations use a Coulomb-shaped potential rather than the constant barrier used here.