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Rectangular Barrier Transmission Calculator
Calculate the transmission probability below, at and above a rectangular barrier. Choose input units and calculate one value within the stated conditions.
Inputs
Calculations run in this browser. Scientific notation such as 1.2e-3 is supported. Do not group thousands.
Result
Barrier transmission probability T
Enter values and select Calculate.
E<U: T=[1+U²sinh²(κa)/(4E(U−E))]⁻¹; E=U: T=[1+mUa²/(2ℏ²)]⁻¹; E>U: T=[1+U²sin²(qa)/(4E(E−U))]⁻¹
Assumptions
Required: m,E>0 and U,a≥0. The exterior potential is zero on both sides and the mass is identical in all regions. Use one-dimensional stationary nonrelativistic scattering with max(E,U)/(mc²)≤0.01. Here κ=√(2m(U−E))/ℏ and q=√(2m(E−U))/ℏ. The threshold limit and large hyperbolic sine are evaluated stably. Above-barrier phases qa>10¹² are rejected to preserve numerical accuracy. Positive probabilities below the numeric range are reported as errors. Above-barrier resonance inputs that are sensitive to phase-rounding uncertainty are also rejected with a precision error.
Displayed values are floating-point approximations. Design safety factors and unprovided error contributions are not calculated.
Formula references and credits
NaNa Lab created the code, descriptions, icons and preview images. Text, figures and code from the linked references have not been reproduced.
Physical constants: NIST / CODATA 2022. The values of μ₀ and ε₀ are measured recommendations, not exact defining constants. Values and precision notes are in the source record.
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