TUNNELING
One exact stationary wave, matched twice: density may decay inside the barrier while probability current crosses and R + T stays one.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Barrier + incident state
A unit-amplitude stationary matter wave approaches a finite rectangular barrier from the selected side. The height and width are literal model parameters. This is the exact sharp-edged one-dimensional potential, with the nonrelativistic approximation and its live wavenumber correction stated openly.
Match both boundaries
The complete complex wave is solved in all three spatial regions. Both the wavefunction and its slope join at x equals zero and x equals a; the visible real and dashed imaginary curves come from those same coefficients. Reversing incidence mirrors the solution with the phase needed to keep the incoming coefficient exactly one.
Density + probability current
Inside the barrier the stationary solution is evanescent, yet its complex combination carries a nonzero conserved probability current. The density decays; no energy is borrowed. For probabilities below subtraction precision, the equivalent transmitted-amplitude and Wronskian forms preserve that current honestly. The arrows have the computed magnitudes one, R, and T.
Stationary phase evolution
One exact global phase period is displayed in three seconds. The real and imaginary components rotate, while probability density and every current remain fixed. The physical period h over E is printed in femtoseconds. This is not a localized wave packet, a particle path, or a traversal-time animation.
Continuity + conservation audit
The phase returns exactly to one plus zero i. Both boundary pairs close, probability current is constant, R plus T returns one, and the amplitude result agrees with the independent closed transmission expression. Zero height, zero width, threshold, resonance, and reversed incidence are all named branches rather than hidden exceptions.