Thicker Barrier, Smaller Tail - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

Here the barrier gets thicker by repeating the same toy attenuation step, so the tail shrinks in a visible sequence instead of by assertion. The fractions are exact for the declared model layers; real tunneling data would still need the barrier profile, incoming energy spread, calibration, and uncertainty beside the arithmetic.

A repeated exact attenuation makes the final toy transmission smaller. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Each layer multiplies amplitude

Each toy layer multiplies tail amplitude by 1/2.

a_{\text{next}}=\frac{1}{2}a_{\text{now}}

Each layer squares the smaller amplitude into probability

The one-layer row has a larger tail than the two-layer row because probability squares the amplitude.

\begin{array}{c|c|c}\text{layer row}&a&P\\\text{one layer}&\frac{1}{2}&\frac{1}{4}\\\text{two layers}&\frac{1}{4}&\frac{1}{16}\\\text{ratio}&\frac{1}{2}&\frac{1}{4}\\\end{array}

Two layers give a smaller probability

After two layers the tail amplitude is 1/4 and the probability is 1/16.

a=\frac{1}{4},\quad P=\frac{1}{16}