Barrier Attenuation Three-Row Scan - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

A barrier attenuation scan holds the layer count fixed at two and instead varies how much each layer attenuates the tail amplitude. The three rows use per-layer attenuation 1/2 (tail probability 1/16), 1/3 (tail probability 1/81), and 2/3 (tail probability 16/81) exactly for the stated toy two-layer rule; real barrier leakiness depends on material composition, barrier width, and particle energy, none of which this toy attenuation constant models.

Three per-layer attenuation strengths at a fixed layer count show how barrier leakiness alone changes the tail probability. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A leakier barrier keeps more tail probability

Chapter four's attenuation scan fixed the per-layer loss and varied layer count. Here the layer count stays fixed at two and the per-layer attenuation itself changes. A weak attenuation of 2/3 per layer leaves a large tail after two layers.

\frac{2}{3}^2=\frac{16}{81}

Three attenuation strengths give three tail probabilities

All three rows keep the same two-layer barrier and the same reflected-transmitted split at the boundary; only the per-layer attenuation strength changes.

\begin{array}{c|c}\text{attenuation}&\text{tail}\\\frac{1}{2}&\frac{1}{16}\\\frac{1}{3}&\frac{1}{81}\\\frac{2}{3}&\frac{16}{81}\\\end{array}

A tighter attenuation leaves almost no tail

The strongest attenuation, 1/3 per layer, leaves the smallest tail of the three rows after the same two layers.

\frac{1}{3}^2=\frac{1}{81}