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

A toy tunneling attenuation model multiplies the tail amplitude by 1/2 at each layer, then squares the result for probability, so each layer cuts the tunneling probability by a factor of 1/4. The three rows give amplitudes 1/2, 1/4, and 1/8 with probabilities 1/4, 1/16, and 1/64 exactly for the stated toy amplitude rule; real tunneling depends on barrier shape, energy spread, material constants, and detector resolution.

A toy barrier becomes clearer when one, two, and three layers are compared exactly. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One layer leaves a one-quarter toy tail

The toy attenuation is an amplitude rule: each layer multiplies tail amplitude by 1/2. One layer gives probability 1/4.

a_1=\frac{1}{2},\quad P_1=\frac{1}{4}

Layer count gives a three-row decay scan

The model does not claim a real material constant. It only claims this exact repeated attenuation, then squares the final amplitude.

\begin{array}{c|c|c|c}\text{layers}&a_{\text{tail}}&P_{\text{tail}}&\text{ratio}\\\text{one}&\frac{1}{2}&\frac{1}{4}&\frac{1}{4}\\\text{two}&\frac{1}{4}&\frac{1}{16}&\frac{1}{4}\\\text{three}&\frac{1}{8}&\frac{1}{64}&\frac{1}{4}\\\end{array}

The third row is smaller by another quarter in probability

The final amplitude after three layers is 1/8. Squaring gives 1/64, so the probability drops by another factor of 1/4.

\left(\frac{1}{8}\right)^2=\frac{1}{64}