Tunneling and Barriers

Reflected and Transmitted

one probability budget

A toy barrier splits probability into reflected and transmitted parts. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Reflection and transmission share one budget

The reflected probability is 3/4 and the transmitted probability is 1/4.

R=34,T=14R=\frac{3}{4},\quad T=\frac{1}{4}
Reflected and transmittedThe two probabilities are checked as one budget.R=3/4T=1/41/2tail P=1/4

The split rows are the whole barrier budget

The scan keeps each side of the barrier in its own row before adding them.

sideProleleft return34reflectedright tail14transmittedsum1budget\begin{array}{c|c|c}\text{side}&P&\text{role}\\\text{left return}&\frac{3}{4}&\text{reflected}\\\text{right tail}&\frac{1}{4}&\text{transmitted}\\\text{sum}&1&\text{budget}\\\end{array}
Tunneling split scanThe renderer checks the split as one budget.R=3/4T=1/41/2tail P=1/4

The probabilities add to one

The budget closes: 3/4 plus 1/4 equals 1.

34+14=1\frac{3}{4}+\frac{1}{4}=1
Closed tunneling budgetThe renderer rejects a split that does not close.R=3/4T=1/41/2tail P=1/4