Reflection and Transmission of Matter Waves at a 1D Step Potential

Problem Statement

A particle of mass $m$m moves along a one-dimensional space (the $x$x-axis). The potential energy $V(x)$V(x) is defined as:

$$V(x) = \begin{cases} 0 & (x < 0) \\ V_0 & (x \ge 0) \end{cases}$$V(x)={0V0​​(x<0)(x≥0)​

A steady stream of particles with energy $E$E (where $E > V_0$E>V0​) is incident from the $-x$−x direction. Find the reflection coefficient $R$R from the boundary conditions of the Schrödinger equation.

Constraints

• Particle mass: $m = 9.1 \times 10^{-31}\text{ kg}$m=9.1×10−31 kg
• Incident particle energy: $E = 289\text{ eV}$E=289 eV
• Potential height: $V_0 = 225\text{ eV}$V0​=225 eV
• Dirac constant: $\hbar = 1.054 \times 10^{-34}\text{ J}\cdot\text{s}$ℏ=1.054×10−34 J⋅s

Input Format

The reflection coefficient $R$R is expressed as an irreducible fraction $A/B$A/B. Find the value of $A+B$A+B, where $A$A and $B$B are positive integers.