Reflection and Transmission of Wave Functions
Definition
The reflection coefficient $R$ and transmission coefficient $T$ of a wave function are defined as follows.
$$ R = \left| \frac{j_{\text{ref}}}{j_{\text{inc}}} \right|,\quad T = \left| \frac{j_{\text{trans}}}{j_{\text{inc}}}\right| \tag{1} $$
Here, $j_{\text{inc}}$ represents the probability flux of the incident wave, $j_{\text{ref}}$ represents the probability flux of the reflection wave, and $j_{\text{trans}}$ represents the probability flux of the transmission wave.
Explanation
When a particle with energy $E$ encounters a potential barrier higher than its energy, reflection and transmission occur. From a classical perspective, the particle does not penetrate the barrier and only reflects (like a ball bouncing off a wall). However, in the microscopic world, due to the wave nature of particles, probabilistic transmission occurs. This is called quantum tunneling or the tunnel effect.
Flux
The amount of a certain physical quantity passing through a point per unit time is called flux. Therefore, when the wave function reflects and transmits, the ratios can be defined concerning the flux of the incident wave, reflection wave, and transmission wave.
$$ \text{반사율} = \dfrac{\text{반사파의 선속}}{\text{입사파의 선속}},\quad \text{투과율} = \dfrac{\text{투과파의 선속}}{\text{입사파의 선속}} $$
However, in quantum mechanics, the concept corresponding to the flux of the wave function is the probability flux $j(x,t)$, and therefore the reflection coefficient and transmission coefficient are defined as $(1)$.