13、Quantum Mechanics: Linear Potential and WKB Approximation

Quantum Mechanics: Linear Potential and WKB Approximation

1. Introduction to the Linear Potential

In quantum mechanics, the linear potential is a fascinating concept. Consider a potential energy function that is linear in (x) for (x > 0) and infinite for (x \leq 0). This can be represented as:
[U (x) =
\begin{cases}
eFx, & x > 0 \
\infty, & x \leq 0
\end{cases}
]
This situation can be visualized in two physical scenarios. One is a particle falling in a uniform gravitational field and being elastically reflected at the surface. The other is a charged particle, like an electron, being forced against an infinite potential wall by a constant electric field of magnitude (F).

The force for (x > 0) is given