WKB Transmission Coefficient for Barrier Penetration

Find the transmission coefficient using the WKB approximation for the case when the energy is less than the potential.

Integrate over the momentum $p(x)$ and calculate $e^{-2\gamma}$, where $\gamma = \frac{1}{\hbar} \int p(x) \, dx$.

In solving this problem, you are dealing with a situation where the energy of a particle is lower than the potential barrier it encounters. This is a classic setup for studying quantum tunneling using the WKB, or Wentzel-Kramers-Brillouin, approximation. The WKB method is a semi-classical approach that is used to approximate the solution to the Schrödinger equation in cases where the potential varies slowly compared to the wavelength of a particle. In such instances, exact solutions can be difficult to find, and the WKB approximation provides a useful tool for estimating probabilities associated with quantum mechanical phenomena like tunneling.

The transmission coefficient, in this context, quantifies the probability that a particle will tunnel through a potential barrier instead of being reflected. By using this approximation, you calculate an integral of the momentum over space, which is crucial in determining the likelihood of tunneling when classical paths are forbidden due to a lack of sufficient energy to overcome the barrier classically. This integration leads to the evaluation of the expression $e^{-2\gamma}$, which directly ties to the probability amplitude for tunneling.

Understanding the WKB approximation and its applications extend to various areas in quantum mechanics, particularly in fields dealing with quantum states that involve potential barriers, such as in nuclear physics, quantum chemistry, and condensed matter physics. Mastery of these concepts is essential for deeper insights into quantum mechanical systems as they apply to real-world physical systems.