WKB approximation

The WKB approximation is a powerful technique in mathematical physics. It is used to find approximate solutions to linear differential equations. These equations often have coefficients that change based on space. In the field of quantum mechanics, this method is essential for semiclassical calculations. It allows scientists to study how wave functions behave. A wave function describes the state of a quantum particle. The WKB method recasts this function as an exponential function. It then uses a semiclassical expansion to simplify the math. This approach works best when the amplitude or phase changes slowly.

To understand the mechanism, imagine a wave moving through a changing environment. The method assumes a specific form for the solution, called an ansatz. This ansatz is an asymptotic series expansion. This expansion relies on a small parameter, often denoted as $\hbar$. When you substitute this expansion into a differential equation, you can cancel out exponential terms. This allows you to solve for many terms in the series. The WKB theory is actually a special case of multiple scale analysis. By using this method, physicists can approximate complex systems without solving the exact equations. This is helpful because many real-world equations are too difficult to solve perfectly.

There are two main types of regions the method must describe. The first is the classically allowed region. In this area, the particle has enough energy to move. The wave function here is oscillatory, meaning it looks like a wave. The second is the classically forbidden region. In this area, the particle's energy is lower than the potential energy. The wave function does not wave here; instead, it either grows or decays exponentially. A critical part of the method involves the turning points. A turning point is the exact location where a particle changes direction. At these points, the potential energy equals the particle's total energy. The WKB method must bridge the gap between the oscillatory and exponential solutions.

The history of this method involves several brilliant minds. The name WKB is an initialism for Wentzel, Kramers, and Brillouin. These three physicists developed the method in 1926. However, they were not the only ones working on these ideas. In 1923, the mathematician Harold Jeffreys developed a similar method. Because of his work, the technique is sometimes called the JWKB method. Some researchers even use the term WKBJ to include Jeffreys. Earlier versions of the method appeared as far back as 1817 with Francesco Carlini. Joseph Liouville and George Green also developed equivalent methods in 1837. Because of them, the technique is often called the Liouville–Green or LG method.

Precision is a major factor when using the WKB approximation. The asymptotic series used in the method is usually a divergent series. This means the terms do not get smaller forever. After a certain point, the terms actually start to increase. Because of this, the smallest error you can achieve is limited. The error is roughly the size of the last term you include in your calculation. If a function changes very slowly, the error can be exponentially small. This makes the method extremely useful for many scientific applications.

One notable application is in the one-dimensional, time-independent Schrödinger equation. This equation is a fundamental part of quantum mechanics. When applying WKB, the wave function is split into two parts. One part handles the phase, and the other handles the amplitude. Near the turning points, the standard WKB solution becomes singular. This means the math breaks down and gives infinite results. To fix this, scientists use the Airy equation. The Airy equation provides a way to describe the wave behavior near the turning point. By using Airy functions, they can connect the allowed and forbidden regions smoothly.

This connection process is vital for finding energy levels. For a wave function to be valid, it must be square-integrable. This means the total probability of finding the particle must be finite. To achieve this, the solution must decay in the forbidden regions. By matching the solutions from different regions, scientists find specific conditions. These conditions are known as quantization conditions. They help determine the allowed energy levels of a quantum system. This process is a version of the Bohr–Sommerfeld quantization condition. It even includes what is called a Maslov correction of 1/2. This connects modern quantum mechanics back to older theories of the atom.