Einstein–Brillouin–Keller method
The Einstein–Brillouin–Keller (EBK) method is a semiclassical way to find the energy levels of quantum systems that are close to classical, especially when the system can be separated into independent motions. EBK improves on the older Bohr–Sommerfeld quantization by including extra phase jumps that occur at turning points in the classical motion (caustics). This correction allows EBK to reproduce exact results for several well-known problems, such as the three-dimensional harmonic oscillator, a particle in a box, and even the relativistic fine structure of the hydrogen atom.
The idea has a long history. Einstein first broadened Bohr–Sommerfeld ideas in 1917, but he noted that the method didn’t generalize to chaotic systems. Brillouin added refinements and, in the same period, developed the WKB approximation. Later, Keller independently reformulated the approach in 1958 by incorporating the caustic corrections. In the 1970s, Maslov formalized the mathematical underpinning with Maslov indices, and Percival helped popularize the EBK name. Berry and Tabor later extended the method to connect with the density of states for integrable systems starting from EBK quantization.
How EBK works, in short, is for a separable system described by coordinates (qi, pi). Each degree of freedom has an action variable obtained by integrating the momentum over one closed orbit of its coordinate. The resulting action values are quantized using integers, with additional indices that count the number of turning points (caustics) and wall reflections the trajectory encounters. These corrections are what make EBK more accurate than older approaches.
Concrete examples help. For the harmonic oscillator, the EBK conditions account for the two turning points and yield the exact energy levels. In the hydrogen atom, EBK treats the radial motion between turning points and handles the angular part in a way that aligns with the usual angular-momentum quantization; in three dimensions, this connects to the Langer correction. For the simple two-dimensional hydrogen case, the method can recover the expected spectrum as well.
In summary, EBK provides a practical bridge between classical trajectories and quantum spectra for many separable systems, explaining why semiclassical ideas can give exact or highly accurate results for a wide range of fundamental problems.
This page was last edited on 2 February 2026, at 12:59 (CET).