The Bohr-Sommerfeld quantization rule, also known as the Bohr-Sommerfeld condition, is an early quantization rule that was proposed by Niels Bohr and Arnold Sommerfeld in the early 20th century to describe the quantized energy levels of atomic systems. It was developed as an extension of Bohr's model of the hydrogen atom, which incorporated the principles of quantization proposed by Max Planck.
According to the Bohr-Sommerfeld quantization rule, the orbital angular momentum of an electron in an atom is quantized, meaning it can only take certain discrete values. The quantization condition is given by:
∮ p(r) dr = nh
Here, p(r) represents the momentum of the electron as a function of its position r, ∮ denotes integration along a closed path in the electron's orbit, n is an integer (known as the principal quantum number), and h is Planck's constant divided by 2π. The integral represents the action, which is a measure of the motion of the electron along its orbit.
The Bohr-Sommerfeld quantization rule extends the concept of quantization beyond discrete energy levels to include the quantization of the orbital angular momentum. It suggests that the electron's angular momentum is quantized in units of h/2π, leading to the familiar quantization of angular momentum in quantum mechanics.
While the Bohr-Sommerfeld quantization rule was an important step in the development of quantum theory, it was eventually superseded by more rigorous and comprehensive approaches such as wave mechanics and the Schrödinger equation, which provided a more complete and accurate description of atomic and subatomic systems.