WKB Approximation

The WKB approximation is a semiclassical method for solving the Schrodinger equation when the potential varies slowly; it builds the wavefunction from a locally defined wavelength and yields the Bohr-Sommerfeld quantization condition and exponential tunneling estimates.

Definition

The WKB approximation is a semiclassical technique for approximating solutions of the Schrodinger equation when the potential changes little over a de Broglie wavelength, representing the wavefunction as an exponential of a slowly varying phase whose leading term is the classical action.

Scope

The topic covers the semiclassical expansion of the wavefunction in powers of the quantum of action, the local wavelength and amplitude in classically allowed regions, exponential growth and decay in forbidden regions, the connection formulas that join solutions across turning points, the Bohr-Sommerfeld quantization condition for bound states, and the exponential WKB estimate for tunneling probabilities.

Core questions

• When is a potential slowly varying enough for the WKB approximation to be valid?
• How does the wavefunction behave in classically allowed versus forbidden regions?
• What connection formulas join the solutions across classical turning points?
• How does WKB reproduce the Bohr-Sommerfeld quantization condition and tunneling rates?

Key concepts

• semiclassical expansion
• local wavelength
• turning points
• connection formulas
• Bohr-Sommerfeld quantization
• tunneling exponent

Key theories

Semiclassical wavefunction

In a slowly varying potential the wavefunction oscillates with a local wavelength set by the classical momentum and an amplitude that grows where the particle moves slowly, while in forbidden regions it grows or decays exponentially, the form underlying both quantization and tunneling.

Bohr-Sommerfeld quantization

Requiring the WKB phase accumulated between turning points to be a half-integer multiple of the action quantum reproduces the old Bohr-Sommerfeld quantization condition, giving accurate energy levels for smooth potentials and large quantum numbers.

Clinical relevance

The WKB method provides quick, physically transparent estimates across physics: it gives nuclear alpha-decay lifetimes through its tunneling exponent, field-emission and scanning-tunneling currents, vibrational levels of molecules, and the semiclassical quantization that bridges classical and quantum descriptions.

History

Wentzel, Kramers, and Brillouin each introduced the approximation in 1926, building on an earlier mathematical treatment by Jeffreys; it connected the new wave mechanics to the older Bohr-Sommerfeld quantization and was soon applied by Gamow to tunneling in alpha decay.

Key figures

• Gregor Wentzel
• Hendrik Kramers
• Leon Brillouin
• Harold Jeffreys

Seminal works

• landau1977
• griffiths2018

Frequently asked questions

When is the WKB approximation accurate?

It is accurate when the potential changes little over a de Broglie wavelength, which typically means high energies or large quantum numbers; it becomes unreliable near classical turning points, where connection formulas must be used to patch the solutions together.

How does WKB describe tunneling?

In the classically forbidden region the WKB wavefunction decays exponentially, and the tunneling probability is approximately the exponential of minus twice the integral of the decay rate across the barrier, the standard semiclassical estimate used for decay and emission rates.