Quantum Mechanics Course

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← Part III/WKB Approximation

7. WKB Approximation

Reading time: ~20 minutes | Pages: 5

A semiclassical approximation valid when the potential varies slowly on the scale of the de Broglie wavelength.

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Video Lecture

WKB Approximation - Semiclassical Quantum Mechanics

Detailed explanation of the WKB approximation method, connection formulas, and applications to tunneling and bound states

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

The Method

WKB = Wentzel-Kramers-Brillouin (also known as JWKB or Liouville-Green method)

Assumption: Wave function varies much more rapidly than potential

$$\lambda(x) = \frac{2\pi\hbar}{p(x)} \ll L$$

where $L$ is the characteristic length scale over which $V(x)$ changes

WKB Wave Function

In classically allowed region ($E > V(x)$):

$$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\left(\pm\frac{i}{\hbar}\int^x p(x')dx'\right)$$

where $p(x) = \sqrt{2m(E - V(x))}$ is the classical momentum

Classically Forbidden Region

In region where $E < V(x)$:

$$\psi(x) \approx \frac{C}{\sqrt{|p(x)|}}\exp\left(\pm\frac{1}{\hbar}\int^x |p(x')|dx'\right)$$

Exponential growth or decay

Connection Formulas

At classical turning points ($E = V(x)$), WKB breaks down. Use connection formulas:

From allowed to forbidden:

$$\frac{C}{\sqrt{p}}\cos\left(\frac{1}{\hbar}\int^x p\,dx' - \frac{\pi}{4}\right) \leftrightarrow \frac{C}{\sqrt{|p|}}e^{-\frac{1}{\hbar}\int^x |p|\,dx'}$$

Bohr-Sommerfeld Quantization

For bound states between two turning points $x_1$ and $x_2$:

$$\oint p(x)dx = 2\pi\hbar\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$

Integral over one complete classical cycle

This reproduces exact energy levels for harmonic oscillator!

Tunneling Probability

Transmission through a barrier from $x_1$ to $x_2$:

$$T \approx \exp\left(-2\int_{x_1}^{x_2}\frac{|p(x)|}{\hbar}dx\right)$$

Valid for thick barriers ($T \ll 1$)

Validity Conditions

WKB approximation is valid when:

$$\left|\frac{d\lambda}{dx}\right| \ll 1$$

Equivalently:

$$\left|\frac{dV/dx}{[2m(E-V)]^{3/2}}\right| \ll 1$$

Breaks down near classical turning points

Applications

  • Alpha decay: Tunneling through Coulomb barrier
  • Field emission: Electrons escape from metals
  • Semiclassical quantization: Energy levels for complex potentials
  • Path integral formulation: Leading term in $\hbar$ expansion

Example: Linear Potential

For $V(x) = Fx$ (constant force):

$$E_n \approx \left(\frac{3\pi\hbar F}{2m}\right)^{2/3}\left(n + \frac{3}{4}\right)^{2/3}$$

Compare with exact Airy function solution

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