Table of Contents
- Introduction
- Motivation and Physical Context
- Basic Idea of the WKB Method
- Mathematical Derivation
- The WKB Ansatz
- Validity and Conditions
- Classical Turning Points
- Matching at Turning Points
- Connection Formulas
- Bohr-Sommerfeld Quantization
- Example: Particle in a Linear Potential
- Example: Harmonic Oscillator (WKB vs Exact)
- Tunneling and Barrier Penetration
- Application in Alpha Decay
- Limitations and Failures
- Extensions and Modern Uses
- Conclusion
1. Introduction
The WKB (Wentzel–Kramers–Brillouin) approximation is a semi-classical method in quantum mechanics used to approximate solutions to the Schrödinger equation in the limit of slowly varying potentials. It bridges the classical and quantum descriptions, offering insight into wave-like behavior in nearly classical systems.
2. Motivation and Physical Context
Many quantum systems exhibit behavior that resembles classical motion in some regimes, especially when the action is large compared to \( \hbar \). In such cases, exact quantum solutions may be difficult to obtain, but WKB provides an elegant approximation.
3. Basic Idea of the WKB Method
The WKB approximation assumes that the quantum wavefunction varies rapidly compared to the potential. The method transforms the Schrödinger equation into a form similar to classical mechanics, exploiting the idea of locally plane wave solutions in classically allowed regions.
4. Mathematical Derivation
Start from the time-independent Schrödinger equation in one dimension:
\[
-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)
\]
Rewriting:
\[
\frac{d^2\psi(x)}{dx^2} + \frac{2m}{\hbar^2}(E – V(x))\psi(x) = 0
\]
Define the local wavenumber:
\[
k(x) = \frac{\sqrt{2m(E – V(x))}}{\hbar}
\]
5. The WKB Ansatz
Assume a solution of the form:
\[
\psi(x) = A(x) e^{i S(x)/\hbar}
\]
Substitute into the Schrödinger equation and expand in powers of \( \hbar \). Retaining leading order yields the WKB form:
\[
\psi(x) \approx \frac{C}{\sqrt{k(x)}} \exp\left(\pm i \int^x k(x’) dx’\right)
\]
This is valid in classically allowed regions where \( E > V(x) \).
6. Validity and Conditions
The WKB approximation is valid when:
\[
\left| \frac{d\lambda(x)}{dx} \right| \ll 1 \quad \text{or} \quad \left| \frac{dV}{dx} \right| \ll \left(2m(E – V(x))^3\right)^{1/2}
\]
i.e., the potential must vary slowly over a de Broglie wavelength.
7. Classical Turning Points
At points where \( E = V(x) \), \( k(x) = 0 \) and the WKB solution diverges. These points are known as turning points and require special treatment using connection formulas.
8. Matching at Turning Points
To patch WKB solutions across turning points, we use Airy function solutions and match asymptotics. The result leads to phase shifts and quantization conditions.
9. Connection Formulas
Near a turning point \( x_0 \), define:
- \( x < x_0 \): classically forbidden
- \( x > x_0 \): classically allowed
The connection formula is:
\[
\psi(x) \sim \frac{C}{|k(x)|^{1/2}} \exp\left( \pm \int |k(x)| dx \right) \leftrightarrow \frac{C’}{k(x)^{1/2}} \cos\left( \int k(x) dx – \frac{\pi}{4} \right)
\]
10. Bohr-Sommerfeld Quantization
For bound states between turning points \( x_1 \) and \( x_2 \), the quantization condition is:
\[
\int_{x_1}^{x_2} k(x) dx = \left(n + \frac{1}{2}\right)\pi \hbar
\]
This provides approximate energy levels in 1D potentials.
11. Example: Particle in a Linear Potential
For \( V(x) = Fx \), the turning point is \( x_0 = E/F \). The WKB solution yields Airy function approximations and matches asymptotically with the exact solution.
12. Example: Harmonic Oscillator (WKB vs Exact)
For \( V(x) = \frac{1}{2} m \omega^2 x^2 \), WKB gives:
\[
E_n = \hbar \omega \left(n + \frac{1}{2}\right)
\]
which matches the exact result—showing WKB’s power in symmetric potentials.
13. Tunneling and Barrier Penetration
In classically forbidden regions (\( E < V(x) \)):
\[
\psi(x) \approx \frac{C}{\sqrt{|k(x)|}} \exp\left( -\int |k(x)| dx \right)
\]
This yields the tunneling probability:
\[
T \approx \exp\left(-2 \int_{x_1}^{x_2} |k(x)| dx\right)
\]
14. Application in Alpha Decay
Gamow used the WKB approximation to calculate alpha decay rates. The alpha particle tunnels through the nuclear potential barrier with probability governed by the exponential decay from the WKB expression.
15. Limitations and Failures
- Not valid near sharp potential changes.
- Breaks down at or very close to turning points without careful matching.
- Not useful for highly quantum systems (e.g., low-energy states in deep wells).
16. Extensions and Modern Uses
- Multidimensional WKB in molecular physics.
- Maslov indices and complex WKB paths.
- Quantum chaos and semiclassical approximations.
- Path integral interpretations in field theory.
17. Conclusion
The WKB approximation is a cornerstone of semiclassical analysis in quantum mechanics. By approximating wavefunctions in slowly varying potentials, it connects quantum phenomena with classical intuition. From quantization rules to tunneling, WKB remains an essential analytical tool across atomic, nuclear, and particle physics.
