WKB Approximation

The WKB approximation is a cornerstone semiclassical technique in quantum mechanics, enabling approximate solutions to the Schrödinger equation for potentials that change slowly compared to the particle's wavelength. Without it, systems like quantum tunneling in chemical reactions or energy levels in complex potential wells would remain analytically intractable. This method provides a powerful bridge, revealing how classical concepts like momentum and action resurface in the quantum realm with crucial modifications.

Deriving the WKB Solution for Slowly Varying Potentials

We begin with the one-dimensional, time-independent Schrödinger equation: $$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+V(x)\psi (x)=E\psi (x)$$ The core idea is that when the potential $V(x)$ changes gradually, the wavefunction's phase should accumulate much like a classical wave. To formalize this, we introduce the semiclassical approximation by assuming the solution takes the form $\psi (x)=e^{iS(x)/\hslash }$. Substituting this ansatz yields an equation for $S(x)$. The key step is to expand $S(x)$ in a power series in $\hslash$: $S(x)=S_0(x)+\hslash S_1(x)+...$.

Collecting terms of the same order in $\hslash$ provides a systematic derivation. The leading order (${\hslash }^0$) recovers the classical local momentum $p(x)=\sqrt{2m(E-V(x))}$. The next order (${\hslash }^1$) gives an amplitude correction. The resulting approximate wavefunction, valid in regions where $E>V(x)$ (the classically allowed region), is: $$\psi (x)\approx \frac{C}{\sqrt{|p(x)|}}\exp \left(\pm \frac{i}{\hslash }\int p(x)dx\right)$$ In regions where $E<V(x)$ (the classically forbidden region), $p(x)$ becomes imaginary, and the solution exponentially decays or grows: $\psi (x)\approx \frac{C^{\prime }}{\sqrt{|p(x)|}}\exp \left(\pm \frac{1}{\hslash }\int |p(x)|dx\right)$. The validity condition for this approximation is that the potential's variation length scale is much larger than the local de Broglie wavelength, mathematically expressed as $\hslash |dV/dx|/|p(x){|}^3\ll 1$.

Applying Connection Formulas at Turning Points

The solutions above diverge at turning points $x_t$ where $E=V(x_t)$ and $p(x_t)=0$, violating the slow-variation condition. To connect wavefunctions across these points, we must solve the Schrödinger equation in a small region around $x_t$ where the potential is approximated as linear, $V(x)\approx E+V^{\prime }(x_t)(x-x_t)$. The exact solution here involves Airy functions.

Matching the asymptotic forms of the Airy function to the WKB solutions on either side yields the connection formulas. For a turning point where the classically allowed region is to the left (lower $x$) and the forbidden region to the right, the correct connection is: $$\frac{2}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_x^{x_t}p(x^{\prime })d{x}^{\prime }-\frac{\pi }{4}\right)\leftrightarrow \frac{1}{\sqrt{\kappa (x)}}\exp \left(-\frac{1}{\hslash }{\int }_{x_t}^x\kappa (x^{\prime })d{x}^{\prime }\right)$$ where $\kappa (x)=\sqrt{2m(V(x)-E)}/\hslash =|p(x)|/\hslash$. The phase shift of $-\pi /4$ is crucial. The reverse connection, from an exponentially decaying solution in the forbidden region to an oscillatory one in the allowed region, uses a similar rule with a $+\pi /4$ phase. You must apply these formulas in the direction of exponential decay to avoid introducing spurious growing exponentials.

Estimating Tunneling Probabilities Through Barriers

A prime application of WKB theory is calculating the probability for a particle to quantum tunnel through a potential barrier where $E<V(x)$. Consider a barrier between points $x_1$ and $x_2$, the classical turning points. The typical scenario involves an incident wave from the left, a forbidden barrier region, and an transmitted wave on the right.

The transmission probability $T$ is approximately the square of the ratio of the wavefunction amplitude after the barrier to its amplitude before. The WKB method yields the celebrated formula: $$T\approx \exp \left(-2{\int }_{x_1}^{x_2}\kappa (x)dx\right)=\exp \left(-\frac{2}{\hslash }{\int }_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx\right)$$ This integral is over the classically forbidden region. For instance, to estimate alpha decay rates, you would model the nuclear potential barrier and compute this integral. The accuracy of this estimate depends on the barrier being wide and smooth enough for the WKB condition to hold; thin or low barriers may require more precise matching.

Computing Bound State Energies with the Bohr-Sommerfeld Condition

For a particle confined in a potential well with two turning points at $x=a$ and $x=b$ (where $E=V$), the wavefunction must be single-valued and decay exponentially outside the well. Applying the connection formulas at both turning points enforces a quantization condition on the phase accumulated in the allowed region.

This leads to the Bohr-Sommerfeld quantization condition: $$\oint p(x)dx=2\pi \hslash \left(n+\frac{1}{2}\right)$$ where the cyclic integral is over a full classical orbit, or equivalently, twice the integral between the turning points: ${\int }_a^{b}p(x)dx=\pi \hslash (n+1/2)$. Here, $n$ is a non-negative integer quantum number. The term $1/2$ arises directly from the $-\pi /4$ phase shifts at both turning points, a purely quantum correction to the old quantum theory. For example, applying this to a harmonic oscillator potential $V(x)=\frac{1}{2}m{\omega }^2{x}^2$ yields $E_n=\hslash \omega (n+1/2)$, which is exact. For anharmonic wells, it provides excellent approximate energies.

Common Pitfalls

1. Ignoring the Validity Condition: Applying WKB when the potential changes abruptly (e.g., at a sharp step) gives incorrect results. Always check that $\hslash |V^{\prime }|/|p{|}^3\ll 1$. For a rough guide, the local de Broglie wavelength $\lambda (x)=h/p(x)$ should vary slowly over a distance of itself.
2. Misapplying Connection Formulas: A frequent error is connecting solutions in the wrong direction, accidentally including an exponentially growing solution in a forbidden region. Remember, physical problems typically require exponential decay away from allowed regions, so connect from the oscillatory side toward the decaying side.
3. Sign Errors in Phase Accumulation: When using the Bohr-Sommerfeld condition, ensure the momentum integral is for the correct limits and that the phase from connection formulas is consistently $-\pi /4$ or $+\pi /4$. Double-check the ordering of turning points.
4. Overlooking the Prefactor: While the exponential factor in tunneling dominates, the prefactor $1/\sqrt{p(x)}$ in the wavefunction amplitude is essential for calculating accurate reflection coefficients or normalization constants in bound states. Neglecting it can lead to errors in probability densities.

Summary

• The WKB approximation provides semiclassical wavefunctions $\psi \sim \frac{1}{\sqrt{p}}{e}^{\pm i\int pdx/\hslash }$ in allowed regions, valid when the potential $V(x)$ varies slowly compared to the local de Broglie wavelength.
• At turning points where $E=V(x)$, connection formulas derived from linear potential matching must be used to link oscillatory and exponential solutions, introducing critical $-\pi /4$ phase shifts.
• The tunneling probability through a smooth barrier is estimated by $T\approx \exp (-2\int \kappa dx)$, where the integral is over the forbidden region.
• Bound state energies in a potential well are approximated by the Bohr-Sommerfeld quantization rule: ${\int }_a^{b}p(x)dx=\pi \hslash (n+1/2)$, which incorporates quantum phase corrections.
• Success hinges on respecting the method's validity conditions and carefully applying connection formulas to avoid unphysical, exponentially growing solutions.