3.3.2 WKB Approximation#

Prompts

What is the WKB ansatz \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\), and how does it reduce the Schr”{o}dinger equation to the Hamilton-Jacobi equation?
How does WKB describe tunneling through a classically forbidden barrier?
What are turning points, and why do connection formulas require a \(\pi/4\) phase shift?
When does the WKB approximation break down, and what is the precise validity criterion?

Lecture Notes#

Overview#

The WKB (Wentzel-Kramers-Brillouin) approximation solves the Schrödinger equation semiclassically when the potential varies slowly compared to the de Broglie wavelength. The ansatz \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\) splits the wavefunction into a rapidly oscillating phase and a slowly varying amplitude. At leading order in \(\hbar\), the phase satisfies the Hamilton-Jacobi equation — classical mechanics — while the amplitude is fixed by probability conservation.

The WKB Ansatz#

Write the wavefunction as

\[ \psi(x, t) = A(x, t) \, \mathrm{e}^{\mathrm{i}S(x, t)/\hbar} \]

Substituting into the Schrödinger equation and expanding in powers of \(\hbar\):

WKB Hierarchy

Leading order (\(\hbar^0\)): the phase \(S\) satisfies the Hamilton-Jacobi equation

\[ \frac{\partial S}{\partial t} + \frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^2 + V(x) = 0 \]

Next order (\(\hbar^1\)): the amplitude \(A\) satisfies a continuity equation — probability conservation.

Derivation: WKB Equations from Schrödinger

Substitute \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\) into \(\mathrm{i}\hbar\partial_t\psi = -\frac{\hbar^2}{2m}\partial_x^2\psi + V\psi\). Computing derivatives:

\[ \frac{\partial^2\psi}{\partial x^2} = \left(A'' + \frac{2\mathrm{i}}{\hbar}A'S' + \frac{\mathrm{i}}{\hbar}AS'' - \frac{A}{\hbar^2}(S')^2\right) \mathrm{e}^{\mathrm{i}S/\hbar} \]

Collecting by powers of \(\hbar\) after canceling \(\mathrm{e}^{\mathrm{i}S/\hbar}\):

Order \(\hbar^0\) (imaginary part): Hamilton-Jacobi equation for \(S\).

Order \(\hbar^1\) (real part): \(\partial_t A + \frac{1}{m}A'S' + \frac{A}{2m}S'' = 0\), a continuity equation for \(A^2\) (probability density).

Stationary States: Allowed and Forbidden Regions#

For energy eigenstates with \(S(x,t) = S_0(x) - Et\), the Hamilton-Jacobi equation gives \(S_0'(x) = \pm p(x)\) where \(p(x) = \sqrt{2m(E - V(x))}\) is the local classical momentum.

WKB Wavefunctions

Classically allowed (\(E > V\), real momentum):

\[ \psi(x) \sim \frac{1}{\sqrt{p(x)}} \exp\left(\pm\frac{\mathrm{i}}{\hbar}\int^x p(x')\,\mathrm{d}x'\right) \]

Classically forbidden (\(E < V\), imaginary momentum \(p \to \mathrm{i}\kappa\)):

\[ \psi(x) \sim \frac{1}{\sqrt{\kappa(x)}} \exp\left(\mp\frac{1}{\hbar}\int^x \kappa(x')\,\mathrm{d}x'\right) \]

where \(\kappa(x) = \sqrt{2m(V(x) - E)}\).

The amplitude \(1/\sqrt{p}\) reflects probability conservation: where the particle moves slowly (small \(p\)), it spends more time, so the probability density is larger.

Connection Formulas at Turning Points#

At a turning point \(x_0\) where \(E = V(x_0)\), the momentum vanishes and the WKB solutions diverge. The exact solution near \(x_0\) (an Airy function) bridges the two regions:

Allowed side (\(x < x_0\)):

\[ \psi \sim \frac{C}{\sqrt{p}} \sin\left(\frac{1}{\hbar}\int_x^{x_0} p\,\mathrm{d}x' + \frac{\pi}{4}\right) \]

Forbidden side (\(x > x_0\)):

\[ \psi \sim \frac{C}{\sqrt{\kappa}} \exp\left(-\frac{1}{\hbar}\int_{x_0}^x \kappa\,\mathrm{d}x'\right) \]

The \(\pi/4\) phase shift is essential — it accounts for the smooth transition through the turning point and determines the Maslov index in quantization conditions (§3.3.3).

Discussion: WKB Exactness

WKB is an approximation, yet it gives exact energy levels for the harmonic oscillator. This raises questions:

For which potentials is WKB exact? What special property must they have?
WKB predicts tunneling rates via \(T \propto \mathrm{e}^{-2\gamma}\). Can you design a potential where WKB gives a very wrong tunneling rate?
Why does WKB work well for atomic energy levels when \(\hbar\) is not truly small? Is there a hidden small parameter?

Tunneling#

For a particle incident on a barrier where \(V > E\) between turning points \(a\) and \(b\):

WKB Tunneling Probability

\[ T \approx \mathrm{e}^{-2\gamma}, \quad \gamma = \frac{1}{\hbar}\int_a^b \kappa(x)\,\mathrm{d}x = \frac{1}{\hbar}\int_a^b \sqrt{2m(V(x) - E)}\,\mathrm{d}x \]

The tunneling probability is exponentially sensitive to the barrier width and height. The penetration depth into the forbidden region is \(\lambda_\mathrm{decay} = \hbar/\kappa\).

Example: Electron Tunneling

Problem. Estimate the tunneling probability for an electron through a 1 nm barrier with height 1 eV above its energy.

Solution. \(\gamma \approx (10^{-9}\;\mathrm{m})\sqrt{2(9.1\times 10^{-31}\;\mathrm{kg})(1.6\times 10^{-19}\;\mathrm{J})}/(10^{-34}\;\mathrm{J{\cdot}s}) \approx 5\), giving \(T \sim \mathrm{e}^{-10} \approx 10^{-4.3}\). Rare but non-negligible at atomic scales — this is why quantum tunneling drives processes like alpha decay and scanning tunneling microscopy.

Validity of WKB#

The WKB approximation requires the potential to vary slowly on the scale of the local de Broglie wavelength:

\[ \left|\frac{\mathrm{d}\ln p}{\mathrm{d}x}\right| = \left|\frac{V'(x)}{2(E - V(x))}\right| \ll 1 \]

This breaks down at turning points (remedied by connection formulas), potential discontinuities, and rapidly oscillating potentials.

Summary#

WKB ansatz \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\): phase obeys Hamilton-Jacobi (classical mechanics); amplitude fixed by probability conservation
Connection formulas: match oscillating and decaying solutions across turning points with \(\pi/4\) phase shifts
Tunneling: \(T \approx \mathrm{e}^{-2\gamma}\) with \(\gamma = \hbar^{-1}\int \kappa\,\mathrm{d}x\) — exponentially suppressed by barrier width and height
Validity: excellent for smooth, slowly varying potentials; breaks down at turning points and sharp features

Homework#

1. WKB Validity Condition
A particle of mass \(m\) moves in a potential \(V(x) = V_0 \mathrm{e}^{-x^2/a^2}\) (Gaussian potential) with energy \(E = V_0/4\). Estimate the region where the WKB approximation is valid using the condition \(\left|\frac{\mathrm{d}\ln p}{dx}\right| \ll 1\). At what value of \(x\) does WKB validity break down?

2. WKB Ansatz and Hamilton-Jacobi
Show that substituting the WKB ansatz \(\psi(x) = A(x) \mathrm{e}^{\mathrm{i}S(x)/\hbar}\) into the time-independent Schrödinger equation \(-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi}{dx^2} + V(x)\psi = E\psi\) yields the Hamilton-Jacobi equation \(\frac{1}{2m}\left(\frac{\mathrm{d}S}{dx}\right)^2 + V(x) = E\) to leading order in \(\hbar\). What does the next-order correction tell you about the amplitude \(A(x)\)?

3. Classically Allowed vs. Forbidden Regions
For a particle with energy \(E\) in a potential \(V(x)\), write the WKB wavefunction in (a) a classically allowed region where \(E > V(x)\), and (b) a classically forbidden region where \(E < V(x)\). Explain physically why the amplitude behaves as \(1/\sqrt{p}\) in the allowed region (hint: probability conservation) and why the wavefunction decays exponentially in the forbidden region.

4. Connection Formulas at a Turning Point
A classical turning point occurs at \(x_0\) where \(E = V(x_0)\). Sketch the WKB wavefunction near this point, showing:

The oscillatory form on the left (\(x < x_0\), allowed region)
The exponentially decaying form on the right (\(x > x_0\), forbidden region)
The phase shift of \(\pi/4\) that appears in the connection formula

Why is this phase shift necessary, and what happens to the wavefunction if you ignore it?

5. Tunneling Through a Square Barrier
A particle of mass \(m\) and energy \(E\) encounters a square potential barrier: \(V(x) = V_0\) for \(0 < x < a\), and \(V(x) = 0\) elsewhere, with \(0 < E < V_0\).
(a) Using the WKB formula, show that the tunneling probability is approximately:

\[T \approx \exp\left(-\frac{2a}{\hbar}\sqrt{2m(V_0 - E)}\right)\]

(b) For an electron (\(m = 9.1 \times 10^{-31}\) kg) with \(V_0 - E = 2\) eV and \(a = 2 \times 10^{-10}\) m (atomic scale), calculate the tunneling probability. Is tunneling rare or common at this scale?

6. Penetration Depth in the Forbidden Region
The characteristic penetration depth of a wavefunction into a classically forbidden region is \(\lambda_{\text{decay}} = \hbar/\kappa\), where \(\kappa = \sqrt{2m(V - E)}\). For an electron near the surface of a metal (work function \(\phi = 5\) eV), how deep does the electron wavefunction penetrate into the vacuum (region where the electron is classically forbidden)? Express your answer in Ångströms. What does this imply for electron emission from metals?

7. Bohr-Sommerfeld Quantization
A particle is bound in a potential well with classical turning points at \(x = a\) and \(x = b\) (where \(V(a) = V(b) = E\)). Using the WKB connection formulas at both turning points, derive the Bohr-Sommerfeld quantization condition:

\[\int_a^b p(x) dx = \pi\hbar\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots\]

Explain where the factor \(1/2\) comes from (hint: phase shifts at both turning points).

8. Harmonic Oscillator via Bohr-Sommerfeld
For a harmonic oscillator \(V(x) = \frac{1}{2}m\omega^2 x^2\), use the Bohr-Sommerfeld quantization condition to derive the energy levels \(E_n = \hbar\omega(n + 1/2)\). Show all steps of the integral. Why does WKB give an exact result for this potential when it is supposed to be an approximation?