3.3.2 WKB Approximation#

Prompts

What is the WKB ansatz \(\psi = A(x)\,\mathrm{e}^{\mathrm{i}W(x)/\hbar}\,\mathrm{e}^{-\mathrm{i}Et/\hbar}\), and how does it reduce the time-independent Schrödinger equation to the Hamilton-Jacobi equation?
Why does the WKB amplitude take the form \(A \propto 1/\sqrt{p(x)}\) in the classically allowed region, and how does this reflect probability conservation?
How does WKB describe tunneling through a classically forbidden barrier, and where does the factor \(T \approx \mathrm{e}^{-2\gamma}\) come from?
What are turning points, and why does the connection formula across a soft turning point require a \(\pi/4\) phase shift?
When does the WKB approximation break down, and what is the dimensionful validity criterion?

Lecture Notes#

Overview#

Section 3.3.1 showed that stationary phase turns the path integral into a semiclassical propagator: a rapidly varying classical-action phase multiplied by a slowly varying fluctuation prefactor. For an energy eigenstate the classical action separates into a spatial part and a time part, \(S_\mathrm{cl}=W(x)-Et\), so the same structure becomes a wavefunction with phase \(W(x)/\hbar\) and amplitude \(A(x)\). This is the WKB approximation. Its physical picture is direct: the phase records the local classical action, while the amplitude records how nearby paths spread or concentrate. In this note we turn that picture into a practical approximation by substituting the WKB form into the Schrödinger equation: the leading term gives Hamilton-Jacobi mechanics, the next term fixes the amplitude, and continuing to imaginary momentum describes tunneling.

The WKB Ansatz#

Motivated by the semiclassical propagator, look for stationary states of energy \(E\) in the form

(100)#\[ \psi(x,t) = A(x)\,\exp\!\left[\frac{\mathrm{i}\,W(x)}{\hbar}\right]\,\mathrm{e}^{-\mathrm{i}Et/\hbar}. \]

Substituting into the time-independent Schrödinger equation \(-(\hbar^{2}/2m)\psi'' + V\psi = E\psi\) and collecting terms order by order in \(\hbar\) produces a hierarchy of equations:

WKB equations

Leading order (\(\hbar^{0}\)): \(W(x)\) obeys the (time-independent) Hamilton-Jacobi equation

(101)#\[ \frac{1}{2m}\!\left(\frac{\mathrm{d}W}{\mathrm{d}x}\right)^{2} + V(x) = E. \]
Next order (\(\hbar^{1}\)): \(A(x)\) obeys a continuity equation

(102)#\[ \frac{\mathrm{d}}{\mathrm{d}x}\!\left(A^{2}\,W'\right) = 0. \]

Derivation: WKB equations from Schrödinger

Substitute the WKB ansatz (100) into the time-independent Schrödinger equation \(-(\hbar^{2}/2m)\psi'' + V\psi = E\psi\). Compute the second derivative,

\[ \psi'' = \left(A'' + \frac{2\mathrm{i}}{\hbar}A'\,W' + \frac{\mathrm{i}}{\hbar}A\,W'' - \frac{1}{\hbar^{2}}A\,(W')^{2}\right)\,\mathrm{e}^{\mathrm{i}W/\hbar}\,\mathrm{e}^{-\mathrm{i}Et/\hbar}. \]

Multiply by \(-\hbar^{2}/(2m)\), add \((V-E)\psi\), cancel the common phase \(\mathrm{e}^{\mathrm{i}W/\hbar}\,\mathrm{e}^{-\mathrm{i}Et/\hbar}\), and multiply through by \(2m\) to expose the ordering in \(\hbar\):

(103)#\[ \underbrace{A\left[(W')^{2}+2m(V-E)\right]}_{\hbar^{0}} \;-\;\mathrm{i}\hbar\,\underbrace{\bigl(2A'W'+AW''\bigr)}_{\hbar^{1}} \;-\;\hbar^{2}\,\underbrace{A''}_{\hbar^{2}} \;=\;0. \]

For the equation to hold at every order in \(\hbar\) (treating \(\hbar\) as a small parameter), each bracketed group must vanish separately.

Order \(\hbar^{0}\). Setting the first bracket to zero gives the Hamilton-Jacobi equation (101). Solving for the phase gradient defines the local momentum (104),

\[ W'(x) \;=\; p(x) \;=\; \pm\sqrt{2m\,(E - V(x))}. \]

Order \(\hbar^{1}\). Setting the second bracket to zero gives the amplitude equation (102),

\[ 2A'\,W' + A\,W'' = 0. \]

Substituting \(W' = p\) (so \(W'' = p'\)),

\[ 2A'\,p + A\,p' = 0. \]

Multiplying by \(A\) turns the left-hand side into a total derivative,

\[ 2A\,A'\,p + A^{2}\,p' \;=\; \frac{\mathrm{d}}{\mathrm{d}x}\bigl(A^{2}\,p\bigr) \;=\; 0, \]

so \(A^{2}p\) is constant in \(x\) and the WKB amplitude takes the form \(A(x) = C/\sqrt{p(x)}\) — Eq. (106). Constancy of \(A^{2}p\) is exactly probability-current conservation \(j = (\hbar/m)\,A^{2}p/\hbar = A^{2}p/m\).

Order \(\hbar^{2}\). The remaining \(-\hbar^{2}A''/(2m)\) piece is dropped — it would couple \(A\) back into the Hamilton-Jacobi equation as a quantum correction (the Bohm potential) and lies beyond the leading-order WKB approximation.

The Hamilton-Jacobi equation says: the phase gradient is the local momentum of a classical particle of energy \(E\),

(104)#\[ p(x) \;\equiv\; \frac{\mathrm{d}W}{\mathrm{d}x} \;=\; \pm\sqrt{2m\,\bigl(E - V(x)\bigr)}. \]

The two signs in (104) give the two WKB branches, corresponding to waves whose phase accumulates with positive or negative local momentum.

Substituting \(W'=p\) into the amplitude equation (102) gives \(\frac{\mathrm{d}}{\mathrm{d}x}(A^{2}p)=0\). The phase and amplitude are therefore fixed together:

WKB solutions

The phase \(W(x)\) accumulates the classical action along the orbit:

(105)#\[ W(x) = W(x_{*}) + \int_{x_{*}}^{x} p(x')\,\mathrm{d}x'. \]
The amplitude \(A(x)\) follows from probability-current conservation \(A^{2}p=\text{const}\):

(106)#\[ A(x) = \frac{C}{\sqrt{p(x)}}. \]

So \(\vert\psi\vert^{2}\propto 1/p\) is largest where the particle moves slowly — classically, the particle spends more time at small \(p\).

Allowed and Forbidden Regions#

Equation (104) gives a real \(p(x)\) where \(E > V(x)\) — the classically allowed region — and an imaginary \(p\) where \(E < V(x)\) — the classically forbidden region.

WKB wavefunctions

In the allowed region (\(E > V\)), with \(p(x) = \sqrt{2m(E-V)}\):

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

In the forbidden region (\(E < V\)), with \(\kappa(x) = \sqrt{2m(V-E)}\):

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

The two signs in each formula give independent solutions — right- vs left-moving in the allowed region; exponentially decaying vs growing in the forbidden region.

Validity of WKB approximation

WKB assumes a local classical momentum: over one de Broglie wavelength, the potential should look almost constant. Equivalently, the local wavelength

\[ \lambda(x)=\frac{2\pi\hbar}{p(x)} \]

should change by much less than one wavelength over a wavelength:

(109)#\[ \left|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\right| \;\ll\; 2\pi. \]

Using \(p(x)=\sqrt{2m(E-V)}\), this becomes the dimensionful condition

(110)#\[ \hbar\,\bigl|\,V'(x)\bigr| \;\ll\; 2\sqrt{2m}\,\bigl(E - V(x)\bigr)^{3/2}. \]

This criterion makes the failure modes easy to diagnose:

Near a turning point, \(E=V(x)\), the right-hand side vanishes and the WKB amplitude \(1/\sqrt{p}\) diverges. The local plane-wave picture cannot be trusted there.
At a discontinuity, \(V'(x)\) is undefined, so there is no slowly varying potential to expand around.
Where the potential changes on the scale of a wavelength, neighboring points no longer have nearly the same local momentum, so the slowly varying amplitude assumption breaks down.

The cure near a smooth turning point is not to abandon semiclassics, but to replace the local WKB form by an Airy-function matching region.

Tunneling Through a Smooth Barrier#

Consider a particle of energy \(E\) incident on a smooth barrier with \(V > E\) between turning points \(a < b\) (Fig. Fig. 16).

WKB tunneling probability

Stitching the WKB solutions across both turning points gives the celebrated tunneling formula:

(111)#\[ T \;\approx\; \mathrm{e}^{-2\gamma}. \]

(112)#\[\begin{split} \begin{split} \gamma &\;=\; \frac{1}{\hbar}\int_{a}^{b}\kappa(x)\,\mathrm{d}x\\ &\;=\; \frac{1}{\hbar}\int_{a}^{b}\sqrt{2m\,\bigl(V(x)-E\bigr)}\,\mathrm{d}x. \end{split} \end{split}\]

Transmission is exponentially sensitive to the integrated \(\sqrt{V-E}\) across the barrier.

Derivation: tunneling exponent from connection formulas

Pick the WKB solution with only an outgoing (right-moving) wave on the right of the barrier, \(x > b\), with amplitude \(F\):

\[ \psi(x) \sim \frac{F}{\sqrt{p}}\,\mathrm{e}^{\mathrm{i}\int_{b}^{x}p\,\mathrm{d}x'/\hbar}. \]

Apply the connection formula (113) at \(x = b\) (allowed \(\to\) forbidden, going leftward into the barrier): the outgoing oscillation matches a purely decaying exponential just inside the barrier,

\[ \psi(x) \sim \frac{F'}{\sqrt{\kappa}}\,\mathrm{e}^{-\frac{1}{\hbar}\int_{b}^{x}\kappa\,\mathrm{d}x'}\qquad (a<x<b), \]

with \(|F'| = |F|\) up to factors of order unity from the connection. Continuing this decaying solution to \(x = a\), its magnitude has shrunk by the factor \(\mathrm{e}^{-\gamma}\) where \(\gamma = \frac{1}{\hbar}\int_{a}^{b}\kappa\,\mathrm{d}x\).

A second connection at \(x = a\) (forbidden \(\to\) allowed, leftward) maps this exponentially small value into an oscillatory wave on the left whose amplitude is therefore \(\mathrm{e}^{-\gamma}\) times the original outgoing amplitude. The transmission probability is the squared amplitude ratio,

\[ T = \frac{|F|^{2}}{|A_\text{in}|^{2}} \approx \mathrm{e}^{-2\gamma}. \]

The factor of two comes from squaring; the connection-formula prefactors give corrections of order unity that are ignored at WKB accuracy.

The penetration depth at energy \(E\) is \(\lambda_\text{decay}(x) = \hbar/\kappa(x)\) — the length scale on which the wavefunction decays inside the barrier.

Example: electron tunneling

Problem. Estimate \(T\) for an electron through a \(1\,\text{nm}\) barrier with \(V - E = 1\,\text{eV}\).

Solution. With \(a = 1\,\text{nm}\), \(V - E = 1.6\times 10^{-19}\,\text{J}\), \(m = 9.1\times 10^{-31}\,\text{kg}\), and a constant \(\kappa = \sqrt{2m(V-E)}\),

\[ \gamma = \frac{a\,\sqrt{2m(V-E)}}{\hbar} \approx 5, \]

so \(T \sim \mathrm{e}^{-10} \approx 5\times 10^{-5}\). Rare but observable — this is why scanning tunneling microscopy and alpha decay work.

Poll: WKB in a forbidden region

In the classically forbidden region where \(V > E\), the local momentum \(p = \sqrt{2m(E-V)}\) becomes imaginary. What does WKB predict for \(\psi(x)\) there?

(A) It oscillates with increasing frequency the deeper one penetrates.

(B) It decays (or grows) exponentially with rate \(\kappa(x)/\hbar = \sqrt{2m(V-E)}/\hbar\), consistent with quantum tunneling.

(D) It is constant, since classically the particle never enters the forbidden region.

Connection at a Turning Point#

At a soft turning point, the wavefunction changes character: oscillatory in the allowed region, exponentially decaying in the forbidden region. We need a rule to connect these two pieces continuously across the boundary, and that rule is what brings the famous \(\pi/4\) phase shift into WKB.

Where does \(\pi/4\) come from? Matching wave to exponential

The two regions host different functions — sinusoid on one side, exponential on the other. Continuity of \(\psi\) and \(\psi'\) across the boundary alone fixes the phase of the allowed-side wave. Consider Fig. Fig. 17 configuration:

Forbidden side. A decaying exponential \(\psi\propto\mathrm{e}^{+\kappa x}\) has the defining property \(\psi'/\psi=+\kappa\): the slope equals the value (up to \(\kappa\) factor rescaling).
Allowed side. A sinusoid \(\psi\propto\sin(kx+\phi)\) has \(\psi'/\psi=k\cot(kx+\phi)\). Continuity of \(\psi\) and \(\psi'\) at \(x=0\) forces \(k\cot\phi=\kappa\). Near the turning point \(k\approx\kappa\), so \(\cot\phi=1\), giving \(\phi=\pi/4\).

The matching forces the allowed-side wave to be at the \(\pi/4\) phase of its cycle at the boundary — precisely where \(\sin=\cos\) (slope equals value).

Soft turning point with forbidden region on the left and allowed region on the right. The decaying exponential in the forbidden region matches a sin(kx+pi/4) wave in the allowed region. The wave's extrapolated node sits inside the forbidden region at kx = -pi/4, marking a "ghost wall." — Fig. 17 The allowed-side wave \(\sin(kx+\pi/4)\) extrapolates to a node at \(kx=-\pi/4\) — the **ghost wall** (hatched line). The wave overshoots the classical turning point at \(x=0\) by \(\pi/4\) (in phase units) before being “reflected.”#

Ghost wall: \(\pi/2\) effective extra phase per soft turn

Equivalently: a soft turning point behaves like a hard wall retracted by \(\pi/4\) into the forbidden region. The wave overshoots the classical boundary by \(\pi/4\) (in phase units) before “hitting” the ghost wall.

A soft reflection has two halves: \(\pi/4\) in to the ghost wall, U-turn, \(\pi/4\) out. The wave acquires an effective extra phase of

\[ \boxed{\;\Delta\Phi_{\text{soft turn}} \;=\; \tfrac{\pi}{4}+\tfrac{\pi}{4} \;=\; \tfrac{\pi}{2}\;} \]

per soft turning point. This per-turn result is the input that fixes the Bohr-Sommerfeld quantization rule of §3.3.3.

Connection formula at a soft turning point

For a turning point at \(x_0\) with the allowed region on the right and the forbidden region on the left (orientation of Fig. Fig. 17),

(113)#\[ \frac{1}{\sqrt{\kappa}}\,\exp\!\left[-\frac{1}{\hbar}\!\int_{x}^{x_0}\!\kappa\,\mathrm{d}x'\right] \;\longleftrightarrow\; \frac{2}{\sqrt{p}}\,\sin\!\left[\frac{1}{\hbar}\!\int_{x_0}^{x}\!p\,\mathrm{d}x' + \frac{\pi}{4}\right]. \]

The matching argument above is the heuristic content; a more rigorous derivation linearizes \(V(x)\) near \(x_0\), reducing the Schrödinger equation to the Airy equation, and matches the Airy function’s asymptotic forms in the two regions (HW 3.3.2.6).

Summary#

WKB ansatz: \(\psi = A(x)\,\mathrm{e}^{\mathrm{i}W(x)/\hbar}\,\mathrm{e}^{-\mathrm{i}Et/\hbar}\).
Phase obeys Hamilton-Jacobi; phase gradient is the local momentum \(p(x) = \pm\sqrt{2m(E-V)}\).
Amplitude: \(A(x) = C/\sqrt{p}\) from probability conservation — large where the particle moves slowly.
Allowed vs forbidden: real \(p\) → oscillation; imaginary \(p\) → exponential decay.
Soft matching: at a turning point, continuity of \(\psi\) and \(\psi'\) forces the allowed-side wave to be at the \(\pi/4\) point of its cycle at the boundary.
Ghost wall: soft turn \(=\) hard wall retracted by \(\pi/4\) into the forbidden region; effective extra phase \(\pi/2\) per soft reflection.
Tunneling: \(T \approx \mathrm{e}^{-2\gamma}\) with \(\gamma = \int\kappa\,\mathrm{d}x/\hbar\).
Validity: \(\hbar|V'| \ll 2\sqrt{2m}(E-V)^{3/2}\); fails at turning points and discontinuities.

Homework#

1. Hamilton-Jacobi from WKB. Substitute the WKB ansatz \(\psi(x) = A(x)\,\mathrm{e}^{\mathrm{i}W(x)/\hbar}\) into the time-independent Schrödinger equation \(-\frac{\hbar^{2}}{2m}\psi'' + V(x)\psi = E\psi\).

(a) Collect terms by powers of \(\hbar\) and show that at leading order (\(\hbar^{0}\)) the phase \(W(x)\) satisfies the Hamilton-Jacobi equation \(\frac{1}{2m}(W')^{2} + V(x) = E\).

(b) At next order (\(\hbar^{1}\)), show that \(2A'W' + AW'' = 0\). Integrate this to obtain \(A \propto 1/\sqrt{W'} = 1/\sqrt{p(x)}\).

2. Allowed and forbidden regions. A particle of energy \(E\) moves in a slowly varying potential \(V(x)\).

(a) In the classically allowed region (\(E > V\)), write the WKB wavefunction \(\psi \sim p^{-1/2}\,\exp(\pm\frac{\mathrm{i}}{\hbar}\int^{x}p\,\mathrm{d}x')\) and explain physically why the amplitude \(1/\sqrt{p}\) means the particle is more likely found where it moves slowly.

(b) In the classically forbidden region (\(E < V\)), set \(p \to \mathrm{i}\kappa\) with \(\kappa = \sqrt{2m(V-E)}\) and show that the wavefunction decays as \(\psi \sim \kappa^{-1/2}\,\exp(-\frac{1}{\hbar}\int^{x}\kappa\,\mathrm{d}x')\).

3. Validity criterion. The WKB approximation requires \(\vert\mathrm{d}\lambda/\mathrm{d}x\vert \ll 2\pi\), where \(\lambda(x) = 2\pi\hbar/p(x)\) is the local de Broglie wavelength.

(a) Show that this is equivalent to the dimensionful condition \(\hbar\,\vert V'(x)\vert \ll 2\sqrt{2m}\,(E - V(x))^{3/2}\) in the allowed region.

(b) For a linear potential \(V(x) = Fx\) (constant force), identify where the criterion fails. Relate this to the turning point \(x_{0} = E/F\).

(c) For a Gaussian barrier \(V(x) = V_{0}\,\mathrm{e}^{-x^{2}/a^{2}}\) with energy \(E = V_{0}/4\), identify the classical turning points and discuss whether WKB is reliable away from them.

4. Square-barrier tunneling (heuristic). A particle of mass \(m\) and energy \(E < V_{0}\) encounters a rectangular barrier \(V(x) = V_{0}\) for \(0 < x < a\) and \(V = 0\) elsewhere. The discontinuous walls violate the WKB validity criterion (110), so the formula below captures only the dominant exponential factor — the true transmission coefficient has an additional reflection-coefficient prefactor of order unity.

(a) Apply \(T \approx \mathrm{e}^{-2\gamma}\) with \(\gamma = \frac{1}{\hbar}\int_{0}^{a}\kappa\,\mathrm{d}x\) to obtain \(T \approx \exp\!\left(-\frac{2a}{\hbar}\sqrt{2m(V_{0} - E)}\right)\).

(b) Evaluate \(T\) for an electron (\(m \approx 9.1\times 10^{-31}\,\text{kg}\)) with \(V_{0} - E = 2\,\text{eV}\) and \(a = 0.2\,\text{nm}\). Is tunneling significant?

(c) Show that doubling the barrier width \(a\to 2a\) replaces \(T\) by \(T^{2}\) (much smaller). Explain why tunneling is exponentially sensitive to barrier width.

5. Penetration depth. The characteristic decay length of the wavefunction in a forbidden region is \(\lambda_\text{decay} = \hbar/\kappa\) with \(\kappa = \sqrt{2m(V - E)}\).

(a) Compute \(\lambda_\text{decay}\) for an electron at a metal surface with work function \(V - E = 5\,\text{eV}\). Express the answer in angstroms.

(b) Explain why this length scale makes scanning tunneling microscopy (STM) work: the tunneling current depends exponentially on the tip-surface distance.

(c) Repeat for a proton (\(m \approx 1.67\times 10^{-27}\,\text{kg}\)) with the same barrier. Why is proton tunneling much harder than electron tunneling?

6. Airy connection formula. At a soft turning point \(x_{0}\) with \(V'(x_{0}) > 0\), linearize the potential as \(V(x) \approx E + V'(x_{0})(x - x_{0})\).

(a) Show that the Schrödinger equation reduces to the Airy equation \(\psi'' = z\psi\) with \(z = (x - x_{0})/\ell\), and identify the length scale \(\ell = \bigl(\hbar^{2}/(2m\,V'(x_{0}))\bigr)^{1/3}\).

(b) The Airy function \(\mathrm{Ai}(z)\) decays exponentially for \(z\to+\infty\) (forbidden side) and oscillates for \(z\to-\infty\) (allowed side). Use its asymptotic forms to recover the connection formula (113) with the \(\pi/4\) phase shift.

(c) Explain in one sentence why omitting the \(\pi/4\) shift in the bound-state quantization rule of §3.3.3 would mis-predict the harmonic-oscillator zero-point energy.

7. Double-well tunnel splitting. Consider a symmetric double-well potential with two minima separated by a barrier of width \(\sim a\). Let \(\omega\) be the small-oscillation frequency in either well and \(\gamma = \frac{1}{\hbar}\int_\text{barrier}\kappa\,\mathrm{d}x\).

(a) Estimate the tunneling amplitude \(\Delta \sim \hbar\omega\,\mathrm{e}^{-\gamma}\) from the WKB tunneling formula.

(b) Explain why \(\Delta\) lifts the degeneracy of the two single-well ground states, splitting them by \(\Delta E \approx 2\Delta\).

(c) For the ammonia molecule NH\(_{3}\) the nitrogen tunnels with \(\gamma \approx 5\). Estimate \(\Delta E/(\hbar\omega)\). Is the splitting large or small compared to a single-well excitation?