The WKB Approximation

There are two ways to arrive at the same semiclassical wavefunction, and they illuminate different faces of it. The direct route — expand $\psi=e^{iS/\hbar}$ in powers of $\hbar$ and match the Schrödinger equation order by order — is developed in the companion article The WKB Approximation, which owns the eikonal equation and the amplitude $p(x)^{-1/2}$ (the "transport" recursion for the amplitude). This article takes the complementary route: the WKB wavefunction is what the Feynman path integral collapses to when $\hbar\to 0$ . The exponential of the classical action, $e^{iS_{\text{cl}}/\hbar}$ , comes from the stationary-phase condition $\delta S=0$ ; the amplitude $p(x)^{-1/2}$ comes from integrating the Gaussian fluctuations around the classical path. In one dimension that fluctuation integral is a Van Vleck determinant, and it is the WKB prefactor. Why bother with a second derivation of a result you already have? Because it tells you what WKB means. The direct method makes the approximation look like a formal series in $\hbar$ ; the path-integral method reveals it as the statement that quantum mechanics reduces to classical mechanics plus one Gaussian layer of quantum fuzz. The stationary-phase picture also generalizes cleanly — to many dimensions, to chaotic systems (the Gutzwiller trace formula), to field theory — where the direct eikonal expansion becomes awkward. You are ready for this because you command the path integral, the classical action, and the WKB wavefunction from the direct approach. Here we prove that the 1D Van Vleck determinant equals $1/p(x)$ , so that $|\psi|^2\propto 1/p$ — the probability of finding the particle at $x$ is proportional to the time it classically dwells there. That single identity ties the sum-over-paths to the oldest intuition in quantum mechanics.

Key takeaways

The semiclassical propagator is the stationary-phase limit of the path integral: $K\approx\sum_{\text{cl}}\sqrt{\tfrac{1}{2\pi i\hbar}\left|\tfrac{\partial^2 S_{\text{cl}}}{\partial x_i\partial x_f}\right|}\,e^{iS_{\text{cl}}/\hbar}$ .
The phase $e^{iS_{\text{cl}}/\hbar}$ is fixed by $\delta S=0$ (the classical path); the amplitude is the Van Vleck determinant of Gaussian fluctuations around it.
In one dimension the Van Vleck determinant reduces to $\partial^2 S_{\text{cl}}/\partial x_i\partial x_f = -m/T$ for a free segment, and in a general potential to the local $1/p(x)$ — reproducing the direct-method amplitude $p^{-1/2}$ .
Hence $|\psi_{\text{WKB}}|^2\propto 1/p(x)$ : probability density is proportional to classical dwell time — the particle spends longest, and is most likely found, where it moves slowest.
The two roads to WKB are complementary: the direct Schrödinger expansion (owned by The WKB Approximation) and this path-integral/stationary-phase route.
KEY IDEA: WKB is the $\hbar\to0$ saddle point of the path integral — the classical action sets the phase, and one Gaussian fluctuation integral (the Van Vleck determinant) sets the amplitude $p^{-1/2}$ .

The core insight

The WKB wavefunction is the stationary-phase evaluation of $\int\mathcal{D}x\,e^{iS/\hbar}$ : the classical action gives the phase and the Van Vleck determinant of quadratic fluctuations gives the amplitude $p(x)^{-1/2}$ .

Where this lives today

The stationary-phase view of semiclassics is the working language of quantum chaos. When a system has no separable coordinates, the direct eikonal expansion has nothing to expand around, but the path integral still has classical periodic orbits — and Gutzwiller's 1971 trace formula sums their Van Vleck amplitudes and Maslov phases to reproduce the quantum spectrum. This is how physicists compute spectral statistics of chaotic billiards, mesoscopic conductance fluctuations, and the level correlations that connect quantum mechanics to random-matrix theory. In lattice gauge theory and instanton physics the same saddle-point logic — classical action in the exponent, fluctuation determinant in the amplitude — evaluates tunnelling amplitudes that no perturbation series can reach.

The physical picture

Imagine the sum over all paths from $x_i$ to $x_f$ . Away from the classical limit every path counts. But as $\hbar\to0$ the phase $S[x]/\hbar$ becomes enormous, and neighbouring paths interfere destructively — except near a path where $S$ is stationary, $\delta S=0$ , because there the phase is momentarily flat and the paths add constructively. That stationary path is the classical trajectory. The path integral thus concentrates onto classical mechanics, and the leading correction is the width of the constructive band around the classical path: the Gaussian fluctuations. The amplitude of the surviving contribution measures how tightly the paths bunch. If a small change in the endpoint $x_f$ forces a large change in the required classical action, the constructive band is narrow and the amplitude large; if the action is insensitive to the endpoint, the band is broad and the amplitude small. That sensitivity is exactly $\partial^2 S_{\text{cl}}/\partial x_i\partial x_f$ — the Van Vleck determinant. In one dimension it turns out to be the reciprocal of the local classical momentum, and this is the whole content of the WKB amplitude.

Figure

Stationary phase selects the classical path

The punchline of the whole method is a single curve: the probability density of a semiclassical state piles up at the turning points, because $|\psi|^2\propto 1/p$ and $p\to0$ there. This is the quantum echo of the fact that a pendulum spends most of its time near the extremes of its swing, where it is moving slowly.

Figure

Probability density piles up where the particle moves slowly

The math: WKB as a saddle point

The starting point is the propagator written as a path integral, $K(x_f,T;x_i,0)=\int\mathcal{D}x\,e^{iS[x]/\hbar}$ with $S=\int_0^T(\tfrac12 m\dot x^2-V)\,dt$ — the construction owned by The Feynman Path Integral Formulation. We now evaluate it in the semiclassical limit by expanding the action about the classical path. Write $x(t)=x_{\text{cl}}(t)+\eta(t)$ with $\eta(0)=\eta(T)=0$ . Because $x_{\text{cl}}$ extremizes the action, $S$ has no linear term in $\eta$ , and to leading order in $\hbar$ only the quadratic term survives:

S[x] = S_{\text{cl}} + \tfrac12\int_0^T dt\,\eta\!\left(-m\tfrac{d^2}{dt^2}-V''(x_{\text{cl}})\right)\eta + \mathcal{O}(\eta^3).

where: $S_{\text{cl}}=S[x_{\text{cl}}]$ is the classical action along the trajectory obeying the boundary conditions, and the operator $\hat F=-m\,d^2/dt^2-V''(x_{\text{cl}})$ is the second variation of the action (the Jacobi operator). Physically, the first term sets the overall phase and the second term — a Gaussian integral — sets the amplitude. Doing the Gaussian integral over $\eta$ gives the Van Vleck-Pauli-Morette form of the semiclassical propagator, one term per classical path:

K_{\text{sc}}(x_f,T;x_i,0)=\sqrt{\frac{1}{2\pi i\hbar}\left|\frac{\partial^2 S_{\text{cl}}}{\partial x_i\,\partial x_f}\right|}\;\exp\!\left(\frac{i}{\hbar}S_{\text{cl}}-\frac{i\pi\mu}{2}\right).

where: $\partial^2 S_{\text{cl}}/\partial x_i\partial x_f$ is the Van Vleck determinant (in 1D a single number), and $\mu$ is the Maslov index counting the caustics the trajectory passes — the phase bookkeeping developed in Bohr-Sommerfeld and Maslov Quantization. The determinant measures how the classical action responds to shifting the endpoints; a large response means paths bunch tightly and the amplitude is large. To recover the stationary-state wavefunction from this propagator you fix the energy rather than the time, which turns the amplitude $\sqrt{\partial^2 S/\partial x_i\partial x_f}$ into the local factor $1/\sqrt{p(x)}$ . The next derivation makes this precise in one dimension, and shows it equals exactly the $p^{-1/2}$ amplitude that The WKB Approximation obtains from the direct $\hbar$ -expansion.

Derivation AThe 1D Van Vleck determinant equals 1/p

We want to prove that the endpoint sensitivity of the classical action — the Van Vleck determinant — reduces, for a one-dimensional energy eigenstate, to the reciprocal of the local classical momentum $p(x)=\sqrt{2m(E-V(x))}$ . The strategy is to compute $\partial^2 S_{\text{cl}}/\partial x_i\partial x_f$ for a free segment first (where everything is explicit), then pass to fixed energy via the abbreviated action $W(x,E)=\int p\,dx$ . Watch the difference between the time-dependent action $S(x_f,x_i;T)$ and the energy-dependent action $W$ ; they are Legendre transforms of each other, and the WKB amplitude lives in $W$ .

Step 1. For a free particle the classical action between endpoints in time $T$ is $S_{\text{cl}}=m(x_f-x_i)^2/2T$ . Differentiate twice: $\partial S_{\text{cl}}/\partial x_f=m(x_f-x_i)/T$ , then $\partial^2 S_{\text{cl}}/\partial x_i\partial x_f=-m/T$ . Why: this is the exact prefactor of the free propagator $\sqrt{m/2\pi i\hbar T}$ — the magnitude $|{-m/T}|=m/T$ under the square root gives precisely $\sqrt{m/2\pi i\hbar T}$ . The Van Vleck formula reproduces the free propagator with no approximation, because a free particle has no fluctuation-inducing curvature.

Step 2. Pass to fixed energy. The energy-dependent (abbreviated) action is $W(x_i,x_f;E)=\int_{x_i}^{x_f}p(x')\,dx'$ with $p=\sqrt{2m(E-V)}$ , related to $S$ by $W=S+ET$ . The stationary-state amplitude is governed by $\partial^2 W/\partial x_i\partial x_f$ together with the energy-derivative $\partial^2 W/\partial E^2$ . Why: fixing energy rather than time is the Legendre transform $t\to E$ . The WKB stationary state $\psi\sim A(x)e^{iW/\hbar}$ carries the amplitude in $A(x)$ , and the semiclassical rule is $A(x)\propto(\partial^2 W/\partial x\,\partial E)^{1/2}$ — the mixed sensitivity to endpoint and energy.

Step 3. Compute the mixed derivative. Since $\partial W/\partial x_f=p(x_f)$ and $\partial W/\partial E=\int dx/v = \int m\,dx/p$ (the travel time), we get $\partial^2 W/\partial x_f\partial E=\partial p(x_f)/\partial E=m/p(x_f)$ . Why: $\partial p/\partial E=m/p$ because $p^2=2m(E-V)$ gives $2p\,\partial p/\partial E=2m$ . So the amplitude factor is $A(x)\propto\sqrt{m/p(x)}\propto p(x)^{-1/2}$ — exactly the WKB amplitude.

Step 4. Read off the probability density: $|\psi_{\text{WKB}}(x)|^2\propto 1/p(x)$ . Since $p(x)=mv(x)$ , this is $1/v(x)$ — proportional to the time $dt=dx/v$ the classical particle dwells in $[x,x+dx]$ . Why: this is the classical probability distribution. A classical particle observed at a random time is found in $dx$ with probability proportional to the fraction of the period it spends there, $dt/T=dx/(vT)$ . WKB reproduces exactly this classical density, with quantum oscillations superimposed. *Verified: the free-segment prefactor $|{-m/T}|=m/T$ matches the exact free propagator, and $\partial p/\partial E=m/p$ gives $A\propto p^{-1/2}$ ; both checked symbolically in the verification script (residual 0). The result is identical to the $p^{-1/2}$ amplitude owned by The WKB Approximation.*

The identity $|\psi|^2\propto 1/p$ is the bridge between the sum-over-paths and Bohr's old intuition. It also explains a fact the direct method leaves mysterious: why the amplitude blows up at turning points. In the path-integral picture the answer is geometric — the classical paths focus there (a caustic), the Van Vleck determinant diverges, and the semiclassical approximation must be repaired by the connection formulas of the next articles.

Go deeper · Derivation B · 1 minStationary phase in one integral: the model saddle

Derivation BStationary phase in one integral: the model saddle

Before trusting the full path integral it helps to see the stationary-phase mechanism on a single ordinary integral, because every step of the WKB amplitude is already visible there. The goal is to evaluate $I(\hbar)=\int dx\,g(x)\,e^{if(x)/\hbar}$ as $\hbar\to0$ and read off both the phase and the $\sqrt{\hbar}$ amplitude, then match the pattern to the path integral. Watch the sign of $f''$ at the saddle: it becomes the Maslov phase.

Step 1. As $\hbar\to0$ the integrand oscillates wildly except where $f'(x)=0$ . Expand about a stationary point $x_0$ : $f(x)=f(x_0)+\tfrac12 f''(x_0)(x-x_0)^2+\cdots$ . Why: away from $x_0$ the phase sweeps through many cycles over any finite interval and cancels; only the neighbourhood of $x_0$ , where the phase is stationary, contributes coherently. This is Kelvin's stationary-phase principle.

Step 2. Do the Gaussian integral $\int d\xi\,e^{if''\xi^2/2\hbar}=\sqrt{2\pi\hbar/|f''|}\,e^{i(\pi/4)\,\text{sgn}f''}$ . Why: the Fresnel integral $\int e^{ia\xi^2}d\xi=\sqrt{\pi/|a|}\,e^{i(\pi/4)\text{sgn}a}$ supplies both the $\sqrt{\hbar}$ scale and the $\pm\pi/4$ phase. The sign of $f''$ decides whether the saddle contributes $+\pi/4$ or $-\pi/4$ .

Step 3. Assemble: $I(\hbar)\approx g(x_0)\sqrt{2\pi\hbar/|f''(x_0)|}\;e^{if(x_0)/\hbar+i(\pi/4)\text{sgn}f''}$ . Why: the phase is the stationary value $f(x_0)/\hbar$ (the "classical action"), the amplitude is $|f''|^{-1/2}$ (the "Van Vleck determinant"), and the $\pi/4$ is one caustic's worth of Maslov phase. The path integral is the same computation with $f\to S[x]/\hbar$ and the ordinary second derivative replaced by the Jacobi operator. *Verified: for $f(x)=x^2/2$ , $g=1$ , the exact Fresnel integral $\int e^{ix^2/2\hbar}dx=\sqrt{2\pi\hbar}\,e^{i\pi/4}$ matches the formula to machine precision (checked numerically).*

Limiting cases

case	result	meaning
$\hbar\to0$	$K\to\sum_{\text{cl}}$ (saddle points)	only classical paths survive; the path integral becomes classical mechanics
far from turning points	$\psi\sim p^{-1/2}e^{\pm i\int p\,dx/\hbar}$	ordinary oscillatory WKB; Van Vleck determinant finite
at a turning point $p\to0$	Van Vleck determinant diverges (caustic)	WKB fails; paths focus, connection formulas needed
free particle	$\partial^2 S/\partial x_i\partial x_f=-m/T$	exact free propagator, no correction
one caustic passed	extra phase $-\pi/2$ ( $\mu\to\mu+1$ )	Maslov index increments — the origin of the Bohr $\tfrac12$

Worked examples

Example 1 — The 1D Van Vleck determinant for a linear potential.

Step 1. Take the linear ramp $V(x)=Fx$ (constant force $-F$ ), so $p(x)=\sqrt{2m(E-Fx)}$ . The WKB amplitude is $A(x)\propto p(x)^{-1/2}=[2m(E-Fx)]^{-1/4}$ .

Step 2. Check the free limit $F\to0$ : $p\to\sqrt{2mE}=$ const, so $A\to$ const — a flat amplitude, exactly the free-particle plane wave. Good.

Step 3. Evaluate at two points for $m=1$ , $E=1$ , $F=0.5$ (units $\hbar=1$ ). At $x=0$ : $p=\sqrt{2}=1.414$ , $A=2^{-1/4}=0.841$ . At $x=1$ : $p=\sqrt{2(1-0.5)}=\sqrt{1}=1$ , $A=(2\cdot0.5)^{-1/4}=1^{-1/4}=1.000$ .

Step 4. The amplitude has grown from $0.841$ to $1.000$ as the particle slows on the ramp. The ratio of densities $|\psi(1)|^2/|\psi(0)|^2 = p(0)/p(1)=1.414/1.000=\mathbf{1.414}$ — the particle is $\sqrt2\times$ more likely to be found where it moves $\sqrt2\times$ slower.

Example 2 — Dwell-time density for the classical oscillator.

Step 1. For $V=\tfrac12 m\omega^2 x^2$ at energy $E$ , the turning points are $x_{\max}=\sqrt{2E/m\omega^2}$ and $p(x)=m\omega\sqrt{x_{\max}^2-x^2}$ . Thus $|\psi_{\text{WKB}}|^2\propto 1/\sqrt{x_{\max}^2-x^2}$ .

Step 2. Evaluate the density ratio between $x=0.9\,x_{\max}$ and $x=0$ . At the centre, $1/\sqrt{x_{\max}^2}=1/x_{\max}$ . At $0.9x_{\max}$ , $1/\sqrt{x_{\max}^2-0.81x_{\max}^2}=1/(x_{\max}\sqrt{0.19})$ .

Step 3. The ratio is $1/\sqrt{0.19}=1/0.4359=\mathbf{2.294}$ .

Step 4. Interpretation: near the turning point the semiclassical particle is $2.29\times$ more likely to be found than at the centre — a striking contrast with the ground state, whose Gaussian is peaked at the centre. WKB describes highly excited states, where the quantum probability, averaged over its rapid oscillations, tracks the classical dwell time. The mismatch with the ground state is precisely the regime where WKB is not meant to apply.

Origins · 1 minStationary Phase Approximation and how this was discovered

Stationary Phase Approximation

Kelvin 1887; Stokes 1847 · The mathematical engine of the semiclassical limit — the dominant contribution to a rapidly oscillating integral comes from the neighbourhood of a stationary point of the phase, where waves add coherently.

History

1828

— Green and Liouville independently develop the asymptotic method for second-order ODEs that underlies WKB

1887

— Kelvin formulates the stationary-phase principle for wave-group propagation

1926

— Wentzel, Kramers, and Brillouin apply the phase-integral method to the Schrödinger equation

1928

— Van Vleck derives the semiclassical density-of-states determinant from the correspondence principle

1948

— Feynman formulates the path integral, from which stationary phase yields WKB directly

1951

— Morette gives the rigorous fluctuation-determinant (Van Vleck-Pauli-Morette) prefactor

Why does the amplitude involve ∂²S/∂x_i∂x_f rather than just ∂S/∂x?

Because the amplitude measures how the bundle of near-classical paths converges or diverges, and that is a second-order property of the action. $\partial S/\partial x_f = p_f$ is just the final momentum (first order). The mixed second derivative $\partial^2 S/\partial x_i\partial x_f$ tells you how the final momentum changes when you shift the initial point — i.e. how strongly the trajectory family focuses. When it diverges, an infinitesimal spread of initial data collapses to a point (a caustic), the density of classical arrivals blows up, and so does the semiclassical amplitude.

The ground-state Gaussian is peaked in the middle, but WKB says density piles up at the edges. Contradiction?

No — they describe different regimes. WKB is the large-quantum-number approximation; it tracks the classical dwell-time density, which for an oscillator does pile up at the turning points. The ground state has $n=0$ , the least classical state possible, and there the single de Broglie wavelength spans the whole well — the WKB validity condition $|d\lambda/dx|\ll1$ fails everywhere. As $n$ grows the quantum $|\psi_n|^2$ , averaged over its $n$ oscillations, converges to the classical $1/p$ curve. The apparent contradiction is just WKB being applied outside its domain.

What exactly is a "caustic" here, and why does it matter?

A caustic is a point where neighbouring classical trajectories focus — the same phenomenon that makes the bright cusp in a coffee cup or the edge of a rainbow. In phase space it is where the Lagrangian manifold folds so that $\partial x_f/\partial p_i=0$ , which makes the Van Vleck determinant $\partial^2 S/\partial x_i\partial x_f$ diverge. In 1D the caustics are exactly the classical turning points. They matter because the semiclassical amplitude fails there and each caustic contributes a $-\pi/2$ phase jump (a Maslov unit), which is the ultimate origin of the $\tfrac12$ in Bohr-Sommerfeld quantization.

How does the Maslov index appear in this derivation?

It appears as the $\pi/4$ phase from the sign of the second derivative in each Gaussian (stationary-phase) integral. Each time the classical trajectory passes a caustic, the fluctuation operator $\hat F=-m\,d^2/dt^2-V''$ acquires a negative eigenvalue, the corresponding Gaussian integral flips its Fresnel phase by $\pi/2$ , and the propagator picks up $e^{-i\pi/2}$ . Summed over the caustics on the path this is the Maslov index $\mu$ , giving the total phase $-i\pi\mu/2$ . The systematic accounting is developed in Bohr-Sommerfeld and Maslov Quantization.

Does this route also give the Gamow tunnelling factor?

Yes, and it is the cleaner route for tunnelling. In the classically forbidden region there is no real stationary path, but there is a complex (or imaginary-time) one, and its action is imaginary: $S_{\text{cl}}\to iS_E$ with $S_E=\int\sqrt{2m(V-E)}\,dx$ . The propagator amplitude then carries $e^{-S_E/\hbar}$ — exactly the Gamow suppression owned by WKB Tunneling and the Gamow Factor. Pushing this to the double well gives the instanton, developed in Instantons and Quantum Tunneling. The stationary-phase framework thus unifies oscillation, tunnelling, and instantons under one saddle-point principle.

Is the classical path always unique?

No, and that is where semiclassics gets interesting. For a free particle or a monotone ramp there is one classical path between given endpoints, but a general potential can have several — think of a ball reaching a target by different arcs. Each contributes a term $\sqrt{\text{Van Vleck}}\,e^{iS_{\text{cl}}/\hbar}$ , and their interference produces semiclassical fringes (the analogue of multi-path interference in optics). Summing over the classical periodic orbits is precisely what Gutzwiller's trace formula does to build a chaotic spectrum.