3.3.1 Stationary Phase Approximation#

Prompts

What is the stationary phase approximation? Under what conditions does it apply to oscillatory integrals?
Why does the classical path (\(\delta S = 0\)) dominate the path integral as \(\hbar \to 0\)?
How do quantum fluctuations around the classical path contribute corrections at finite \(\hbar\)?
How does stationary phase make the correspondence principle precise — classical mechanics as a limit of quantum mechanics?

Lecture Notes#

Overview#

The stationary phase approximation (SPA) is the mathematical mechanism by which classical mechanics emerges from quantum mechanics. For integrals of the form \(I = \int f(x)\, \mathrm{e}^{\mathrm{i}\Phi(x)/\hbar}\,\mathrm{d}x\) with small \(\hbar\), the dominant contribution comes from stationary points where \(\Phi'(x_0) = 0\). Applied to the path integral, this selects the classical path — the trajectory where \(\delta S = 0\). This is the correspondence principle made precise.

The Mathematical Result#

Many integrals in quantum mechanics take the form

\[ I = \int \mathrm{e}^{\mathrm{i}\Phi(x)/\hbar} f(x) \, \mathrm{d}x \]

where \(\Phi(x)\) is a phase function, \(f(x)\) is a slowly-varying amplitude, and \(\hbar\) is small. When \(\hbar\) is tiny, the integrand oscillates wildly — contributions cancel everywhere except near points where \(\Phi\) varies slowly.

Stationary Phase Formula

For an oscillatory integral with a single stationary point \(\Phi'(x_0) = 0\), the leading approximation is

\[ I \approx f(x_0) \sqrt{\frac{2\pi\hbar}{\mathrm{i}\Phi''(x_0)}} \, \mathrm{e}^{\mathrm{i}\Phi(x_0)/\hbar} \]

Valid when: (1) isolated stationary point with \(|\Phi''(x_0)|\) not too small, (2) \(f(x)\) slowly varying, (3) \(\Phi \gg \hbar\).

Derivation: Stationary Phase Formula

Near the stationary point \(x_0\), expand the phase to second order:

\[ \Phi(x) \approx \Phi(x_0) + \frac{1}{2}\Phi''(x_0)(x - x_0)^2 \]

The integral becomes Gaussian:

\[ I \approx \mathrm{e}^{\mathrm{i}\Phi(x_0)/\hbar} f(x_0) \int \mathrm{e}^{\mathrm{i}\Phi''(x_0)(x-x_0)^2/(2\hbar)} \mathrm{d}x \]

Using the Fresnel integral \(\int \mathrm{e}^{\mathrm{i}ax^2} \mathrm{d}x = \sqrt{\pi/(\mathrm{i}a)}\), the result follows.

The width of the region that contributes is \(\Delta x \sim \sqrt{\hbar/|\Phi''|}\). As \(\hbar \to 0\), this region shrinks to zero — only the stationary point survives.

Why the Classical Path Dominates#

In the path integral

\[ K(\boldsymbol{x}_f, t_f; \boldsymbol{x}_i, t_i) = \int \mathrm{e}^{\mathrm{i}S[\boldsymbol{x}]/\hbar} \, \mathcal{D}[\boldsymbol{x}] \]

the phase function is the classical action \(S[\boldsymbol{x}]\). The stationary point is the classical path \(\boldsymbol{x}_\mathrm{cl}(t)\), defined by \(\delta S/\delta \boldsymbol{x} = 0\) — the Euler-Lagrange (Newton’s) equations.

Constructive interference at \(\boldsymbol{x}_\mathrm{cl}\): Nearby paths have actions differing only at second order, so their phases nearly agree — they add coherently.

Destructive interference away from \(\boldsymbol{x}_\mathrm{cl}\): Paths far from the classical trajectory have first-order action differences \(\Delta S \gg \hbar\), producing rapidly oscillating phases that cancel.

Discussion: Classical Emergence

The stationary phase approximation says that as \(\hbar \to 0\), only classical paths survive. This raises several questions:

In the limit \(\hbar \to 0\), does stationary phase become exact or just increasingly accurate? What controls the error?
For a potential with multiple classical paths (e.g., reflection and transmission), do they interfere constructively or destructively?
Tunneling has no classical path, yet it occurs. How does the path integral handle contributions from classically forbidden trajectories?

Quantum Fluctuations: The Second Variation#

Expanding the action to second order around the classical path:

\[ S[\boldsymbol{x}_\mathrm{cl} + \delta\boldsymbol{x}] \approx S_\mathrm{cl} + \frac{1}{2}\int \mathrm{d}t \, \delta\boldsymbol{x} \cdot M \cdot \delta\boldsymbol{x} \]

where \(M = -m\frac{\mathrm{d}^2}{\mathrm{d}t^2} - V''(\boldsymbol{x}_\mathrm{cl})\) is the second-variation operator. The Gaussian integral over fluctuations gives the semiclassical propagator:

Semiclassical Propagator

\[ K \approx \mathrm{e}^{\mathrm{i}S_\mathrm{cl}/\hbar} \cdot \frac{1}{\sqrt{\det M}} \]

The phase comes from the classical action; the prefactor encodes quantum fluctuations.

Example: Electron vs Baseball

Problem. Compare \(S/\hbar\) for an electron moving 1 cm in 1 s versus a baseball (0.1 kg) moving 1 m in 1 s.

Solution. The classical action is \(S \sim md^2/t\).

Electron: \(S \sim (10^{-30})(10^{-2})^2/1 = 10^{-34}\) J·s \(\approx \hbar\). Quantum regime — all paths matter.
Baseball: \(S \sim (0.1)(1)^2/1 = 0.1\) J·s \(= 10^{33}\hbar\). Classical regime — only the classical path survives.

The Correspondence Principle#

Stationary phase quantifies the correspondence principle: quantum mechanics reduces to classical mechanics when \(S \gg \hbar\).

Regime	\(S/\hbar\)	Behavior
Classical	\(\gg 1\)	Single path dominates; definite trajectory
Semiclassical	\(\sim 1\)	Classical path + quantum corrections
Quantum	\(\lesssim 1\)	All paths contribute; wave-like behavior

Summary#

Stationary phase: oscillatory integrals are dominated by points where \(\Phi'(x_0) = 0\); the contributing region has width \(\sim \sqrt{\hbar}\)
Classical path selection: the path integral selects \(\delta S = 0\) through constructive interference; all other paths cancel
Semiclassical propagator: \(K \approx \mathrm{e}^{\mathrm{i}S_\mathrm{cl}/\hbar}/\sqrt{\det \hat{D}}\) includes quantum corrections from the second variation
Correspondence principle: classical mechanics emerges when \(S \gg \hbar\); the ratio \(S/\hbar\) controls the quantum-to-classical transition

Homework#

1. Stationary Phase Integral (Conceptual)

Consider the integral \(I = \int_{-\infty}^{\infty} \mathrm{e}^{\mathrm{i}\Phi(x)/\hbar} f(x) \, \mathrm{d}x\) where \(\Phi(x) = x^2 + x^3\) and \(f(x) = 1\).

(a) Find all stationary points where \(d\Phi/dx = 0\).

(b) For which stationary point(s) is the second derivative \(\Phi''(x_0)\) positive? Negative? Explain why the sign matters for the SPA.

2. Gaussian Integral via SPA

Verify the stationary phase result for a pure Gaussian. Let \(\Phi(x) = ax^2\) (with \(a > 0\) real) and \(f(x) = 1\).

(a) Apply the SPA formula to compute \(I = \int_{-\infty}^{\infty} \mathrm{e}^{\mathrm{i}ax^2/\hbar} dx\).

(b) Compare with the exact Gaussian integral formula \(\int_{-\infty}^{\infty} \mathrm{e}^{\mathrm{i}ax^2} dx = \sqrt{\pi/(ia)}\).

3. Width of the Stationary Region

In the SPA, the main contribution comes from a region where the phase variation is \(\sim \hbar\).

(a) Use the condition \(\Phi''(x - x_0)^2 \sim \hbar\) to estimate the width of the stationary region in terms of \(\hbar\), \(\Phi(x_0)\), and \(\Phi''(x_0)\).

(b) For a free particle path integral with action \(S = mv^2 t / 2\) (where \(v\) is velocity and \(t\) is the time interval), estimate the width in position space. How does it depend on \(\hbar\), \(m\), \(v\), and \(t\)?

(c) For an electron and a baseball moving at the same velocity over the same time interval, which has a larger quantum uncertainty region? Explain physically.

4. Classical Path Dominance

Consider two paths in a simple 1D potential: the classical path \(x_{\text{cl}}(t)\) and a neighboring path that differs by a small fixed displacement \(\delta x\) at all times.

(a) Explain why the action difference \(\Delta S\) between these two paths depends on whether \(\delta x\) is first-order or second-order in deviations from \(x_{\text{cl}}\).

(b) If \(\delta x \sim 1\) (macroscopic), what is the approximate action difference \(\Delta S\) in units of \(\hbar\) for a macroscopic system (where \(S/\hbar \gg 1\))?

(c) Convert this action difference to a phase difference \(\Delta\phi = \Delta S/\hbar\). Why does this lead to destructive interference?

(d) What happens in the classical limit \(\hbar \to 0\)?

5. The Correspondence Principle in the Path Integral

Make a quantitative estimate using the correspondence principle.

(a) For a particle of mass \(m\) moving distance \(d\) in time \(t\), the classical action is \(S \sim md^2/t\). Show that the ratio \(S/\hbar\) is a dimensionless measure of how “classical” the system is.

(b) For an electron (\(m = 10^{-30}\) kg) moving 1 nm (\(10^{-9}\) m) in 1 ns (\(10^{-9}\) s), compute \(S/\hbar\). Is this electron classical or quantum?

(d) How would the electron’s \(S/\hbar\) change if it were confined to a region \(\sim\) 0.1 nm and observed over time \(\sim\) 0.1 ns? Does this favor quantum or classical behavior?

6. Second Variation of the Action

For a particle in a potential \(V(x)\), the classical action is \(S = \int_0^T \left[\frac{1}{2}m\dot{x}^2 - V(x)\right] dt\).

(a) The second variation is \(\delta^2 S / \delta x(t)^2 = -m d^2/dt^2 - V''(x_{\text{cl}})\). Explain what each term represents: the first involves kinetic energy, the second involves the potential curvature at the classical path.

(b) For a harmonic oscillator, \(V(x) = \frac{1}{2}m\omega^2 x^2\), show that \(\delta^2 S / \delta x(t)^2 = -m(d^2/dt^2 + \omega^2)\).

(c) Is this second-variation operator positive or negative definite? What does this tell you about the stability of the classical path to small perturbations?

7. Validity Conditions for SPA

A student wants to apply the SPA to compute \(I = \int_{-\infty}^{\infty} \mathrm{e}^{\mathrm{i} x^4 / \hbar} \sin(x) \, \mathrm{d}x\).

(a) Identify the stationary points of \(\Phi(x) = x^4\).

(b) Check the validity conditions: (i) Is there a clear, isolated stationary point? (ii) How large is \(\Phi''(x_0)\)? (iii) Is the amplitude \(f(x) = \sin(x)\) slowly varying near the stationary point?

8. Destructive Interference (Qualitative Reasoning)

In the path integral, most paths give phase factors \(\mathrm{e}^{\mathrm{i}S[\text{path}]/\hbar}\) that point in different directions in the complex plane.

(a) Why do paths that are far from the classical path tend to have very different phases?

(b) Sketch a phasor diagram for five paths: the classical path and two pairs of neighbors symmetrically displaced from it. Explain how the phasors for non-classical paths tend to cancel.

(c) For a macroscopic object, how many distinct “orientations” of phasors (phase differences) occur among all possible paths? Why does this lead to nearly complete cancellation?

9. Semiclassical Propagator

The SPA gives the propagator as

\[K(x_f, t_f; x_i, t_i) \approx \mathrm{e}^{\mathrm{i}S_{\text{cl}}/\hbar} \sqrt{\frac{1}{\det(\delta^2 S/\delta x^2)}}\]

(a) The exponent \(S_{\text{cl}}/\hbar\) oscillates rapidly as the final position \(x_f\) changes. What physical phenomenon does this oscillation describe?

(b) The prefactor \(\sqrt{1/\det(\delta^2 S/\delta x^2)}\) is a smooth, slowly-varying function. What does it represent?