Free-Particle and Harmonic-Oscillator Path Integrals

The path integral is a formal object until you can actually do one, and there are exactly two nontrivial cases where the sum over infinitely many paths collapses to a single closed formula: the free particle and the harmonic oscillator. Both are Gaussian — the action is quadratic in the path — so the functional integral is an infinite-dimensional Gaussian integral, and Gaussian integrals are the one kind we can always evaluate. The payoff is not just two propagators. It is a template: for any quadratic-in-fluctuations problem, the answer has the shape $K=(\text{prefactor})\,e^{iS_{\rm cl}/\hbar}$ , where $S_{\rm cl}$ is the action of the classical path and the prefactor is a determinant that counts how neighbouring classical paths spread. That prefactor has a name and a beautiful meaning: the Van Vleck–Pauli–Morette determinant, $\sqrt{(1/2\pi i\hbar)\,\partial^2 S_{\rm cl}/\partial x_i\partial x_f}$ . It measures the density of classical trajectories arriving at $x_f$ — where they bunch up, the amplitude is large. And it has a spectacular failure mode. For the oscillator the prefactor is $\sqrt{m\omega/2\pi i\hbar\sin\omega T}$ , which diverges every half-period, at $\omega T=n\pi$ . Those are caustics: instants when every trajectory launched from $x_i$ refocuses to a single point, the classical density blows up, and the semiclassical formula breaks. The phase the propagator picks up crossing each caustic is exactly the Maslov phase that the WKB chapter charges for a turning point. You are ready: you know Gaussian integrals, the free kernel, and the oscillator spectrum $E_n=\hbar\omega(n+\tfrac12)$ . Here we evaluate both path integrals exactly, read off the Van Vleck prefactor, locate the caustics, and — as a stringent consistency check — resum the oscillator kernel into its energy eigenstates and recover that same spectrum. The caustics we find here are the thread that reappears as the Maslov index in the last article of the chapter.

Key takeaways

Both exactly-solvable path integrals have the semiclassical form $K=\sqrt{\dfrac{1}{2\pi i\hbar}\dfrac{\partial^2 S_{\rm cl}}{\partial x_i\partial x_f}}\;e^{iS_{\rm cl}/\hbar}$ — the Van Vleck–Pauli–Morette prefactor times the classical phase.
Free particle: $K_0=\sqrt{m/2\pi i\hbar T}\,e^{im(x_f-x_i)^2/2\hbar T}$ , with $S_{\rm cl}=m(\Delta x)^2/2T$ .
Oscillator: $K=\sqrt{\dfrac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\!\Big[\dfrac{im\omega}{2\hbar\sin\omega T}\big((x_i^2+x_f^2)\cos\omega T-2x_ix_f\big)\Big]$ .
The prefactor diverges at caustics $\omega T=n\pi$ : every path refocuses, the classical density blows up, and the propagator picks up a Maslov phase.
Resumming the oscillator kernel over its eigenstates returns $K=\sum_n e^{-i(n+\frac12)\omega T}\psi_n(x_f)\psi_n^*(x_i)$ — an independent recovery of $E_n=\hbar\omega(n+\tfrac12)$ .
KEY IDEA: for a quadratic action the path integral is a Gaussian and collapses to (Van Vleck prefactor) × $e^{iS_{\rm cl}/\hbar}$ ; the prefactor's zeros of $\sin\omega T$ are caustics — the seeds of the Maslov index.

The core insight

For any action quadratic in the path, the path integral equals the Van Vleck prefactor $\sqrt{(1/2\pi i\hbar)\,\partial^2 S_{\rm cl}/\partial x_i\partial x_f}$ times $e^{iS_{\rm cl}/\hbar}$ ; its divergences are the caustics where classical trajectories refocus.

Where this lives today

The Van Vleck prefactor and its caustics are the backbone of semiclassical physics. In molecular dynamics, semiclassical initial-value representations propagate wavepackets using exactly $\sqrt{\text{Van Vleck}}\,e^{iS_{\rm cl}/\hbar}$ summed over classical trajectories, with the Maslov phase tracked as each trajectory passes a caustic — this is how large molecules' vibrational spectra are computed without solving the full Schrödinger equation. The same caustics are visible optics: the bright cusp at the bottom of a teacup and the shimmering network on a pool floor are light's Van Vleck determinant diverging, and the finite brightness there (rather than true infinity) is diffraction smoothing the caustic exactly as it smooths the quantum propagator.

The physical picture

The free particle is the warm-up: released from $x_i$ with a spread of momenta, the classical trajectories fan out and never come back, so the amplitude to arrive at $x_f$ is a smooth spreading Gaussian — this is the "spreading Gaussian kernel" that The Quantum Propagator flagged as belonging to this article. The oscillator is where it gets dramatic. Every trajectory launched from $x_i$ , whatever its initial momentum, has the same period $2\pi/\omega$ . So after a half period, $\omega T=\pi$ , all of them arrive back at $-x_i$ simultaneously, no matter how they started. The bundle of paths, spread out across the well, collapses to a single point.

Figure

Oscillator trajectories refocus at half-period caustics

At that refocusing instant the classical density of trajectories is infinite — geometrically, the paths form a caustic, the same bright cusp you see at the bottom of a coffee cup or in a swimming pool. The Van Vleck prefactor, which is literally the square root of that density, diverges there. This is the punchline of the article, and the next figure shows it directly: $|{\rm prefactor}|=\sqrt{m\omega/2\pi\hbar|\sin\omega T|}$ spikes to infinity at every $\omega T=n\pi$ .

Figure

The Van Vleck prefactor diverges at every caustic ωT = nπ

The core equations

We keep the chapter convention $K=\int\mathcal D x\,e^{iS/\hbar}$ . Because the action is quadratic, the integral over the fluctuation $y(t)=x(t)-x_{\rm cl}(t)$ factors off entirely, leaving $K=F(T)\,e^{iS_{\rm cl}/\hbar}$ : the classical action sets the phase, and a fluctuation determinant $F(T)$ sets the prefactor. The first equation states that universal semiclassical shape, and identifies $F$ with the Van Vleck determinant.

K(x_f,T;x_i,0)=\sqrt{\frac{1}{2\pi i\hbar}\,\frac{\partial^2 S_{\rm cl}}{\partial x_i\,\partial x_f}}\;e^{iS_{\rm cl}(x_f,x_i;T)/\hbar} \qquad (1)

where: $S_{\rm cl}$ — the action evaluated on the classical path with the given endpoints; $\partial^2 S_{\rm cl}/\partial x_i\partial x_f$ — the Van Vleck determinant (in 1D, a single second derivative); $\hbar$ — as usual. Read physically: $\partial^2 S_{\rm cl}/\partial x_i\partial x_f=-\partial p_i/\partial x_f$ measures how the required launch momentum changes with the destination — small change means many launch momenta funnel into a small target, i.e. a high trajectory density and large amplitude. Where the destination becomes insensitive to launch conditions (a focus), the determinant blows up. For the free particle the classical path is a straight line and the action is elementary; this is the case The Quantum Propagator reserved for us.

K_0(x_f,T;x_i,0)=\sqrt{\frac{m}{2\pi i\hbar T}}\;e^{\,im(x_f-x_i)^2/2\hbar T},\qquad S_{\rm cl}=\frac{m(x_f-x_i)^2}{2T} \qquad (2)

where: the prefactor $\sqrt{m/2\pi i\hbar T}$ is $\sqrt{(1/2\pi i\hbar)\,|\partial^2 S_{\rm cl}/\partial x_i\partial x_f|}$ with $\partial^2 S_{\rm cl}/\partial x_i\partial x_f=-m/T$ . Read it: the free particle has no caustics — $1/T$ diverges only at $T=0$ (the delta limit), never at finite time, because free trajectories fan out monotonically and never refocus. The oscillator is the star. Its classical path is a sinusoid pinned at both ends, and its action is the classic Feynman result — the one whose $1/\sin\omega T$ carries the caustics.

K(x_f,T;x_i,0)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\,\exp\!\Big[\frac{im\omega}{2\hbar\sin\omega T}\big((x_i^2+x_f^2)\cos\omega T-2x_ix_f\big)\Big] \qquad (3)

where: $\omega$ — oscillator frequency; the prefactor is again the Van Vleck form with $\partial^2 S_{\rm cl}/\partial x_i\partial x_f=-m\omega/\sin\omega T$ . Read physically: as $\omega\to0$ , (3) reduces smoothly to the free kernel (2); as $\omega T\to n\pi$ , $\sin\omega T\to0$ and the prefactor diverges — the caustics of the figure.

Derivation AThe two exact kernels and the Van Vleck prefactor

The goal is to obtain (2) and (3) and to read off the Van Vleck prefactor in each, rather than quoting it. The strategy exploits that a quadratic action makes the path integral a shifted Gaussian: write $x=x_{\rm cl}+y$ with $y$ vanishing at both endpoints, so $S[x]=S_{\rm cl}+S_2[y]$ with no cross term (the first variation vanishes on $x_{\rm cl}$ ). The classical action gives the phase; the $y$ -integral gives the prefactor, and for the oscillator that determinant is what produces $1/\sin\omega T$ . The one thing to watch is that the fluctuation integral is endpoint-independent — it depends only on $T$ — which is why the prefactor is a pure function of $T$ .

Step 1. Solve the classical equation. Free: $x_{\rm cl}(t)=x_i+(x_f-x_i)t/T$ . Oscillator: $x_{\rm cl}(t)=[x_i\sin\omega(T-t)+x_f\sin\omega t]/\sin\omega T$ . Why: these are the unique solutions of $\ddot x=0$ and $\ddot x=-\omega^2 x$ hitting $x_i$ at $t=0$ and $x_f$ at $t=T$ ; the verifier confirms both endpoints and both equations of motion.

Step 2. Evaluate the classical action. For a quadratic Lagrangian, integrate by parts and use the equation of motion to get $S_{\rm cl}=\tfrac{m}{2}[x\dot x]_0^T$ . Why: the bulk term vanishes on the classical path, leaving only boundary data — a shortcut that gives $S_{\rm cl}=m(\Delta x)^2/2T$ (free) and the $\cos\omega T$ , $x_ix_f$ combination in (3). The verifier confirms $S_{\rm cl}=\tfrac{m}{2}[x\dot x]_0^T$ equals the closed form (3).

Step 3. Take the Van Vleck determinant $\partial^2 S_{\rm cl}/\partial x_i\partial x_f$ . Free: $-m/T$ . Oscillator: $-m\omega/\sin\omega T$ . Why: these are the prefactors of (2) and (3) up to the universal $\sqrt{1/2\pi i\hbar}$ ; the verifier confirms both second derivatives symbolically.

Step 4. The fluctuation integral $\int\mathcal D y\,e^{iS_2[y]/\hbar}$ reproduces exactly $\sqrt{(1/2\pi i\hbar)\,\partial^2 S_{\rm cl}/\partial x_i\partial x_f}$ — the Van Vleck–Pauli–Morette theorem. Why: for a quadratic action the fluctuation determinant and the Van Vleck determinant are the same object (the Gelfand–Yaglom result); this is why we can read the prefactor straight off the classical action without doing the functional determinant by hand. *Verified: the free and oscillator second derivatives, the classical-path endpoints and equations of motion, and the boundary-term action are all confirmed symbolically; the $\omega\to0$ limit of (3) reduces to (2).*

The prefactor's divergences are not a defect — they are physics, and they demand a phase convention. When $\omega T$ passes through $n\pi$ , $\sin\omega T$ changes sign, so $\sqrt{1/\sin\omega T}$ jumps by $\sqrt{-1}=e^{-i\pi/2}$ per caustic. This $-\pi/2$ per refocusing is the Maslov phase.

Go deeper · Derivation B · 2 minCaustics, the Maslov phase, and the spectral check

Derivation BCaustics, the Maslov phase, and the spectral check

We now show the two things that make (3) trustworthy: that its divergences sit exactly at the caustics, with a definite phase jump, and that resumming (3) over energy eigenstates returns the known oscillator spectrum. The strategy for the phase is to track $\sqrt{1/\sin\omega T}$ across a zero of $\sin$ ; for the spectrum, to recognise (3) as the Mehler kernel, whose generating-function identity is $\sum_n e^{-i(n+\frac12)\omega T}\psi_n(x_f)\psi_n^*(x_i)$ . The subtlety is that the spectral sum converges only conditionally on the real- $T$ axis, so a small imaginary-time regulator is needed to sum it cleanly.

Step 1. Locate the caustics: the prefactor $\sqrt{m\omega/2\pi i\hbar\sin\omega T}$ diverges when $\sin\omega T=0$ , i.e. $\omega T=n\pi$ , $n=1,2,3,\dots$ Why: these are exactly the refocusing times of Step-1's classical bundle — every trajectory from $x_i$ returns to a common point, so the Van Vleck density is infinite. The verifier confirms $\sin(n\pi)=0$ at $n\pi$ .

Step 2. Track the phase across a caustic: as $\omega T$ increases through $n\pi$ , $\sin\omega T$ changes sign, so the propagator acquires a factor $e^{-i\pi/2}$ — the Maslov phase, $-\pi/2$ per turning point. Why: this is the same $\pi/4$ -per-turning-point (hence $\pi/2$ per full pass) bookkeeping that the WKB Connection Formulas article derives from the Airy function; here it appears as a branch cut of $\sqrt{1/\sin}$ . The last article of the chapter, Bohr-Sommerfeld and Maslov Quantization, promotes it to a topological index.

Step 3. Resum (3) as the Mehler kernel: $K=\sum_{n=0}^\infty e^{-i(n+\frac12)\omega T}\psi_n(x_f)\psi_n^*(x_i)$ , with $\psi_n$ the oscillator eigenfunctions. Why: this is the spectral form of the propagator (owned by The Quantum Propagator), and the phases $e^{-i(n+\frac12)\omega T}$ display the spectrum $E_n=\hbar\omega(n+\tfrac12)$ — the ground-state $\tfrac12$ visible as the $n=0$ phase. The result is recalled, not re-derived, from Quantum Harmonic Oscillator: Analytic Solution.

Step 4. Verify numerically that (3) equals the spectral sum. Why: with the imaginary-time regulator $T\to T-i(0.4/\omega)$ the Mehler sum converges geometrically and reproduces (3) to $\sim10^{-12}$ in 60 terms — an independent confirmation that the path-integral kernel carries the correct spectrum. *Verified: caustics at $\omega T=n\pi$ ; the oscillator kernel equals the regulated Mehler spectral sum to $<10^{-8}$ ; the $n=0$ phase encodes the zero-point $\tfrac12\hbar\omega$ .*

Limiting cases

case	result	meaning
$\omega\to0$	(3) $\to$ (2)	oscillator becomes free particle
$\omega T\to n\pi$	prefactor $\to\infty$	caustic: all paths refocus
$T\to0$	$K_0\to\delta(x_f-x_i)$	free delta limit
across a caustic	phase jump $e^{-i\pi/2}$	the Maslov phase, $-\pi/2$ per focus
resum over $n$	$\sum_n e^{-i(n+\frac12)\omega T}\psi_n\psi_n^*$	recovers $E_n=\hbar\omega(n+\tfrac12)$
$\hbar\to0$	prefactor $\times e^{iS_{\rm cl}/\hbar}$	this is the semiclassical form for any quadratic action

Common trapThe prefactor divergence is a caustic, not a breakdown of quantum mechanics

It is tempting to read $\sin\omega T\to0$ as the propagator becoming infinite and conclude something is broken. Nothing is broken. The semiclassical formula diverges because the classical trajectory density diverges — a real feature of the focusing bundle — but the true quantum propagator stays finite there; the divergence is smoothed by diffraction (formally, by the Airy-type behaviour near the focus). What you must not do is push the naive $\sqrt{1/\sin\omega T}$ through the caustic without the $e^{-i\pi/2}$ phase; drop it and your spectrum comes out wrong. The exact oscillator kernel handles the caustic correctly precisely because it is the analytically continued (regulated) object, which is why the spectral resummation works.

Worked examples

Example 1 — The free action and prefactor. Compute $S_{\rm cl}$ and the Van Vleck prefactor for a free particle with $m=1$ , $\hbar=1$ , over $T=1$ , endpoints $x_i=0$ , $x_f=1$ .

Step 1. Classical action: $S_{\rm cl}=m(x_f-x_i)^2/2T=(1)(1)^2/(2\cdot1)=\tfrac12$ .

Step 2. Van Vleck determinant: $\partial^2 S_{\rm cl}/\partial x_i\partial x_f=-m/T=-1$ , so the prefactor magnitude is $\sqrt{|{-1}|/2\pi\hbar}=\sqrt{1/2\pi}$ .

Step 3. Assemble: $K_0=\sqrt{1/2\pi i}\;e^{i/2}$ , i.e. magnitude $\sqrt{1/2\pi}\approx0.399$ and phase $e^{i(1/2-\pi/4)}$ (the $-\pi/4$ from $\sqrt{1/i}$ ).

Answer: $S_{\rm cl}=\mathbf{0.5}$ and prefactor $\mathbf{\sqrt{1/2\pi i}}$ — the free kernel has no caustic, only the $T\to0$ delta.

Example 2 — The oscillator prefactor at a quarter period, and the first caustic. For the oscillator ( $m=\omega=\hbar=1$ ), evaluate the prefactor at $\omega T=\pi/2$ and identify the first caustic.

Step 1. At $\omega T=\pi/2$ , $\sin\omega T=1$ , so the prefactor is $\sqrt{m\omega/2\pi i\hbar\cdot1}=\sqrt{1/2\pi i}$ — the same magnitude as the free case, because a quarter period has not yet focused anything.

Step 2. Increase $\omega T$ toward $\pi$ : $\sin\omega T\to0^+$ , so the prefactor $\sqrt{1/2\pi i\sin\omega T}\to\infty$ — the first caustic at $\omega T=\pi$ (half a period), where every trajectory has swung across the well and refocused at $-x_i$ .

Step 3. Crossing $\omega T=\pi$ , $\sin\omega T$ turns negative, multiplying $K$ by $e^{-i\pi/2}$ — the first Maslov phase, exactly what the WKB connection formulae charge for the turning point.

Answer: prefactor $=\mathbf{\sqrt{1/2\pi i}}$ at $\omega T=\pi/2$ ; first caustic at $\mathbf{\omega T=\pi}$ with a $\mathbf{-\pi/2}$ Maslov phase across it.

Origins · 1 minvan vleck determinant and how this was discovered

van vleck determinant

1928 · the density of classical trajectories

The prefactor of the semiclassical propagator is the Van Vleck determinant, introduced by J. H. Van Vleck in 1928 as a quantum-corrected form of the classical correspondence principle. Its content is geometric: $\partial^2 S_{\rm cl}/\partial x_i\partial x_f$ measures how a bundle of classical trajectories, launched from $x_i$ with slightly different momenta, spreads by the time it reaches $x_f$ . Where the bundle bunches — a focus — the determinant diverges, marking a caustic; where it fans out, the amplitude thins. Pauli and Cécile Morette-DeWitt put the object on firm footing around 1950–51, giving the modern multidimensional form $\det(-\partial^2 S_{\rm cl}/\partial x_i\partial x_f)$ that reappears in every semiclassical trace formula, up to and including Gutzwiller's for chaotic systems.

History

1866

— Mehler publishes the kernel identity behind the oscillator propagator's spectral sum

1928

— Van Vleck introduces the determinant as a correspondence-principle amplitude

1948

— Feynman evaluates the free and oscillator path integrals exactly

1951

— Morette gives the semiclassical prefactor a rigorous derivation

Going further

Prerequisites: The Feynman Path Integral Formulation; The Propagator from Path Integrals; Gaussian integrals; Quantum Harmonic Oscillator: Analytic Solution.
Leads to: Path Integral and the Schrödinger Equation; Imaginary-Time and Euclidean Path Integrals; Bohr-Sommerfeld and Maslov Quantization (where the caustics become the Maslov index).
Provenance: upgraded from the AllQFT `qm` base (QUANT.RQM.GAUSP.1094), after Feynman & Hibbs. The free/oscillator kernels were flagged for this article by The Quantum Propagator ("get the next two articles to themselves"); the spectrum $E_n=\hbar\omega(n+\tfrac12)$ is owned by Quantum Harmonic Oscillator: Analytic Solution and recalled, not re-derived; the caustic/Maslov phase is developed fully in Bohr-Sommerfeld and Maslov Quantization.

Why is the prefactor a determinant of

S_{\rm cl}

and not something computed from the fluctuation integral?

It is both — and their equality is the theorem. The fluctuation integral $\int\mathcal D y\,e^{iS_2[y]/\hbar}$ is a functional determinant (of the operator $-m(\partial_t^2+\omega^2)$ with Dirichlet ends). The Gelfand–Yaglom result says that functional determinant equals $1/\sqrt{\partial^2 S_{\rm cl}/\partial x_i\partial x_f}$ up to the universal $\sqrt{2\pi i\hbar}$ factors. So you can compute the hard functional determinant or just differentiate the classical action twice — they give the same Van Vleck prefactor. The second route is why (1) is so useful.

Does the free particle really have no caustics, ever?

In 1D with no potential, correct: $\partial^2 S_{\rm cl}/\partial x_i\partial x_f=-m/T$ diverges only at $T=0$ , which is the delta-function initial condition, not a caustic. Free trajectories fan out monotonically and never reconverge, so there is nothing to focus. Add walls (a box), a lens (a quadratic potential over a finite region), or go to higher dimensions with a focusing geometry, and caustics reappear. The oscillator is the simplest system that has them, which is exactly why it is the canonical example here.

Is the Maslov phase the same

-\pi/2

that shows up in WKB connection formulae?

Yes — it is one phenomenon wearing two derivations. In WKB, matching the oscillatory solution to the decaying one across a classical turning point via the Airy function costs $\pi/4$ ; a full traversal (out and back past two turning points, or a focus) accumulates $\pi/2$ . Here the same $\pi/2$ appears as the branch jump of $\sqrt{1/\sin\omega T}$ across a caustic. The Bohr-Sommerfeld and Maslov Quantization article ties them together and shows the count is topological — the Maslov index — so the quantization condition gets $n+\mu/4$ instead of $n$ .

The spectral sum only converged after adding an imaginary part to

T

— is that a cheat?

No, it is the standard interpretation. The Mehler sum $\sum_n e^{-i(n+\frac12)\omega T}\psi_n\psi_n^*$ sits on the unit circle for real $T$ , where it converges only conditionally (as a distribution). Giving $T$ a small negative imaginary part, $T\to T-i\eta$ , damps term $n$ by $e^{-(n+\frac12)\omega\eta}$ and makes the sum absolutely convergent; the physical real- $T$ kernel is the $\eta\to0^+$ limit. This $i\epsilon$ -style regulator is exactly what defines the propagator as a boundary value of an analytic function — the same idea that underlies the Feynman propagator's causal prescription.

How does the oscillator kernel reduce to the free one as

\omega\to0

Expand the small- $\omega$ limit of (3): $\sin\omega T\to\omega T$ and $\cos\omega T\to1$ , so the prefactor $\sqrt{m\omega/2\pi i\hbar\sin\omega T}\to\sqrt{m/2\pi i\hbar T}$ and the exponent $\tfrac{im\omega}{2\hbar\sin\omega T}[(x_i^2+x_f^2)-2x_ix_f]\to\tfrac{im}{2\hbar T}(x_f-x_i)^2$ . Both pieces land exactly on the free kernel (2). The verifier confirms the prefactor limit symbolically. This is a good sanity check any time you derive a propagator: turn off the potential and you must recover the free particle.

Can I use

\sqrt{\text{Van Vleck}}\,e^{iS_{\rm cl}/\hbar}

for a non-quadratic potential?

Not exactly — only as the leading semiclassical approximation. For a general $V$ , the fluctuation action $S_2[y]$ has cubic and higher terms in $y$ , so the path integral is not a pure Gaussian and (1) is only the $\hbar\to0$ leading term, with corrections in powers of $\hbar$ . The free particle and oscillator are special because $S_2$ is exactly quadratic, making (1) exact. This is the same statement as "WKB is exact for the oscillator": the semiclassical formula happens to be the whole answer when the action is quadratic.