Feynman's path-integral idea

For ten chapters we have spoken quantum mechanics in only one language: the wavefunction obeys the Schrödinger equation, $\ii\hbar\partial_t\Psi=\hat H\Psi$, and given the initial state you solve for the future. That language has served us well, but it hides the most naive question of all — “which way did the particle go?” The wavefunction spreads over all of space, and you simply cannot say which route the particle took.

In 1948 Feynman gave us a second language. Its starting point is exactly the glaring fact we left hanging in section 1.6: in the double-slit experiment, the arrival pattern of a single electron somehow “knows” about both slits. Back then we said “the electron’s wave passes through both slits at once.” Feynman pushed that sentence to its logical extreme.

From two slits to infinitely many

Recall the double-slit rule: the total amplitude to arrive at a point on the screen is the sum of the amplitudes of the two routes,

$$A_{\text{tot}}=A_{\text{slit}\,1}+A_{\text{slit}\,2}\tag{11.1.1}$$

The probability is $|A_{\text{tot}}|^2$, and the cross term is the interference fringes.

Now run a thought experiment, upgrading it in three steps:

Step 1: more slits. Cut three slits in the barrier, then four… then $N$. The rule does not change: the total amplitude is the sum over $N$ routes.

Step 2: more screens. Insert a second barrier between source and screen, also riddled with slits. A “path” must now specify which slit of the first barrier the particle threads, then which slit of the second — the number of paths multiplies. The total amplitude is still the sum over all of these zigzag routes.

Step 3: remove the boards. Cut ever more slits in each barrier until the whole board is slit — that is, remove the board; then pack the barriers ever more densely, until there is a “board” at every point of space. The zigzag “which slits in which order” now becomes an arbitrary continuous path from source to screen.

What is left once every board is gone? Vacuum. But the summation rule has no reason to fail at the instant the last board is removed. We are forced to a startling conclusion:

What does each path contribute?

The first half of the rule (sum over paths) was forced on us by the double slit. The second half must answer: what is the amplitude $A[x(t)]$ of a single path?

Feynman’s answer is startlingly short: every path has the same magnitude of amplitude; only the phase differs, and the phase is set by the path’s action:

$$A[x(t)]=\text{(const)}\times\ee^{\ii S[x(t)]/\hbar}\tag{11.1.3}$$

The action is an old friend from classical mechanics: along a given path, accumulate “kinetic minus potential energy” (the Lagrangian $L$) over time,

$$S[x(t)]=\int_{t_a}^{t_b}L\,\dd t=\int_{t_a}^{t_b}\left[\frac{m\dot x^2}{2}-V(x)\right]\dd t\tag{11.1.4}$$

In plain words: $S$ is a “score” assigned to each path — moving fast earns kinetic-energy points, loitering where the potential is high loses points. Classical mechanics’ principle of least action says the true path is the one that makes $S$ stationary (usually a minimum) — Newton’s equation is precisely the mathematical statement of $\delta S=0$.

$S$ has dimensions of energy × time, the same as $\hbar$, so $S/\hbar$ is a pure number and can serve as a phase angle. That is no coincidence — we are about to see that the ratio of $S$ to $\hbar$ is exactly what decides “how quantum” a system is.

Where did the classical world go? Stationary phase

If an electron really takes every route, why does a baseball appear to fly along a single parabola? The answer hides in the cancellation of phases.

The stationary-phase argument: why only the Newtonian path survives as ħ→0basic~7 minexpand

Examine the contributions of two classes of paths.

Class 1: paths far from the classical one. Take a path $x(t)$ with $\delta S\neq0$, meaning any slight deformation changes the action at first order. Call the deformation $\eta$; the phase changes by roughly $\delta S[\eta]/\hbar$.

The key is the $\hbar$ in the denominator. For a macroscopic system the typical $S$ is enormously larger than $\hbar$ (we estimate it below), so even a deformation $\eta$ too small to measure produces a huge phase change: the little arrows of neighbouring paths spin wildly, pointing at every possible angle. Add them up — near-total cancellation.

This is the same phenomenon as the integral $\int\ee^{\ii\lambda f(u)}\dd u$ tending to zero as $\lambda\to\infty$ (the rapidly oscillating positive and negative parts wipe each other out).

Class 2: paths near the classical one. At $x_{\text{cl}}$ we have $\delta S=0$, so the action responds to deformations only at second order:

$$S[x_{\text{cl}}+\eta]=S[x_{\text{cl}}]+O(\eta^2)\tag{11.1.7}$$

There is therefore a whole swath of paths with nearly identical phases; their arrows align and reinforce. The width of this “coherence band” is set by the $O(\eta^2)$ term reaching a phase of order $\pi$: for a free particle one can estimate $\eta\sim\sqrt{\hbar T/m}$ ($T$ being the time of flight).

Conclusion. The total amplitude comes almost entirely from a bundle of paths of width about $\sqrt{\hbar T/m}$ around the classical one. As $\hbar\to0$ (or $S\gg\hbar$) this bundle shrinks to a single line — the particle “takes the Newtonian path”.

Order-of-magnitude check. A baseball ($m=0.145$ kg) flying at 40 m/s for 0.5 s has $S\sim E_k T\approx116\times0.5\approx58$ J·s, so

$$\frac{S}{\hbar}\approx\frac{58}{1.05\times10^{-34}}\approx5\times10^{35}\tag{11.1.8}$$

Stray from the classical path by even $10^{-17}$ m and the phase winds through many full turns. The baseball has no choice. The electron in the double slit, by contrast, has an action difference between its two routes of order $\hbar$ — which is why it “takes both”.

∎Key formulas

• Sum over paths

  $$K(b,a)=\int\mathcal{D}x(t)\,\ee^{\ii S[x(t)]/\hbar}$$

  Every path joining a and b contributes — equal magnitude, differing only in phase

• Action

  $$S[x(t)]=\int_{t_a}^{t_b}\Bigl[\frac{m\dot x^2}{2}-V(x)\Bigr]\dd t$$

  Classical mechanics' scoring function; same dimensions as ħ

• Classical path

  $$\delta S[x_{\text{cl}}]=0$$

  Stationary phase = Newton’s equation; coherence-band width about √(ħT/m)

• Classical-limit criterion

  $$S/\hbar\gg1$$

  About 10³⁶ for a baseball, about 1 for a double-slit electron — that is the dividing line

?Self-check4 questions

1. 1.

   In the path integral, what determines the amplitude of a single path?

   • The magnitude is the same for all paths; the phase equals the path’s action divided by ħ
   • The magnitude is inversely proportional to the path length, so shorter paths contribute more
   • Only paths satisfying Newton’s equation have non-zero amplitude
   • The magnitude is set by the potential energy along the path

   Submit
2. 2.

   Why does only the classical path’s contribution survive as ħ→0 (or S≫ħ)? (Select all that apply.)

   Select all that apply

   • Away from the classical path the action changes at first order, the phase spins rapidly, and contributions cancel
   • Near the classical path δS=0, so a whole bundle of paths shares nearly the same phase and adds coherently
   • The amplitudes of non-classical paths decay exponentially with the deviation
   • The coherence band around the classical path has width about √(ħT/m), which shrinks to a line as ħ→0

   Submit
3. 3.

   In the language of path integrals, the double-slit experiment is:

   • The special case of the sum over all paths in which the barrier leaves only two routes open
   • An example where the path integral fails and one must return to the wavefunction language
   • A proof that only two paths exist
   • A demonstration of the classical limit

   Submit
4. 4.

   A baseball of m=0.145 kg flies at a constant 40 m/s for 0.5 s; take S≈kinetic energy × time. Find log₁₀(S/ħ). (ħ=1.05×10⁻³⁴ J·s)

   50% relative tolerance

   Submit

What comes next

“Sum over all paths” is still only a manifesto. To make it compute actual numbers, we must first define $\int\mathcal{D}x(t)$ properly and then carry the infinite-dimensional integral through completely for the simplest system — the free particle. The result is called the propagator, and it will lock onto the wave-packet spreading of chapter 2 with a satisfying click.