3.2.1 Path Integral Formulation#

Prompts

What is the propagator \(K(x,t;x',t')\)? Why does knowing it give the time evolution of any wavefunction?
State the composition (semigroup) property of the propagator. How does it follow from inserting a position resolution of identity?
Apply the “phase = action” principle to a short time slice \(\delta t\). What is the phase of \(K_{\delta t}\), and why is its amplitude not yet fixed?
What does \(\int\mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S[x]/\hbar}\) actually mean as the \(N\to\infty\) limit of time-sliced integrals?

Lecture Notes#

Overview#

In §3.1.3 we adopted the guiding principle “phase = action”: every quantum amplitude is built from \(\mathrm{e}^{\mathrm{i}S/\hbar}\) summed over paths. So far this is abstract. To extract a concrete, quantitative prediction we need a definite object to compute. That object is the propagator — the position-space matrix element of the time-evolution operator. Our job in §3.2.1 is to set up the problem cleanly: define the propagator, decompose long times into short slices, write down the contribution of one slice from the phase-action principle, and reach a precise definition of the formal “path integral” \(\int \mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S/\hbar}\).

The Propagator#

Propagator

Let \(\hat{U}(t,t')\) denote the unitary that implements time evolution from \(t'\) to \(t\). The propagator is its position-space matrix element:

(62)#\[ K(x,t;x',t') = \langle x\vert \hat{U}(t,t')\vert x'\rangle. \]

It is the quantum amplitude for a particle prepared at \(x'\) at time \(t'\) to be detected at \(x\) at time \(t\).

The propagator is the central object of this chapter for two reasons.

Wave evolution. Inserting the position resolution of identity into \(\vert\psi(t)\rangle = \hat{U}(t,t')\vert\psi(t')\rangle\) gives

(63)#\[ \psi(x,t) = \int K(x,t;x',t')\,\psi(x',t')\,\mathrm{d}x'. \]

So \(K\) is the kernel that propagates any initial wavefunction forward in time. Knowing \(K\) is knowing all of the dynamics.

Composition (semigroup). For any intermediate time \(t'<t_m<t\),

(64)#\[ K(x,t;x',t') = \int K(x,t;x_m,t_m)\,K(x_m,t_m;x',t')\,\mathrm{d}x_m. \]

Derivation: composition from resolution of identity

Start from \(\hat{U}(t,t') = \hat{U}(t,t_m)\,\hat{U}(t_m,t')\) and insert \(\int\vert x_m\rangle\langle x_m\vert\,\mathrm{d}x_m = \hat{I}\) between the two factors:

\[\begin{split} \begin{split} K(x,t;x',t') &= \langle x\vert\hat{U}(t,t_m)\,\hat{U}(t_m,t')\vert x'\rangle\\ &= \int\langle x\vert\hat{U}(t,t_m)\vert x_m\rangle\,\langle x_m\vert\hat{U}(t_m,t')\vert x'\rangle\,\mathrm{d}x_m\\ &= \int K(x,t;x_m,t_m)\,K(x_m,t_m;x',t')\,\mathrm{d}x_m. \end{split} \end{split}\]

The composition property is the engine of everything that follows: a long propagation factorizes into shorter ones, with each intermediate position summed over.

Time Slicing#

Iterate composition \(N\) times. Divide the interval \([t',t]\) into equal slices of duration

\[ \delta t = \frac{t-t'}{N}, \]

with intermediate times \(t_n = t' + n\,\delta t\) and intermediate positions \(x_n\), fixing the endpoints \((x_0, t_0) = (x',t')\) and \((x_N,t_N) = (x,t)\). The composition rule gives

(65)#\[ K(x,t;x',t') = \int \left[\,\prod_{n=1}^{N-1}\mathrm{d}x_n\,\right]\;\prod_{n=0}^{N-1} K_{\delta t}(x_{n+1},x_n), \]

where the slice propagator is the short-time piece

(66)#\[ K_{\delta t}(x_{n+1},x_n) \;\equiv\; K(x_{n+1},t_{n+1};x_n,t_n). \]

Each ordered tuple \((x_0,x_1,\ldots,x_N)\) is a discretized path, so the right-hand side is a sum over piecewise-linear histories connecting \(x'\) to \(x\).

The Slice Propagator from “Phase = Action”

For a short slice, approximate the path by the straight-line segment from \(x_n\) to \(x_{n+1}\) at constant velocity \(v_n = (x_{n+1}-x_n)/\delta t\). The action of a free particle accumulated over this segment is

(67)#\[ S_{\text{slice}} \;=\; \int_{t_n}^{t_{n+1}}\frac{1}{2}m\,v_n^{\,2}\,\mathrm{d}\tau \;=\; \frac{m\,(x_{n+1}-x_n)^2}{2\,\delta t}. \]

Apply the principle “phase = action” from §3.1.3 to this single slice: the slice propagator should carry a phase equal to \(S_{\text{slice}}/\hbar\). The principle fixes only the phase; the overall amplitude is not specified.

Slice propagator (free, ansatz)

(68)#\[ K_{\delta t}(x_{n+1},x_n) \;=\; A(\delta t)\;\exp\!\left[\,\frac{\mathrm{i}}{\hbar}\,\frac{m\,(x_{n+1}-x_n)^2}{2\,\delta t}\,\right], \]

with a \(\delta t\)-dependent normalization \(A(\delta t)\) that the phase-action principle does not determine.

We will pin down \(A(\delta t)\) in §3.2.2 by demanding self-consistency of the one-slice evolution at \(\delta t = 0\).

The Path Integral as a Limit#

Fix the endpoints \(x_0=x'\) and \(x_N=x\). As \(N\to\infty\) the discretized path \((x_0,\ldots,x_N)\) approaches a continuous curve \(x(\tau)\), and the discretized action becomes the classical action functional,

\[ \sum_{n=0}^{N-1} S_{\text{slice}}(x_{n+1},x_n) \;\longrightarrow\; S[x] \;=\; \int_{t'}^{t} L(x,\dot x)\,\mathrm{d}\tau. \]

Bundle the per-slice integrations and amplitude factors into a single symbol \(\mathcal{D}[x]\):

Path integral (definition)

(69)#\[\begin{split} \begin{split} K(x,t;x',t') &\;\equiv\; \lim_{N\to\infty}\int\left[\,\prod_{n=1}^{N-1}\mathrm{d}x_n\,\right]\prod_{n=0}^{N-1}K_{\delta t}(x_{n+1},x_n)\\ &\;=:\;\int_{x(t')=x'}^{x(t)=x}\mathcal{D}[x]\;\mathrm{e}^{\mathrm{i}S[x]/\hbar}. \end{split} \end{split}\]

The Feynman path integral \(\int\mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S[x]/\hbar}\) is the \(N\to\infty\) limit of the time-sliced product (65). It has no other meaning.

Discussion: is the continuum path integral well defined?

The set of continuous paths between two spacetime points is uncountable, and writing \(\int\mathcal{D}[x]\) as a true measure-theoretic integral on path space is delicate. The slicing form sidesteps this difficulty: every quantity is a finite-dimensional ordinary integral, and the continuum object is defined as the \(N\to\infty\) limit. The slicing is not a calculational trick — it is the operational definition.

Strategy: Master One Slice First#

Equation (65) reduces a quantum dynamics problem to an ordinary \((N{-}1)\)-fold integral. For large \(N\) this is still intractable; we cannot evaluate all slices simultaneously. The pedagogical move is to study how one slice transforms the wavefunction. Combining (63) with the definition of \(K_{\delta t}\),

(70)#\[ \psi(x,t+\delta t) \;=\; \int K_{\delta t}(x,x')\,\psi(x',t)\,\mathrm{d}x'. \]

A single slice should advance time by an infinitesimal amount, producing only an infinitesimal change in \(\psi\). Expanding both sides to first order in \(\delta t\) should turn this integral relation into a partial differential equation — which we will recognize as the Schrödinger equation. That is the program of §3.2.2.

Poll: meaning of \(\mathcal{D}[x]\)

In the path integral \(K = \int\mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S[x]/\hbar}\), what does the symbol \(\mathcal{D}[x]\) stand for?

(A) An ordinary one-dimensional measure \(\mathrm{d}x\) over the spatial coordinate \(x\).

(B) A formal limit of \((N{-}1)\)-fold integrals over intermediate positions \(x_1,\ldots,x_{N-1}\), together with normalization factors per slice, as \(N\to\infty\).

(D) A symbol meaning “the classical path”; only one path contributes once the limit is taken.

Summary#

Propagator: \(K(x,t;x',t') = \langle x\vert\hat U(t,t')\vert x'\rangle\) is the kernel that evolves any initial wavefunction () and obeys the composition rule ().
Time slicing: iterating composition turns \(K\) into an \((N{-}1)\)-fold integral over intermediate positions weighted by a product of slice propagators ().
Slice propagator: the principle phase = action fixes the phase of \(K_{\delta t}\) to \(\mathrm{e}^{\mathrm{i}S_{\text{slice}}/\hbar}\); the amplitude \(A(\delta t)\) remains undetermined for now ().
Path integral: \(\int\mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S[x]/\hbar}\) is defined as the \(N\to\infty\) limit of the slicing formula ().
Next step: evaluate the one-slice evolution to first order in \(\delta t\) — the result will be the Schrödinger equation, and the requirement that \(\psi\) be unchanged at \(\delta t = 0\) will fix \(A(\delta t)\).

Homework#

1. Propagating a Gaussian wavepacket. A particle is in the Gaussian state \(\psi(x,0) = (\pi\sigma^{2})^{-1/4}\exp(-x^{2}/(2\sigma^{2}))\) at \(t=0\). The free-particle propagator is

\[ K(x,t;x',0) = \sqrt{\frac{m}{2\pi\mathrm{i}\hbar t}}\,\exp\!\left(\frac{\mathrm{i}m(x-x')^{2}}{2\hbar t}\right). \]

(a) Use \(\psi(x,t) = \int K(x,t;x',0)\,\psi(x',0)\,\mathrm{d}x'\) to compute \(\psi(x,t)\) by Gaussian integration over \(x'\).

(b) Show that \(\vert\psi(x,t)\vert^{2}\) remains Gaussian with time-dependent width \(\sigma_{t} = \sigma\sqrt{1 + (\hbar t/(m\sigma^{2}))^{2}}\).

(c) Identify the spreading time scale \(\tau \sim m\sigma^{2}/\hbar\) and verify it numerically for an electron with \(\sigma = 1\,\text{Å}\).

2. Composition test. Verify the composition property of the free propagator (defined in Problem 1) by direct computation.

(a) Compute \(\int K(x,t;x_{1},t/2)\,K(x_{1},t/2;x',0)\,\mathrm{d}x_{1}\) by completing the square in \(x_{1}\), and show that the result equals \(K(x,t;x',0)\).

(b) Repeat with two intermediate times to verify \(\iint K(x,t;x_{2},2t/3)\,K(x_{2},2t/3;x_{1},t/3)\,K(x_{1},t/3;x',0)\,\mathrm{d}x_{1}\,\mathrm{d}x_{2} = K(x,t;x',0)\).

(c) Argue by induction that any uniform partition of \([0,t]\) into \(N\) slices reproduces the same \(K(x,t;x',0)\). This is the explicit verification that the time-sliced path integral converges to the propagator.

3. Free-particle slice action. Verify Eq. (67) by evaluating \(S = \int_{t_n}^{t_{n+1}}\tfrac{1}{2}m\dot{x}^2\,\mathrm{d}\tau\) along the straight-line path \(x(\tau) = x_n + (x_{n+1}-x_n)(\tau-t_n)/\delta t\).

4. Slice action with a potential. For the Lagrangian \(L = \tfrac{1}{2}m\dot{x}^2 - V(x)\), repeat the straight-line estimate over a single slice and show that, to first order in \(\delta t\),

\[ S_{\text{slice}} \;=\; \frac{m\,(x_{n+1}-x_n)^2}{2\,\delta t} \;-\; V\!\left(\tfrac{x_n+x_{n+1}}{2}\right)\delta t \;+\; O(\delta t^{\,2}). \]

(a) Approximate \(x(\tau)\) by the straight-line segment.

(b) Estimate \(\int_{t_n}^{t_{n+1}} V(x(\tau))\,\mathrm{d}\tau\) using the midpoint value of \(x(\tau)\) and explain why corrections are \(O(\delta t^{\,2})\).

5. Phase difference between nearby slices. Two slice paths share the initial point \(x_n\) but end at \(x_{n+1}\) and \(x_{n+1} + \Delta\) respectively, with \(\vert\Delta\vert\) small.

(a) Compute the difference \(\Delta S_{\text{slice}}\) between their free-particle slice actions to first order in \(\Delta\).

(b) Identify the slice momentum \(p_n = m(x_{n+1}-x_n)/\delta t\) and rewrite \(\Delta S_{\text{slice}}/\hbar\) as \(p_n\,\Delta/\hbar\).

(c) Comment on this result in light of the de Broglie relation \(p = \hbar k\): what is the effective wavelength of the slice propagator as a function of \(x_{n+1}\)?

6. Functional equation from composition. Suppose the slice propagator depends on its arguments only through the displacement: \(K_{\delta t}(x',x) = A(\delta t)\,F(x'-x;\delta t)\).

(a) Apply the composition property (64) to two consecutive slices of duration \(\delta t\) each and show that

\[ F(x'-x;2\delta t) \;=\; A(\delta t)\int F(x'-y;\delta t)\,F(y-x;\delta t)\,\mathrm{d}y. \]

(b) Verify dimensional consistency. (We do not yet know \(A\) or \(F\); the eventual closed form derived in §3.2.2 must satisfy this constraint.)

7. Typical paths in the slicing. Argue that the typical path that contributes coherently to (65) for a free particle has slice displacements of order

\[ \vert x_{n+1} - x_n\vert \;\sim\; \sqrt{\frac{\hbar\,\delta t}{m}}. \]

(a) Show that this is the value of \(\vert x_{n+1}-x_n\vert\) at which the phase \(S_{\text{slice}}/\hbar\) in (68) is of order unity, so that contributions from larger displacements oscillate and cancel.

(b) Compare this to the smooth scaling \(\vert x_{n+1}-x_n\vert \sim v\,\delta t\) that a classical trajectory would have, and conclude that as \(\delta t\to 0\) the typical path is not differentiable. (Quantum paths are more like Brownian trajectories than classical orbits.)