The propagator

The last section laid out the manifesto: the amplitude from $a$ to $b$ is the sum of $\ee^{\ii S/\hbar}$ over all paths. But “sum over all paths” is still just a sentence — there are uncountably many paths, so how do you add them? In this section we actually compute the sum, for the simplest system there is: the free particle ($V=0$).

First, let us give this “amplitude from $a$ to $b$” its official name.

What the propagator is

The propagator $K(x_b,t_b;x_a,t_a)$ is defined as the probability amplitude to find the particle at $x_b$ at time $t_b$, given that it was definitely at $x_a$ at time $t_a$. In the language of chapter 3 it is a matrix element of the evolution operator:

$$K(x_b,t_b;x_a,t_a)=\bra{x_b}\ee^{-\ii\hat H(t_b-t_a)/\hbar}\ket{x_a}\tag{11.2.1}$$

Why does it deserve a name of its own? Because knowing $K$ means knowing all of the dynamics. Any initial state $\Psi(x_a,t_a)$ is a superposition of position eigenstates, each component propagating independently, so

$$\Psi(x_b,t_b)=\int K(x_b,t_b;x_a,t_a)\,\Psi(x_a,t_a)\,\dd x_a\tag{11.2.2}$$

The propagator is the Schrödinger equation’s “universal solution”: solve for $K$ once, and the solution for any initial condition is a single integral away. (Mathematically this is exactly the role of a Green’s function: $K$ satisfies the Schrödinger equation with the initial condition $K\to\delta(x_b-x_a)$ as $t_b\to t_a$ — at the start the particle is at $x_a$, with zero amplitude anywhere else.)

$K$ also has a composition property, inherited directly from the fact that paths can be spliced: going from $a$ to an intermediate point $c$ and then from $c$ to $b$, summed over all intermediate points, equals going from $a$ to $b$ directly:

$$K(b,a)=\int K(b,c)\,K(c,a)\,\dd x_c\tag{11.2.3}$$

This property is about to earn its keep — it lets us chop a long propagation into many short segments.

Defining the path integral properly: time slicing

The free particle: computing the integral to the end

The full derivation of the free-particle propagatoradvanced~12 minexpand

With $V=0$ we face a chain of nested Gaussian-type integrals. Only one tool is needed throughout — the Gaussian integral formula (valid for complex $\alpha$ too, understood by analytic continuation):

$$\int_{-\infty}^{\infty}\ee^{\ii\alpha u^2}\dd u=\sqrt{\frac{\pi\ii}{\alpha}}\tag{11.2.7}$$

Step 1: the single-step propagator. Over a short interval $\varepsilon$ there is effectively only the straight-line “path”, with amplitude

$$K_\varepsilon(x_{j+1},x_j)=\frac1A\exp\!\left[\frac{\ii m(x_{j+1}-x_j)^2}{2\hbar\varepsilon}\right]\tag{11.2.8}$$

Step 2: compose two steps. Use the composition formula to merge two $\varepsilon$ steps into one of $2\varepsilon$, integrating over the intermediate point $x_1$:

$$K_{2\varepsilon}(x_2,x_0)=\frac1{A^2}\int\dd x_1\, \exp\!\left[\frac{\ii m}{2\hbar\varepsilon}\bigl[(x_2-x_1)^2+(x_1-x_0)^2\bigr]\right]\tag{11.2.9}$$

Complete the square in $x_1$ inside the exponent:

$$(x_2-x_1)^2+(x_1-x_0)^2=2\Bigl(x_1-\frac{x_0+x_2}{2}\Bigr)^2+\frac{(x_2-x_0)^2}{2}\tag{11.2.10}$$

The second term is independent of $x_1$ and comes outside; the first is killed by the Gaussian formula (set $u=x_1-\frac{x_0+x_2}2$, $\alpha=\frac{m}{\hbar\varepsilon}$):

$$K_{2\varepsilon}(x_2,x_0)=\frac1{A^2}\sqrt{\frac{\pi\ii\hbar\varepsilon}{m}}\, \exp\!\left[\frac{\ii m(x_2-x_0)^2}{2\hbar(2\varepsilon)}\right]\tag{11.2.11}$$

Key observation: the result has exactly the same shape as the single-step propagator, just with $\varepsilon\to2\varepsilon$. Choosing $A=\sqrt{2\pi\ii\hbar\varepsilon/m}$ makes the prefactor tidy up to exactly $\sqrt{m/2\pi\ii\hbar(2\varepsilon)}$ — this is how the normalisation constant gets fixed “self-consistently”.

Step 3: induction to N steps. Each intermediate point integrated away leaves the propagator with the same shape and one more time step. After $N$ steps (with $N\varepsilon=t$, writing $t=t_b-t_a$, $x=x_b-x_a$):

$$\boxed{\;K(x_b,t_b;x_a,t_a)=\sqrt{\frac{m}{2\pi\ii\hbar t}}\, \exp\!\left[\frac{\ii m(x_b-x_a)^2}{2\hbar t}\right]\;}\tag{11.2.12}$$

The limit $N\to\infty$ is trivial — the shape never depended on $N$ in the first place.

Step 4: checks.

• As $t\to0$ the exponent oscillates infinitely fast and the prefactor diverges — precisely one representation of $\delta(x_b-x_a)$. The initial condition checks out.
• Substitute back into the Schrödinger equation $\ii\hbar\partial_t K=-\frac{\hbar^2}{2m}\partial_{x_b}^2K$ and the two sides match term by term (verify it yourself — five lines).

Squaring the books with wave-packet spreading

If the propagator is right, it has to reproduce physics we already know. Section 2.10 computed, by superposing momenta, that a free Gaussian wave packet of initial width $\sigma_0$ spreads with time. Let us redo the calculation with $K$.

Take the initial state to be a Gaussian packet centred at the origin, $\Psi(x_a,0)\propto\ee^{-x_a^2/4\sigma_0^2}$, and insert it into the evolution formula:

$$\Psi(x,t)=\int K(x,t;x_a,0)\,\Psi(x_a,0)\,\dd x_a \;\propto\;\int\dd x_a\exp\!\left[\frac{\ii m(x-x_a)^2}{2\hbar t}-\frac{x_a^2}{4\sigma_0^2}\right]\tag{11.2.14}$$

The exponent is again quadratic in $x_a$ — yet another Gaussian integral. Complete the square, integrate, and tidy up:

$$|\Psi(x,t)|^2\propto\exp\!\left[-\frac{x^2}{2\sigma(t)^2}\right],\qquad \sigma(t)=\sigma_0\sqrt{1+\Bigl(\frac{\hbar t}{2m\sigma_0^2}\Bigr)^2}\tag{11.2.15}$$

Word for word the result of section 2.10: spreading time $\tau=2m\sigma_0^2/\hbar$, and the narrower the packet, the faster it spreads. Two languages, one physics — the first concrete audit of their equivalence.

∎Key formulas

• Propagator definition

  $$K(b,a)=\bra{x_b}\ee^{-\ii\hat H(t_b-t_a)/\hbar}\ket{x_a}$$

  Position-representation matrix element of the evolution operator; tends to δ(x_b−x_a) as t_b→t_a

• Universal evolution formula

  $$\Psi(x_b,t_b)=\int K(b,a)\,\Psi(x_a,t_a)\,\dd x_a$$

  Solve for K once and any initial condition is one integral away

• Composition property

  $$K(b,a)=\int K(b,c)\,K(c,a)\,\dd x_c$$

  The basis of time slicing; also called the Chapman-Kolmogorov relation

• Free-particle propagator

  $$K=\sqrt{\frac{m}{2\pi\ii\hbar t}}\,\ee^{\ii m(x_b-x_a)^2/2\hbar t}$$

  Phase = classical action m(Δx)²/2t; a universal feature of quadratic Lagrangians

?Self-check4 questions

1. 1.

   The physical meaning of the propagator K(b,a) is:

   • The probability amplitude to find the particle at x_b at time t_b, given it was at x_a at time t_a
   • The probability for the particle to travel from x_a to x_b
   • The amplitude for the particle to follow the classical path from a to b
   • The correlation function of the wavefunction at positions x_a and x_b

   Submit
2. 2.

   The free-particle propagator satisfies K ∝ e^{iS_cl/ħ}, with the phase exactly the classical action. The condition for this "exactly" to hold is:

   • The Lagrangian is quadratic in coordinates and velocities
   • It holds exactly in any potential
   • It holds only in the limit ħ→0
   • It holds only for V=0

   Submit
3. 3.

   Which statements about the time-sliced definition of ∫𝒟x(t) are correct? (Select all that apply.)

   Select all that apply

   • A discretised path is specified by its positions x₁,…,x_{N−1} at the slice times, integrated over in an (N−1)-fold integral
   • The normalisation constant A=√(2πiħε/m) is fixed by composition self-consistency
   • Only differentiable, smooth paths participate in the sum
   • The composition property K(b,a)=∫K(b,c)K(c,a)dx_c is what justifies the slicing method

   Submit
4. 4.

   An electron Gaussian wave packet has initial width σ₀=1.0 nm. What is its spreading time τ=2mσ₀²/ħ in femtoseconds? (m=9.11×10⁻³¹ kg, ħ=1.055×10⁻³⁴ J·s)

   fs150% relative tolerance

   Submit

What comes next

We have now computed a concrete result from the path-integral side and verified that it satisfies the Schrödinger equation. But “verifying one example” is not “proving the two formulations equivalent”. The next section does something more thorough: starting from a short time slice, it derives the Schrödinger equation itself — turning chapter 2’s starting point into a theorem of the path integral.