Categories: Physics, Quantum mechanics.

Path integral formulation

In quantum mechanics, the path integral formulation is an alternative description of quantum mechanics, which is equivalent to the “traditional” Schrödinger equation. Whereas the latter is based on Hamiltonian mechanics, the former comes from Lagrangian mechanics.

It expresses the propagator KK using the following sum over all possible paths x(t)x(t), which all go from the initial position x0x_0 at time t0t_0 to the destination xNx_N at time tNt_N:

K(xN,tN;x0,t0)=Aallx(t)exp(iS[x]/)\begin{aligned} \boxed{ K(x_N, t_N; x_0, t_0) = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) } \end{aligned}

Where AA normalizes. S[x]S[x] is the classical action of the path xx, whose minimization yields the Euler-Lagrange equation from Lagrangian mechanics. Note that each path is given an equal weight, even unrealistic paths that make big detours.

This apparent problem solves itself, thanks to the fact that paths close to the classical optimum xc(t)x_c(t) have an action close to Sc=S[xc]S_c = S[x_c], while the paths far away have very different actions. Since S[x]S[x] is inside a complex exponential, this means that paths close to xcx_c add contructively, and the others add destructively and cancel out.

An interesting way too look at it is by varying \hbar: as its value decreases, minor action differences yield big phase differences, which make the quantum wave function stay closer to xcx_c. In the limit 0\hbar \to 0, quantum mechanics thus turns into classical mechanics.

Time-slicing derivation

The most popular way to derive the path integral formulation proceeds as follows: starting from the definition of the propagator KK, we divide the time interval tNt0t_N - t_0 into NN “slices” of equal width Δt=(tNt0)/N\Delta t = (t_N - t_0) / N, where NN is large:

K(xN,tN;x0,t0)=xNeiH^(tNt0)/x0=xNeiH^Δt/eiH^Δt/x0\begin{aligned} K(x_N, t_N; x_0, t_0) &= \matrixel{x_N}{e^{- i \hat{H} (t_N - t_0) / \hbar}}{x_0} = \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar} \cdots e^{- i \hat{H} \Delta t / \hbar}}{x_0} \end{aligned}

Between the exponentials we insert N ⁣ ⁣1N\!-\!1 identity operators I^=xxdx\hat{I} = \int \Ket{x} \Bra{x} \dd{x}, and define xj=x(tj)x_j = x(t_j) for an arbitrary path x(t)x(t):

K=xNeiH^Δt/xN1x1eiH^Δt/x0dx1dxN1\begin{aligned} K &= \int\cdots\int \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar}}{x_{N-1}} \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta t / \hbar}}{x_0} \dd{x_1} \cdots \dd{x_{N - 1}} \end{aligned}

For sufficiently small time steps Δt\Delta t (i.e. large NN we make the following approximation (which would be exact, were it not for the fact that T^\hat{T} and V^\hat{V} are operators):

eiH^Δt/=ei(T^+V^)Δt/eiT^Δt/eiV^Δt/\begin{aligned} e^{- i \hat{H} \Delta t / \hbar} = e^{- i (\hat{T} + \hat{V}) \Delta t / \hbar} \approx e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar} \end{aligned}

Since V^=V(xj)\hat{V} = V(x_j), we can take it out of the inner product as a constant factor:

xj+1eiT^Δt/eiV^Δt/xj=eiV(xj)Δt/xj+1eiT^Δt/xj\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar}}{x_j} = e^{- i V(x_j) \Delta t / \hbar} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} \end{aligned}

Here we insert the identity operator expanded in the momentum basis I^=ppdp\hat{I} = \int \Ket{p} \Bra{p} \dd{p}, and commute it with the kinetic energy T^=p^2/(2m)\hat{T} = \hat{p}^2 / (2m) to get:

xj+1eiT^Δt/xj=xj+1|pexp ⁣( ⁣ ⁣ip2Δt2m)p|xjdp\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} = \int_{-\infty}^\infty \Inprod{x_{j+1}}{p} \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \Inprod{p}{x_j} \dd{p} \end{aligned}

In the momentum basis p\Ket{p}, the position basis vectors are represented by plane waves:

p|xj=12πexp ⁣( ⁣ ⁣ixjp)xj+1|p=12πexp ⁣(ixj+1p)\begin{aligned} \Inprod{p}{x_j} = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( \!-\! i \frac{x_j p}{\hbar} \Big) \qquad \Inprod{x_{j+1}}{p} = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( i \frac{x_{j+1} p}{\hbar} \Big) \end{aligned}

With this, we return to the inner product and further evaluate the integral:

xj+1eiT^Δt/xj=12πexp ⁣( ⁣ ⁣ip2Δt2m)exp ⁣(i(xj+1xj)p)dp=12π2πmiΔtexp ⁣(im(xj+1xj)22Δt)\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} &= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \exp\!\Big(i \frac{(x_{j+1} - x_j) p}{\hbar}\Big) \:dp \\ &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta t}} \exp\!\Big( i \frac{m (x_{j+1} - x_j)^2}{2 \hbar \Delta t} \Big) \end{aligned}

Inserting this back into the definition of the propagator K(xN,tN;x0,t0)K(x_N, t_N; x_0, t_0) yields:

K=(im2πΔt) ⁣N/2exp ⁣( ⁣j=0N1i(m(xj+1 ⁣ ⁣xj)22ΔtV(xj)Δt) ⁣)dx1dxN1\begin{aligned} K = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \exp\!\bigg(\! \sum_{j = 0}^{N - 1} i \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta t} - \frac{V(x_j) \Delta t}{\hbar} \Big) \!\bigg) \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}

For large NN and small Δt\Delta t, the sum in the exponent becomes an integral:

ij=0N1(m(xj+1 ⁣ ⁣xj)22Δt2V(xj))Δtit0tN(12mx˙2V(x))dτ\begin{aligned} \frac{i}{\hbar} \sum_{j = 0}^{N - 1} \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \Delta t^2} - V(x_j) \Big) \Delta t \quad \to \quad \frac{i}{\hbar} \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} \end{aligned}

Upon closer inspection, this integral turns out to be the classical action S[x]S[x], with the integrand being the Lagrangian LL:

S[x(t)]=t0tNL(x,x˙,τ)dτ=t0tN(12mx˙2V(x))dτ\begin{aligned} S[x(t)] = \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} = \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} \end{aligned}

The definition of the propagator KK is then further reduced to the following:

K=(im2πΔt) ⁣N/2exp(iS[x]/)dx1dxN1\begin{aligned} K = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \exp(i S[x] / \hbar) \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}

Finally, for the purpose of normalization, we define the integral over all paths x(t)x(t) as follows, where we write D[x]D[x] instead of dx\dd{x}:

D[x]limN(im2πΔt) ⁣N/2dx1dxN1\begin{aligned} \int D[x] \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}

We thus arrive at Feynman’s path integral, which sums over all possible paths x(t)x(t):

K=exp(iS[x]/)D[x]=Aallx(t)exp(iS[x]/)\begin{aligned} K = \int \exp(i S[x] / \hbar) \:D[x] = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) \end{aligned}


  1. R. Shankar, Principles of quantum mechanics, 2nd edition, Springer.
  2. L.E. Ballentine, Quantum mechanics: a modern development, 2nd edition, World Scientific.