Categories: Physics, Quantum mechanics.

Path integral formulation

The path integral formulation is an alternative description of quantum mechanics, 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 as the following “sum” over all possible paths x(t)x(t) that take the particle from the starting point (x0,t0)(x_0, t_0) to the destination (xN,tN)(x_N, t_N):

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

Where AA is a normalization constant, and S[x]S[x] is the classical action of the path x(t)x(t), defined as shown below from the system’s Lagrangian LL, and whose minimization would lead to the Euler-Lagrange equation of classical Lagrangian mechanics. Let x˙(t)=dx/dt\dot{x}(t) = \idv{x}{t}:

S[x]t0tNL(x,x˙,τ)dτ\begin{aligned} S[x] \equiv \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} \end{aligned}

Note that KK’s sum gives each path an equal weight, even unrealistic paths taking bigs detours. This apparent problem solves itself as follows: paths close to the classical optimum xc(t)x_c(t) have an action close to Sc=S[xc]S_c = S[x_c], since SS is stationary there. Meanwhile, for paths far away from xcx_c, SS gives very different values, which change by a lot if a small change is made to xx. Because S[x]S[x] is inside a complex exponential, paths close to xcx_c therefore add more or less constructively, while the others add destructively and cancel out.

Consequently, the “quantum path” is still close to xc(t)x_c(t). An interesting way to think about this is by treating \hbar as a parameter: as its value decreases, small action changes result in bigger phase differences, which makes the quantum wavefunction stay closer to xcx_c for the aforementioned reasons. In the limit 0\hbar \to 0, quantum mechanics simply turns into classical mechanics.

In reality, KK’s sum is evaluated as an integral over all paths x(t)x(t), hence this is called the path integral formulation. The proof that the propagator KK’s Schrödinger-picture definition can be rewritten as such an integral is given below.

Time-slicing derivation

For a time-independent Hamiltonian H^\hat{H}, we start from the definition of the propagator KK, and divide the time interval tN ⁣ ⁣t0t_N \!-\! t_0 into NN “slices” of equal width Δt(tN ⁣ ⁣t0)/N\Delta{t} \equiv (t_N \!-\! t_0) / N:

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 identity operators xxdx\int_{-\infty}^\infty \Ket{x} \Bra{x} \dd{x}, and define xjx(tj)x_j \equiv x(t_j) for an arbitrary path x(t)x(t), where tjt_j is the endpoint of the jjth slice. This is equivalent to splitting KK into a product of all slices’ individual propagators:

K=K(xN,tN;xN1,tN1)K(x2,t2;x1,t1)K(x1,t1;x0,t0)= ⁣ ⁣xNeiH^Δt/xN1x1eiH^Δt/x0dx1dxN1\begin{aligned} K &= K(x_N, t_N; x_{N-1}, t_{N-1}) \cdots K(x_2, t_2; x_1, t_1) \: K(x_1, t_1; x_0, t_0) \\ &= \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 can split the Hamiltonian into its kinetic and potential terms H^=T^+V^\hat{H} = \hat{T} + \hat{V}. Note that this is an approximation, since T^\hat{T} and V^\hat{V} are operators that do not commute, but it becomes exact in the limit Δt0\Delta{t} \to 0:

eiH^Δt/eiT^Δt/eiV^Δt/\begin{aligned} e^{- i \hat{H} \Delta{t} / \hbar} \approx e^{- i \hat{T} \Delta{t} / \hbar} \: e^{- i \hat{V} \Delta{t} / \hbar} \end{aligned}

We substitute V^=V(xj)\hat{V} = V(x_j), and apply it directly to xj\ket{x_j}, such that we can take it out of the inner product as a constant factor:

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

In order to evaluate the remaining inner product, we insert the identity operator again, this time expanded in the momentum basis ppdp\int_{-\infty}^\infty \Ket{p} \Bra{p} \dd{p}, and use T^=p^2/(2m)\hat{T} = \hat{p}^2 / (2m) to get:

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

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

px=eixp/2π\begin{aligned} \inprod{p}{x} = \frac{e^{- i x p / \hbar}}{\sqrt{2 \pi \hbar}} \end{aligned}

Inserting this and looking up the resulting integral, we arrive at:

xj+1eiT^Δt/xj=12πexp ⁣( ⁣ ⁣iΔt2mp2+i(xj+1 ⁣ ⁣xj)p)dp=12π2πmiΔtexp ⁣(im(xj+1 ⁣ ⁣xj)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\!\bigg( \!-\! i \frac{\Delta{t}}{2 m \hbar} p^2 + i \frac{(x_{j+1} \!-\! x_j)}{\hbar} p \bigg) \dd{p} \\ &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta{t}}} \exp\!\bigg( i \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta{t}} \bigg) \end{aligned}

Including the factor due to V^\hat{V}, we find that the propagator of a single time slice is:

xj+1eiH^Δt/xj=im2πΔtexp ⁣(im2(xj+1 ⁣ ⁣xj)2ΔtiV(xj)Δt)\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{H} \Delta t / \hbar}}{x_j} = \sqrt{\frac{- i m}{2 \pi \hbar \Delta{t}}} \exp\!\bigg( \frac{i}{\hbar} \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}} - \frac{i}{\hbar} V(x_j) \: \Delta{t} \bigg) \end{aligned}

This is a “local” result; inserting it into the “global” propagator K(xN,tN;x0,t0)K(x_N, t_N; x_0, t_0) yields:

K=(im2πΔt) ⁣N/2 ⁣ ⁣ ⁣j=0N1exp ⁣(im2(xj+1 ⁣ ⁣xj)2ΔtiV(xj)Δt)dx1dxN1=(im2πΔt) ⁣N/2 ⁣ ⁣ ⁣exp ⁣(iΔtj=0N1(m2(xj+1 ⁣ ⁣xj)2Δt2V(xj)))dx1dxN1\begin{aligned} K &= \bigg( \frac{- i m}{2 \pi \hbar \Delta{t}} \bigg)^{\!N / 2} \!\int\!\cdots\!\int \prod_{j = 0}^{N - 1} \exp\!\bigg( \frac{i}{\hbar} \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}} - \frac{i}{\hbar} V(x_j) \: \Delta{t} \bigg) \dd{x_1} \cdots \dd{x_{N-1}} \\ &= \Big( \frac{- i m}{2 \pi \hbar \Delta{t}} \Big)^{\!N / 2} \!\int\!\cdots\!\int \exp\!\bigg( \frac{i \Delta{t}}{\hbar} \sum_{j = 0}^{N-1} \Big( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \Big) \bigg) \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}

It is worth noting that there are N ⁣ ⁣1N\!-\!1 integrals, but NN factors (im/2πΔt)1/2(-i m / 2 \pi \hbar \Delta{t})^{1/2} i.e. one for each slice. According to convention, N ⁣ ⁣1N\!-\!1 of those factors are said to belong to the integrals, and then the remaining one belongs to the process as a whole.

In the limit Δt0\Delta{t} \to 0 (or NN \to \infty), the sum in the exponent becomes an integral:

limΔt0j=0N1(m2(xj+1 ⁣ ⁣xj)2Δt2V(xj))Δt=t0tN ⁣(12mx˙2V(x))dτ=t0tNL(x,x˙,τ)dτ=S[x]\begin{aligned} \lim_{\Delta{t} \to 0} \sum_{j = 0}^{N - 1} \bigg( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \bigg) \Delta{t} \:\:&=\:\: \int_{t_0}^{t_N} \!\bigg( \frac{1}{2} m \dot{x}^2 - V(x) \bigg) \dd{\tau} \\ \:\:&=\:\: \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} \\ \:\:&=\:\: S[x] \end{aligned}

Where we have recognized the Lagrangian L=TVL = T - V and hence the action S[x]S[x] of the path x(t)x(t). We thus arrive at the following formula for the global propagator KK, known as Feynman’s path integral or sometimes the configuration space path integral:

K=eiS[x]/Dx\begin{aligned} \boxed{ K = \int e^{i S[x] / \hbar} \:\mathcal{D}{x} } \end{aligned}

Where we have introduced the following notation to indicate an integral over all paths, because writing the factor and all those integrals can become tedious:

DxlimN(im2πΔt) ⁣N/2dx1dxN1\begin{aligned} \boxed{ \int \mathcal{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}

It is worth stressing that this is simply an abbreviation; in practice, calculating KK in this way still requires the individual slices to be taken into account.

References

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