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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 $K$ as the following “sum” over all possible paths $x(t)$ that take the particle from the starting point $(x_0, t_0)$ to the destination $(x_N, t_N)$:

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

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

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

Note that $K$’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 $x_c(t)$ have an action close to $S_c = S[x_c]$, since $S$ is stationary there. Meanwhile, for paths far away from $x_c$, $S$ gives very different values, which change by a lot if a small change is made to $x$. Because $S[x]$ is inside a complex exponential, paths close to $x_c$ therefore add more or less constructively, while the others add destructively and cancel out.

Consequently, the “quantum path” is still close to $x_c(t)$. 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 $x_c$ for the aforementioned reasons. In the limit $\hbar \to 0$, quantum mechanics simply turns into classical mechanics.

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

Time-slicing derivation

For a time-independent Hamiltonian $\hat{H}$, we start from the definition of the propagator $K$, and divide the time interval $t_N \!-\! t_0$ into $N$ “slices” of equal width $\Delta{t} \equiv (t_N \!-\! t_0) / N$:

$$\begin{aligned} K(x_N, t_N; x_0, t_0) &= \matrixel{x_N}{e^{- i \hat{H} (t_N - t_0) / \hbar}}{x_0} \\ &= \matrixel{x_N}{e^{- i \hat{H} \Delta{t} / \hbar} \cdots e^{- i \hat{H} \Delta{t} / \hbar}}{x_0} \end{aligned}$$

Between the exponentials we insert identity operators $\int_{-\infty}^\infty \Ket{x} \Bra{x} \dd{x}$, and define $x_j \equiv x(t_j)$ for an arbitrary path $x(t)$, where $t_j$ is the endpoint of the $j$th slice. This is equivalent to splitting $K$ into a product of all slices’ individual propagators:

$$\begin{aligned} K &= K(x_N, t_N; x_{N-1}, t_{N-1}) \cdots K(x_2, t_2; x_1, t_1) \: K(x_1, t_1; x_0, t_0) \\ &= \int \!\cdots \! \int \matrixel{x_N}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_{N-1}} \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_0} \dd{x_1} \cdots \dd{x_{N - 1}} \end{aligned}$$

For sufficiently small time steps $\Delta{t}$ (i.e. large $N$), we can split the Hamiltonian into its kinetic and potential terms $\hat{H} = \hat{T} + \hat{V}$. Note that this is an approximation, since $\hat{T}$ and $\hat{V}$ are operators that do not commute, but it becomes exact in the limit $\Delta{t} \to 0$:

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

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

$$\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_j} &= \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta{t} / \hbar} \: e^{- i \hat{V} \Delta{t} / \hbar}}{x_j} \\ &= e^{- i V(x_j) \Delta{t} / \hbar} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta{t} / \hbar}}{x_j} \end{aligned}$$

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

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

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

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

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

$$\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} &= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty \exp\!\bigg( \!-\! i \frac{\Delta{t}}{2 m \hbar} p^2 + i \frac{(x_{j+1} \!-\! x_j)}{\hbar} p \bigg) \dd{p} \\ &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta{t}}} \exp\!\bigg( i \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta{t}} \bigg) \end{aligned}$$

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

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

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

$$\begin{aligned} K &= \bigg( \frac{- i m}{2 \pi \hbar \Delta{t}} \bigg)^{\!N / 2} \!\int\!\cdots\!\int \prod_{j = 0}^{N - 1} \exp\!\bigg( \frac{i}{\hbar} \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}} - \frac{i}{\hbar} V(x_j) \: \Delta{t} \bigg) \dd{x_1} \cdots \dd{x_{N-1}} \\ &= \Big( \frac{- i m}{2 \pi \hbar \Delta{t}} \Big)^{\!N / 2} \!\int\!\cdots\!\int \exp\!\bigg( \frac{i \Delta{t}}{\hbar} \sum_{j = 0}^{N-1} \Big( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \Big) \bigg) \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}$$

It is worth noting that there are $N\!-\!1$ integrals, but $N$ factors $(-i m / 2 \pi \hbar \Delta{t})^{1/2}$ i.e. one for each slice. According to convention, $N\!-\!1$ 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 $\Delta{t} \to 0$ (or $N \to \infty$), the sum in the exponent becomes an integral:

$$\begin{aligned} \lim_{\Delta{t} \to 0} \sum_{j = 0}^{N - 1} \bigg( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \bigg) \Delta{t} \:\:&=\:\: \int_{t_0}^{t_N} \!\bigg( \frac{1}{2} m \dot{x}^2 - V(x) \bigg) \dd{\tau} \\ \:\:&=\:\: \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} \\ \:\:&=\:\: S[x] \end{aligned}$$

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

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

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:

$$\begin{aligned} \boxed{ \int \mathcal{D}{x} \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \dd{x_1} \cdots \dd{x_{N-1}} } \end{aligned}$$

It is worth stressing that this is simply an abbreviation; in practice, calculating $K$ 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.

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