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+---
+title: "Path integral formulation"
+firstLetter: "P"
+publishDate: 2021-07-03
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-07-03T14:39:50+02:00
+draft: false
+markup: pandoc
+---
+
+# 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](/know/concept/hamiltonian-mechanics/),
+the former comes from [Lagrangian mechanics](/know/concept/lagrangian-mechanics/).
+
+It expresses the [propagator](/know/concept/propagator/) $K$
+using the following sum over all possible paths $x(t)$,
+which all go from the initial position $x_0$ at time $t_0$
+to the destination $x_N$ at time $t_N$:
+
+$$\begin{aligned}
+    \boxed{
+        K(x_N, t_N; x_0, t_0)
+        = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar)
+    }
+\end{aligned}$$
+
+Where $A$ normalizes.
+$S[x]$ is the classical action of the path $x$, 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 $x_c(t)$
+have an action close to $S_c = S[x_c]$,
+while the paths far away have very different actions.
+Since $S[x]$ is inside a complex exponential,
+this means that paths close to $x_c$ 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 $x_c$.
+In the limit $\hbar \to 0$, 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 $K$,
+we divide the time interval $t_N - t_0$ into $N$ "slices"
+of equal width $\Delta t = (t_N - t_0) / N$,
+where $N$ is large:
+
+$$\begin{aligned}
+    K(x_N, t_N; x_0, t_0)
+    &= \matrixel{x_N}{e^{- i \hat{H} (t_N - t_0) / \hbar}}{x_0}
+    = \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar} \cdots e^{- i \hat{H} \Delta t / \hbar}}{x_0}
+\end{aligned}$$
+
+Between the exponentials we insert $N\!-\!1$ identity operators
+$\hat{I} = \int \ket{x} \bra{x} \dd{x}$,
+and define $x_j = x(t_j)$ for an arbitrary path $x(t)$:
+
+$$\begin{aligned}
+    K
+    &= \idotsint \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar}}{x_{N-1}} \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta t / \hbar}}{x_0}
+    \dd{x_1} \cdots \dd{x_{N - 1}}
+\end{aligned}$$
+
+For sufficiently small time steps $\Delta t$ (i.e. large $N$
+we make the following approximation
+(which would be exact, were it not for the fact that
+$\hat{T}$ and $\hat{V}$ are operators):
+
+$$\begin{aligned}
+    e^{- i \hat{H} \Delta t / \hbar}
+    = e^{- i (\hat{T} + \hat{V}) \Delta t / \hbar}
+    \approx e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar}
+\end{aligned}$$
+
+Since $\hat{V} = V(x_j)$,
+we can take it out of the inner product as a constant factor:
+
+$$\begin{aligned}
+    \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar}}{x_j}
+    = e^{- i V(x_j) \Delta t / \hbar} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j}
+\end{aligned}$$
+
+Here we insert the identity operator
+expanded in the momentum basis $\hat{I} = \int \ket{p} \bra{p} \dd{p}$,
+and commute it with the kinetic energy $\hat{T} = \hat{p}^2 / (2m)$ to get:
+
+$$\begin{aligned}
+    \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j}
+    = \int_{-\infty}^\infty \braket{x_{j+1}}{p} \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \braket{p}{x_j} \dd{p}
+\end{aligned}$$
+
+In the momentum basis $\ket{p}$,
+the position basis vectors
+are represented by plane waves:
+
+$$\begin{aligned}
+    \braket{p}{x_j}
+    = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( \!-\! i \frac{x_j p}{\hbar} \Big)
+    \qquad
+    \braket{x_{j+1}}{p}
+    = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( i \frac{x_{j+1} p}{\hbar} \Big)
+\end{aligned}$$
+
+With this, we return to the inner product and further evaluate the integral:
+
+$$\begin{aligned}
+    \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j}
+    &= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty
+    \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \exp\!\Big(i \frac{(x_{j+1} - x_j) p}{\hbar}\Big) \:dp
+    \\
+    &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta t}} \exp\!\Big( i \frac{m (x_{j+1} - x_j)^2}{2 \hbar \Delta t} \Big)
+\end{aligned}$$
+
+Inserting this back into the definition of the propagator $K(x_N, t_N; x_0, t_0)$ yields:
+
+$$\begin{aligned}
+    K
+    = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2}
+    \idotsint
+    \exp\!\bigg(\! \sum_{j = 0}^{N - 1} i \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta t} - \frac{V(x_j) \Delta t}{\hbar} \Big) \!\bigg)
+    \dd{x_1} \cdots \dd{x_{N-1}}
+\end{aligned}$$
+
+For large $N$ and small $\Delta t$, the sum in the exponent becomes an integral:
+
+$$\begin{aligned}
+    \frac{i}{\hbar} \sum_{j = 0}^{N - 1} \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \Delta t^2} - V(x_j) \Big) \Delta t
+    \quad \to \quad
+    \frac{i}{\hbar} \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau}
+\end{aligned}$$
+
+Upon closer inspection, this integral turns out to be the classical action $S[x]$,
+with the integrand being the Lagrangian $L$:
+
+$$\begin{aligned}
+    S[x(t)]
+    = \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau}
+    = \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau}
+\end{aligned}$$
+
+The definition of the propagator $K$ is then further reduced to the following:
+
+$$\begin{aligned}
+    K
+    = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2}
+    \idotsint \exp(i S[x] / \hbar) \dd{x_1} \cdots \dd{x_{N-1}}
+\end{aligned}$$
+
+Finally, for the purpose of normalization,
+we define the integral over all paths $x(t)$ as follows,
+where we write $D[x]$ instead of $\dd{x}$:
+
+$$\begin{aligned}
+    \int D[x]
+    \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \idotsint \dd{x_1} \cdots \dd{x_{N-1}}
+\end{aligned}$$
+
+We thus arrive at **Feynman's path integral**,
+which sums over all possible paths $x(t)$:
+
+$$\begin{aligned}
+    K
+    = \int \exp(i S[x] / \hbar) \:D[x]
+    = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar)
+\end{aligned}$$
+
+
+
+## References
+1.  R. Shankar,
+    *Principles of quantum mechanics*, 2nd edition,
+    Springer.
+2.  L.E. Ballentine,
+    *Quantum mechanics: a modern development*, 2nd edition,
+    World Scientific.