From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../concept/path-integral-formulation/index.md | 182 +++++++++++++++++++++ 1 file changed, 182 insertions(+) create mode 100644 source/know/concept/path-integral-formulation/index.md (limited to 'source/know/concept/path-integral-formulation') diff --git a/source/know/concept/path-integral-formulation/index.md b/source/know/concept/path-integral-formulation/index.md new file mode 100644 index 0000000..9c2e301 --- /dev/null +++ b/source/know/concept/path-integral-formulation/index.md @@ -0,0 +1,182 @@ +--- +title: "Path integral formulation" +date: 2021-07-03 +categories: +- Physics +- Quantum mechanics +layout: "concept" +--- + +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 $x(t)$: + +$$\begin{aligned} + K + &= \int\cdots\int \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar}}{x_{N-1}} \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta t / \hbar}}{x_0} + \dd{x_1} \cdots \dd{x_{N - 1}} +\end{aligned}$$ + +For sufficiently small time steps $\Delta t$ (i.e. large $N$ +we 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: + +$$\begin{aligned} + \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} + = \int_{-\infty}^\infty \Inprod{x_{j+1}}{p} \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \Inprod{p}{x_j} \dd{p} +\end{aligned}$$ + +In the momentum basis $\Ket{p}$, +the position basis vectors +are represented by plane waves: + +$$\begin{aligned} + \Inprod{p}{x_j} + = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( \!-\! i \frac{x_j p}{\hbar} \Big) + \qquad + \Inprod{x_{j+1}}{p} + = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( i \frac{x_{j+1} p}{\hbar} \Big) +\end{aligned}$$ + +With this, we return to the inner product and further evaluate the integral: + +$$\begin{aligned} + \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} + &= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty + \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \exp\!\Big(i \frac{(x_{j+1} - x_j) p}{\hbar}\Big) \:dp + \\ + &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta t}} \exp\!\Big( i \frac{m (x_{j+1} - x_j)^2}{2 \hbar \Delta t} \Big) +\end{aligned}$$ + +Inserting this back into the definition of the propagator $K(x_N, t_N; x_0, t_0)$ yields: + +$$\begin{aligned} + K + = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} + \int\cdots\int + \exp\!\bigg(\! \sum_{j = 0}^{N - 1} i \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta t} - \frac{V(x_j) \Delta t}{\hbar} \Big) \!\bigg) + \dd{x_1} \cdots \dd{x_{N-1}} +\end{aligned}$$ + +For large $N$ 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: + +$$\begin{aligned} + K + = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} + \int\cdots\int \exp(i S[x] / \hbar) \dd{x_1} \cdots \dd{x_{N-1}} +\end{aligned}$$ + +Finally, for the purpose of normalization, +we define the integral over all paths $x(t)$ as follows, +where we write $D[x]$ instead of $\dd{x}$: + +$$\begin{aligned} + \int D[x] + \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \dd{x_1} \cdots \dd{x_{N-1}} +\end{aligned}$$ + +We thus arrive at **Feynman's path integral**, +which sums over all possible paths $x(t)$: + +$$\begin{aligned} + K + = \int \exp(i S[x] / \hbar) \:D[x] + = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) +\end{aligned}$$ + + + +## References +1. R. Shankar, + *Principles of quantum mechanics*, 2nd edition, + Springer. +2. L.E. Ballentine, + *Quantum mechanics: a modern development*, 2nd edition, + World Scientific. -- cgit v1.2.3