From b8f17e01d64b15935053c25e94d816ca01859152 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 20 Oct 2024 16:25:03 +0200 Subject: Improve knowledge base --- .../concept/path-integral-formulation/index.md | 239 +++++++++++++-------- 1 file changed, 147 insertions(+), 92 deletions(-) (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 index a8dcc76..657ff17 100644 --- a/source/know/concept/path-integral-formulation/index.md +++ b/source/know/concept/path-integral-formulation/index.md @@ -8,170 +8,225 @@ categories: layout: "concept" --- -In quantum mechanics, the **path integral formulation** -is an alternative description of quantum mechanics, -which is equivalent to the "traditional" Schrödinger equation. +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](/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 $$t_N$$: +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 $$(x_N, t_N)$$: $$\begin{aligned} - \boxed{ - K(x_N, t_N; x_0, t_0) - = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) - } + K(x_N, t_N; x_0, t_0) + = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) \end{aligned}$$ -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. +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](/know/concept/euler-lagrange-equation/) +of classical Lagrangian mechanics. +Let $$\dot{x}(t) = \idv{x}{t}$$: -This apparent problem solves itself, -thanks to the fact that paths close to the classical optimum $$x_c(t)$$ +$$\begin{aligned} + S[x] + \equiv \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} +\end{aligned}$$ + +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 $$S_c = S[x_c]$$, -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. +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. + -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: +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 $$\Delta{t} \equiv (t_N \!-\! t_0) / N$$: $$\begin{aligned} K(x_N, t_N; x_0, t_0) &= \matrixel{x_N}{e^{- i \hat{H} (t_N - t_0) / \hbar}}{x_0} - = \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar} \cdots e^{- i \hat{H} \Delta t / \hbar}}{x_0} + \\ + &= \matrixel{x_N}{e^{- i \hat{H} \Delta{t} / \hbar} \cdots e^{- i \hat{H} \Delta{t} / \hbar}}{x_0} \end{aligned}$$ -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 $$x(t)$$: +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: $$\begin{aligned} K - &= \int\cdots\int \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar}}{x_{N-1}} \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta t / \hbar}}{x_0} + &= K(x_N, t_N; x_{N-1}, t_{N-1}) + \cdots K(x_2, t_2; x_1, t_1) \: K(x_1, t_1; x_0, t_0) + \\ + &= \int \!\cdots \! \int + \matrixel{x_N}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_{N-1}} + \cdots \matrixel{x_1}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_0} \dd{x_1} \cdots \dd{x_{N - 1}} \end{aligned}$$ -For sufficiently small time steps $$\Delta t$$ (i.e. large $$N$$ -we make the following approximation -(which would be exact, were it not for the fact that -$$\hat{T}$$ and $$\hat{V}$$ are operators): +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 $$\Delta{t} \to 0$$: $$\begin{aligned} - e^{- i \hat{H} \Delta t / \hbar} - = e^{- i (\hat{T} + \hat{V}) \Delta t / \hbar} - \approx e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar} + e^{- i \hat{H} \Delta{t} / \hbar} + \approx e^{- i \hat{T} \Delta{t} / \hbar} \: e^{- i \hat{V} \Delta{t} / \hbar} \end{aligned}$$ -Since $$\hat{V} = V(x_j)$$, -we can take it out of the inner product as a constant factor: +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: $$\begin{aligned} - \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar}}{x_j} - = e^{- i V(x_j) \Delta t / \hbar} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} + \matrixel{x_{j+1}}{e^{- i \hat{H} \Delta{t} / \hbar}}{x_j} + &= \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta{t} / \hbar} \: e^{- i \hat{V} \Delta{t} / \hbar}}{x_j} + \\ + &= e^{- i V(x_j) \Delta{t} / \hbar} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta{t} / \hbar}}{x_j} \end{aligned}$$ -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: +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: $$\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} - = \int_{-\infty}^\infty \Inprod{x_{j+1}}{p} \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \Inprod{p}{x_j} \dd{p} + &= \int_{-\infty}^\infty \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta{t} / \hbar}}{p} \inprod{p}{x_j} \dd{p} + \\ + &= \int_{-\infty}^\infty \exp\!\bigg(\!-\! i \frac{p^2 \Delta{t}}{2 m \hbar} \bigg) \inprod{x_{j+1}}{p} \inprod{p}{x_j} \dd{p} \end{aligned}$$ In the momentum basis $$\Ket{p}$$, -the position basis vectors -are represented by plane waves: +the position basis vectors $$\Ket{x}$$ +are given by plane waves: $$\begin{aligned} - \Inprod{p}{x_j} - = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( \!-\! i \frac{x_j p}{\hbar} \Big) - \qquad - \Inprod{x_{j+1}}{p} - = \frac{1}{\sqrt{2 \pi \hbar}} \exp\!\Big( i \frac{x_{j+1} p}{\hbar} \Big) + \inprod{p}{x} + = \frac{e^{- i x p / \hbar}}{\sqrt{2 \pi \hbar}} \end{aligned}$$ -With this, we return to the inner product and further evaluate the integral: +Inserting this and looking up the resulting integral, +we arrive at: $$\begin{aligned} \matrixel{x_{j+1}}{e^{- i \hat{T} \Delta t / \hbar}}{x_j} &= \frac{1}{2 \pi \hbar} \int_{-\infty}^\infty - \exp\!\Big(\!-\! i \frac{p^2 \Delta t}{2 m \hbar}\Big) \exp\!\Big(i \frac{(x_{j+1} - x_j) p}{\hbar}\Big) \:dp + \exp\!\bigg( \!-\! i \frac{\Delta{t}}{2 m \hbar} p^2 + i \frac{(x_{j+1} \!-\! x_j)}{\hbar} p \bigg) \dd{p} \\ - &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta t}} \exp\!\Big( i \frac{m (x_{j+1} - x_j)^2}{2 \hbar \Delta t} \Big) + &= \frac{1}{2 \pi \hbar} \sqrt{\frac{2 \pi m \hbar}{i \Delta{t}}} + \exp\!\bigg( i \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta{t}} \bigg) \end{aligned}$$ -Inserting this back into the definition of the propagator $$K(x_N, t_N; x_0, t_0)$$ yields: +Including the factor due to $$\hat{V}$$, +we find that the propagator of a single time slice is: $$\begin{aligned} - K - = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} - \int\cdots\int - \exp\!\bigg(\! \sum_{j = 0}^{N - 1} i \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \hbar \Delta t} - \frac{V(x_j) \Delta t}{\hbar} \Big) \!\bigg) - \dd{x_1} \cdots \dd{x_{N-1}} + \matrixel{x_{j+1}}{e^{- i \hat{H} \Delta t / \hbar}}{x_j} + = \sqrt{\frac{- i m}{2 \pi \hbar \Delta{t}}} + \exp\!\bigg( \frac{i}{\hbar} \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}} - \frac{i}{\hbar} V(x_j) \: \Delta{t} \bigg) \end{aligned}$$ -For large $$N$$ and small $$\Delta t$$, the sum in the exponent becomes an integral: +This is a "local" result; +inserting it into the "global" propagator $$K(x_N, t_N; x_0, t_0)$$ yields: $$\begin{aligned} - \frac{i}{\hbar} \sum_{j = 0}^{N - 1} \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \Delta t^2} - V(x_j) \Big) \Delta t - \quad \to \quad - \frac{i}{\hbar} \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} + K + &= \bigg( \frac{- i m}{2 \pi \hbar \Delta{t}} \bigg)^{\!N / 2} + \!\int\!\cdots\!\int \prod_{j = 0}^{N - 1} + \exp\!\bigg( \frac{i}{\hbar} \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}} - \frac{i}{\hbar} V(x_j) \: \Delta{t} \bigg) + \dd{x_1} \cdots \dd{x_{N-1}} + \\ + &= \Big( \frac{- i m}{2 \pi \hbar \Delta{t}} \Big)^{\!N / 2} + \!\int\!\cdots\!\int + \exp\!\bigg( \frac{i \Delta{t}}{\hbar} \sum_{j = 0}^{N-1} + \Big( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \Big) \bigg) + \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}$$ -Upon closer inspection, this integral turns out to be the classical action $$S[x]$$, -with the integrand being the Lagrangian $$L$$: - -$$\begin{aligned} - S[x(t)] - = \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} - = \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} -\end{aligned}$$ +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. -The definition of the propagator $$K$$ is then further reduced to the following: +In the limit $$\Delta{t} \to 0$$ (or $$N \to \infty$$), +the sum in the exponent becomes an integral: $$\begin{aligned} - K - = \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} - \int\cdots\int \exp(i S[x] / \hbar) \dd{x_1} \cdots \dd{x_{N-1}} + \lim_{\Delta{t} \to 0} + \sum_{j = 0}^{N - 1} \bigg( \frac{m}{2} \frac{(x_{j+1} \!-\! x_j)^2}{\Delta{t}^2} - V(x_j) \bigg) \Delta{t} + \:\:&=\:\: + \int_{t_0}^{t_N} \!\bigg( \frac{1}{2} m \dot{x}^2 - V(x) \bigg) \dd{\tau} + \\ + \:\:&=\:\: + \int_{t_0}^{t_N} L(x, \dot{x}, \tau) \dd{\tau} + \\ + \:\:&=\:\: + S[x] \end{aligned}$$ -Finally, for the purpose of normalization, -we define the integral over all paths $$x(t)$$ as follows, -where we write $$D[x]$$ instead of $$\dd{x}$$: +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**: $$\begin{aligned} - \int D[x] - \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \dd{x_1} \cdots \dd{x_{N-1}} + \boxed{ + K + = \int e^{i S[x] / \hbar} \:\mathcal{D}{x} + } \end{aligned}$$ -We thus arrive at **Feynman's path integral**, -which sums over all possible paths $$x(t)$$: +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: $$\begin{aligned} - K - = \int \exp(i S[x] / \hbar) \:D[x] - = A \sum_{\mathrm{all}\:x(t)} \exp(i S[x] / \hbar) + \boxed{ + \int \mathcal{D}{x} + \equiv \lim_{N \to \infty} \Big( \frac{- i m}{2 \pi \hbar \Delta t} \Big)^{\!N / 2} \int\cdots\int \dd{x_1} \cdots \dd{x_{N-1}} + } \end{aligned}$$ +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 -- cgit v1.2.3