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Diffstat (limited to 'source/know/concept/path-integral-formulation')
| -rw-r--r-- | source/know/concept/path-integral-formulation/index.md | 68 |
1 files changed, 34 insertions, 34 deletions
diff --git a/source/know/concept/path-integral-formulation/index.md b/source/know/concept/path-integral-formulation/index.md index b92cf5f..a8dcc76 100644 --- a/source/know/concept/path-integral-formulation/index.md +++ b/source/know/concept/path-integral-formulation/index.md @@ -14,10 +14,10 @@ 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 $t_N$: +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 $$t_N$$: $$\begin{aligned} \boxed{ @@ -26,32 +26,32 @@ $$\begin{aligned} } \end{aligned}$$ -Where $A$ normalizes. -$S[x]$ is the classical action of the path $x$, whose minimization yields +Where $$A$$ normalizes. +$$S[x]$$ is the classical action of the path $$x$$, whose minimization yields the Euler-Lagrange equation from Lagrangian mechanics. Note that each path is given an equal weight, even unrealistic paths that make big detours. This apparent problem solves itself, -thanks to the fact that paths close to the classical optimum $x_c(t)$ -have an action close to $S_c = S[x_c]$, +thanks to the fact that paths close to the classical optimum $$x_c(t)$$ +have an action close to $$S_c = S[x_c]$$, while the paths far away have very different actions. -Since $S[x]$ is inside a complex exponential, -this means that paths close to $x_c$ add contructively, +Since $$S[x]$$ is inside a complex exponential, +this means that paths close to $$x_c$$ add contructively, and the others add destructively and cancel out. -An interesting way too look at it is by varying $\hbar$: +An interesting way too look at it is by varying $$\hbar$$: as its value decreases, minor action differences yield big phase differences, -which make the quantum wave function stay closer to $x_c$. -In the limit $\hbar \to 0$, quantum mechanics thus turns into classical mechanics. +which make the quantum wave function stay closer to $$x_c$$. +In the limit $$\hbar \to 0$$, quantum mechanics thus turns into classical mechanics. ## Time-slicing derivation The most popular way to derive the path integral formulation proceeds as follows: -starting from the definition of the propagator $K$, -we divide the time interval $t_N - t_0$ into $N$ "slices" -of equal width $\Delta t = (t_N - t_0) / N$, -where $N$ is large: +starting from the definition of the propagator $$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) @@ -59,9 +59,9 @@ $$\begin{aligned} = \matrixel{x_N}{e^{- i \hat{H} \Delta t / \hbar} \cdots e^{- i \hat{H} \Delta t / \hbar}}{x_0} \end{aligned}$$ -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 $$N\!-\!1$$ identity operators +$$\hat{I} = \int \Ket{x} \Bra{x} \dd{x}$$, +and define $$x_j = x(t_j)$$ for an arbitrary path $$x(t)$$: $$\begin{aligned} K @@ -69,10 +69,10 @@ $$\begin{aligned} \dd{x_1} \cdots \dd{x_{N - 1}} \end{aligned}$$ -For sufficiently small time steps $\Delta t$ (i.e. large $N$ +For sufficiently small time steps $$\Delta t$$ (i.e. large $$N$$ we make the following approximation (which would be exact, were it not for the fact that -$\hat{T}$ and $\hat{V}$ are operators): +$$\hat{T}$$ and $$\hat{V}$$ are operators): $$\begin{aligned} e^{- i \hat{H} \Delta t / \hbar} @@ -80,7 +80,7 @@ $$\begin{aligned} \approx e^{- i \hat{T} \Delta t / \hbar} e^{- i \hat{V} \Delta t / \hbar} \end{aligned}$$ -Since $\hat{V} = V(x_j)$, +Since $$\hat{V} = V(x_j)$$, we can take it out of the inner product as a constant factor: $$\begin{aligned} @@ -89,15 +89,15 @@ $$\begin{aligned} \end{aligned}$$ Here we insert the identity operator -expanded in the momentum basis $\hat{I} = \int \Ket{p} \Bra{p} \dd{p}$, -and commute it with the kinetic energy $\hat{T} = \hat{p}^2 / (2m)$ to get: +expanded in the momentum basis $$\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}$$ -In the momentum basis $\Ket{p}$, +In the momentum basis $$\Ket{p}$$, the position basis vectors are represented by plane waves: @@ -119,7 +119,7 @@ $$\begin{aligned} &= \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}$$ -Inserting this back into the definition of the propagator $K(x_N, t_N; x_0, t_0)$ yields: +Inserting this back into the definition of the propagator $$K(x_N, t_N; x_0, t_0)$$ yields: $$\begin{aligned} K @@ -129,7 +129,7 @@ $$\begin{aligned} \dd{x_1} \cdots \dd{x_{N-1}} \end{aligned}$$ -For large $N$ and small $\Delta t$, the sum in the exponent becomes an integral: +For large $$N$$ and small $$\Delta t$$, the sum in the exponent becomes an integral: $$\begin{aligned} \frac{i}{\hbar} \sum_{j = 0}^{N - 1} \Big( \frac{m (x_{j+1} \!-\! x_j)^2}{2 \Delta t^2} - V(x_j) \Big) \Delta t @@ -137,8 +137,8 @@ $$\begin{aligned} \frac{i}{\hbar} \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} \end{aligned}$$ -Upon closer inspection, this integral turns out to be the classical action $S[x]$, -with the integrand being the Lagrangian $L$: +Upon closer inspection, this integral turns out to be the classical action $$S[x]$$, +with the integrand being the Lagrangian $$L$$: $$\begin{aligned} S[x(t)] @@ -146,7 +146,7 @@ $$\begin{aligned} = \int_{t_0}^{t_N} \Big( \frac{1}{2} m \dot{x}^2 - V(x) \Big) \dd{\tau} \end{aligned}$$ -The definition of the propagator $K$ is then further reduced to the following: +The definition of the propagator $$K$$ is then further reduced to the following: $$\begin{aligned} K @@ -155,8 +155,8 @@ $$\begin{aligned} \end{aligned}$$ Finally, for the purpose of normalization, -we define the integral over all paths $x(t)$ as follows, -where we write $D[x]$ instead of $\dd{x}$: +we define the integral over all paths $$x(t)$$ as follows, +where we write $$D[x]$$ instead of $$\dd{x}$$: $$\begin{aligned} \int D[x] @@ -164,7 +164,7 @@ $$\begin{aligned} \end{aligned}$$ We thus arrive at **Feynman's path integral**, -which sums over all possible paths $x(t)$: +which sums over all possible paths $$x(t)$$: $$\begin{aligned} K |