From b8f17e01d64b15935053c25e94d816ca01859152 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 20 Oct 2024 16:25:03 +0200 Subject: Improve knowledge base --- source/know/concept/interaction-picture/index.md | 46 ++-- .../concept/path-integral-formulation/index.md | 239 +++++++++++++-------- source/know/concept/propagator/index.md | 79 ++++--- 3 files changed, 220 insertions(+), 144 deletions(-) (limited to 'source') diff --git a/source/know/concept/interaction-picture/index.md b/source/know/concept/interaction-picture/index.md index 8428bf3..a3bb260 100644 --- a/source/know/concept/interaction-picture/index.md +++ b/source/know/concept/interaction-picture/index.md @@ -39,14 +39,14 @@ Basically, any way of splitting $$\hat{H}_S$$ is valid as long as $$\hat{H}_{0, S}$$ is time-independent, but only a few ways are useful. -We now define the unitary conversion operator $$\hat{U}(t)$$ as shown below. -Note its similarity to the [time-evolution operator](/know/concept/time-evolution-operator/) -$$\hat{K}_S(t)$$, but with the opposite sign in the exponent: +We now define the unitary conversion operator $$\hat{U}_0(t)$$ as shown below. +Note its similarity to the +[time-evolution operator](/know/concept/time-evolution-operator/) $$\hat{K}_S(t)$$: $$\begin{aligned} \boxed{ - \hat{U}(t) - \equiv \exp\!\bigg( \frac{i}{\hbar} \hat{H}_{0,S} t \bigg) + \hat{U}_0(t) + \equiv \exp\!\bigg( \!-\! \frac{i}{\hbar} \hat{H}_{0,S} t \bigg) } \end{aligned}$$ @@ -56,17 +56,17 @@ and operators $$\hat{L}_I(t)$$ are then defined as follows: $$\begin{aligned} \boxed{ \Ket{\psi_I(t)} - \equiv \hat{U}(t) \Ket{\psi_S(t)} + \equiv \hat{U}_0^\dagger(t) \Ket{\psi_S(t)} } \qquad\qquad \boxed{ \hat{L}_I(t) - \equiv \hat{U}(t) \: \hat{L}_S(t) \: \hat{U}{}^\dagger(t) + \equiv \hat{U}_0^\dagger(t) \: \hat{L}_S(t) \: \hat{U}{}_0(t) } \end{aligned}$$ Because $$\hat{H}_{0, S}$$ is time-independent, -it commutes with $$\hat{U}(t)$$, +it commutes with $$\hat{U}_0$$, so conveniently $$\hat{H}_{0, I} = \hat{H}_{0, S}$$. @@ -78,25 +78,25 @@ we differentiate it and multiply by $$i \hbar$$: $$\begin{aligned} i \hbar \dv{}{t} \Ket{\psi_I} - &= i \hbar \dv{\hat{U}}{t} \Ket{\psi_S} + \hat{U} \bigg( i \hbar \dv{}{t}\Ket{\psi_S} \bigg) + &= i \hbar \dv{\hat{U}_0^\dagger}{t} \Ket{\psi_S} + \hat{U}_0^\dagger \bigg( i \hbar \dv{}{t}\Ket{\psi_S} \bigg) \end{aligned}$$ -We insert the definition of $$\hat{U}$$ in the first term +We insert the definition of $$\hat{U}_0$$ in the first term and the Schrödinger equation into the second, -and use the fact that $$\comm{\hat{H}_{0, S}}{\hat{U}} = 0$$ +and use the fact that $$\comm{\hat{H}_{0, S}}{\hat{U}_0} = 0$$ thanks to the time-independence of $$\hat{H}_{0, S}$$: $$\begin{aligned} i \hbar \dv{}{t} \Ket{\psi_I} - &= - \hat{H}_{0,S} \hat{U} \Ket{\psi_S} + \hat{U} \hat{H}_S \Ket{\psi_S} + &= - \hat{H}_{0,S} \hat{U}_0^\dagger \Ket{\psi_S} + \hat{U}_0^\dagger \hat{H}_S \Ket{\psi_S} \\ - &= \hat{U} \big( \!-\! \hat{H}_{0,S} + \hat{H}_S \big) \Ket{\psi_S} + &= \hat{U}_0^\dagger \big( \!-\! \hat{H}_{0,S} + \hat{H}_S \big) \Ket{\psi_S} \\ - &= \hat{U} \big( \hat{H}_{1,S} \big) \hat{U}{}^\dagger \hat{U} \Ket{\psi_S} + &= \hat{U}_0^\dagger \hat{H}_{1,S} \big( \hat{U}_0 \hat{U}_0^\dagger \big) \Ket{\psi_S} \end{aligned}$$ Which leads to an analogue of the Schrödinger equation, -with $$\hat{H}_{1,I} = \hat{U} \hat{H}_{1,S} \hat{U}{}^\dagger$$: +with $$\hat{H}_{1,I} = \hat{U}_0^\dagger \hat{H}_{1,S} \hat{U}_0$$: $$\begin{aligned} \boxed{ @@ -110,11 +110,11 @@ in order to describe its evolution in time: $$\begin{aligned} \dv{\hat{L}_I}{t} - &= \dv{\hat{U}}{t} \hat{L}_S \hat{U}{}^\dagger + \hat{U} \hat{L}_S \dv{\hat{U}{}^\dagger}{t} - + \hat{U} \dv{\hat{L}_S}{t} \hat{U}{}^\dagger + &= \dv{\hat{U}_0^\dagger}{t} \hat{L}_S \hat{U}_0 + \hat{U}_0^\dagger \hat{L}_S \dv{\hat{U}_0}{t} + + \hat{U}_0^\dagger \dv{\hat{L}_S}{t} \hat{U}_0 \\ - &= \frac{i}{\hbar} \hat{U} \hat{H}_{0,S} \big( \hat{U}{}^\dagger \hat{U} \big) \hat{L}_S \hat{U}{}^\dagger - - \frac{i}{\hbar} \hat{U} \hat{L}_S \big( \hat{U}{}^\dagger \hat{U} \big) \hat{H}_{0,S} \hat{U}{}^\dagger + &= \frac{i}{\hbar} \hat{U}_0^\dagger \hat{H}_{0,S} \big( \hat{U}_0 \hat{U}_0^\dagger \big) \hat{L}_S \hat{U}_0 + - \frac{i}{\hbar} \hat{U}_0^\dagger \hat{L}_S \big( \hat{U}_0 \hat{U}_0^\dagger \big) \hat{H}_{0,S} \hat{U}_0 + \bigg( \dv{\hat{L}_S}{t} \bigg)_I \\ &= \frac{i}{\hbar} \hat{H}_{0,I} \hat{L}_I @@ -144,13 +144,15 @@ can be solved in isolation in a kind of Schrödinger picture. What about the time evolution operator $$\hat{K}_S(t)$$? Its interaction version $$\hat{K}_I(t)$$ is unsurprisingly obtained by the standard transform -$$\hat{K}_I = \hat{U} \hat{K}_S \hat{U}^\dagger$$: +$$\hat{K}_I = \hat{U}_0^\dagger \hat{K}_S \hat{U}_0$$: $$\begin{aligned} \Ket{\psi_I(t)} - &= \hat{U}(t) \Ket{\psi_S(t)} + &= \hat{U}_0^\dagger(t) \Ket{\psi_S(t)} \\ - &= \hat{U}(t) \: \hat{K}_S(t) \: \hat{U}^\dagger(t) \Ket{\psi_I(0)} + &= \hat{U}_0^\dagger(t) \: \hat{K}_S(t) \Ket{\psi_S(0)} + \\ + &= \hat{U}_0^\dagger(t) \: \hat{K}_S(t) \: \hat{U}_0(t) \: \hat{U}_0^\dagger(t) \Ket{\psi_S(0)} \\ &\equiv \hat{K}_I(t) \Ket{\psi_I(0)} \end{aligned}$$ 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 diff --git a/source/know/concept/propagator/index.md b/source/know/concept/propagator/index.md index 54e9eb6..50228e2 100644 --- a/source/know/concept/propagator/index.md +++ b/source/know/concept/propagator/index.md @@ -8,63 +8,82 @@ categories: layout: "concept" --- -In quantum mechanics, the **propagator** $$K(x_f, t_f; x_i, t_i)$$ -gives the probability amplitude that a particle -starting at $$x_i$$ at $$t_i$$ ends up at position $$x_f$$ at $$t_f$$. -It is defined as follows: +In quantum mechanics, the **propagator** $$K(x, t; x_0, t_0)$$ +gives the probability amplitude that a (spinless) particle +starting at $$(x_0, t_0)$$ ends up at $$(x, t)$$. +It is defined as: $$\begin{aligned} \boxed{ - K(x_f, t_f; x_i, t_i) - \equiv \matrixel{x_f}{\hat{U}(t_f, t_i)}{x_i} + K(x, t; x_0, t_0) + \equiv \matrixel{x}{\hat{U}(t, t_0)}{x_0} } \end{aligned}$$ -Where $$\hat{U} \equiv \exp(- i t \hat{H} / \hbar)$$ is the time-evolution operator. -The probability that a particle travels -from $$(x_i, t_i)$$ to $$(x_f, t_f)$$ is then given by: +With $$\hat{U}$$ the [time evolution operator](/know/concept/time-evolution-operator/), +given by $$\hat{U}(t, t_0) = e^{- i (t - t_0) \hat{H} / \hbar}$$ +for a time-independent $$\hat{H}$$. +Practically, $$K$$ is often calculated using +[path integrals](/know/concept/path-integral-formulation/). -$$\begin{aligned} - P - &= \big| K(x_f, t_f; x_i, t_i) \big|^2 -\end{aligned}$$ - -Given a general (i.e. non-collapsed) initial state $$\psi_i(x) \equiv \psi(x, t_i)$$, -we must integrate over $$x_i$$: +The principle here is straightforward: +evolve the initial state with $$\hat{U}$$, +and project the resulting superposition $$\ket{\psi}$$ onto the queried final state. +The probability density $$P$$ that the particle has travelled +from $$(x_0, t_0)$$ to $$(x, t)$$ is then: $$\begin{aligned} P - &= \bigg| \int_{-\infty}^\infty K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \bigg|^2 + \propto \big| K(x, t; x_0, t_0) \big|^2 \end{aligned}$$ -And if the final state $$\psi_f(x) \equiv \psi(x, t_f)$$ -is not a basis vector either, then we integrate twice: +The propagator is also useful if the particle +starts in a general superposition $$\ket{\psi(t_0)}$$, +in which case the final wavefunction $$\psi(x, t)$$ is as follows: $$\begin{aligned} - P - &= \bigg| \iint_{-\infty}^\infty \psi_f^*(x_f) \: K(x_f, t_f; x_i, t_i) \: \psi_i(x_i) \dd{x_i} \dd{x_f} \bigg|^2 + \psi(x, t) + &= \inprod{x}{\psi(t)} + \\ + &= \matrixel{x}{\hat{U}(t, t_0)}{\psi(t_0)} + \\ + &= \int_{-\infty}^\infty \bra{x} \hat{U}(t, t_0) \Big( \exprod{x_0}{x_0} \Big) \ket{\psi(t_0)} \dd{x_0} \end{aligned}$$ -Given a $$\psi_i(x)$$, the propagator can also be used -to find the full final wave function: +Where we introduced an identity operator +and recognized $$\psi(x_0, t_0) = \inprod{x_0}{\psi(t_0)}$$, so: $$\begin{aligned} \boxed{ - \psi(x_f, t_f) - = \int_{-\infty}^\infty \psi_i(x_i) K(x_f, t_f; x_i, t_i) \:dx_i + \psi(x, t) + = \int_{-\infty}^\infty K(x, t; x_0, t_0) \: \psi(x_0, t_0) \dd{x_0} } \end{aligned}$$ -Sometimes the name "propagator" is also used to refer to +The probability density of finding +the particle at $$(x, t)$$ is then +$$P \propto \big| \psi(x, t) \big|^2 $$ as usual. + +Sometimes the name *propagator* is also used to refer to the [fundamental solution](/know/concept/fundamental-solution/) $$G$$ of the time-dependent Schrödinger equation, which is related to $$K$$ by: $$\begin{aligned} - \boxed{ - G(x_f, t_f; x_i, t_i) - = - \frac{i}{\hbar} \: \Theta(t_f - t_i) \: K(x_f, t_f; x_i, t_i) - } + G(x, t; x_0, t_0) + = - \frac{i}{\hbar} \: \Theta(t - t_0) \: K(x, t; x_0, t_0) \end{aligned}$$ Where $$\Theta(t)$$ is the [Heaviside step function](/know/concept/heaviside-step-function/). +This $$G$$ is a particular example +of a [Green's function](/know/concept/greens-functions/), +but not all Green's functions are fundamental solutions +to the Schrödinger equation. +To add to the confusion, older literature tends to +call *all* fundamental solutions *Green's functions*, +even in classical contexts, + so the term has a distinct (but related) meaning +inside and outside quantum mechanics. +The result is a mess where the terms *propagator*, +*fundamental solution* and *Green's function* +are used more or less interchangeably. -- cgit v1.2.3