Show that the Green's function for the time-independent Klein-Gordon equation $$ \left(\nabla^{2}-m^{2}\right) \phi=\rho(r) $$ can be expressed as the Fourier integral $$ G\left(x-x^{\prime}\right)=-\frac{1}{(2 \pi)^{3}} \iiint d^{3} k \frac{e^{u k\left(x-y^{\prime}\right)}}{k^{2}+m^{2}} $$ Evaluate this integral and show that it results in $$ G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \quad \text { where } \quad \mathbf{R}=\mathbf{x}-\mathbf{x}^{\prime}, \quad R=|\mathbf{R}| $$ Find the solution \(\phi\) correspondmg to a point source $$ \rho(\mathbf{r})=q \delta^{3}(r) $$

Short Answer

Expert verified
The Fourier integral is \( G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \). The Green's function is calculated as \( G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \). Finally, the solution for \(\phi\) is: \( \phi(r) = \frac{q}{4\pi r} e^{-mr} \)

Step by step solution

01

- Showing that the Green's function for Klein-Gordon equation can be represented as the Fourier integral

We know that \( (\nabla^{2}-m^{2}) \phi=\rho(r) \), we substitute \( \phi = \iiint d^{3} k \frac{e^{ik(x-x')}}{k^{2}+m^{2}} \) into Klein-Gordon equation we get \( \rho(r) \). This process shows that the Green's function can indeed be represented as a Fourier integral.
02

- Calculate the Fourier Integral

Now we compute the Fourier integral \( G(x-x')=-\frac{1}{(2 \pi)^{3}} \iiint d^{3} k \frac{e^{ik(x-x')}}{k^{2}+m^{2}} \). This simplifies to \( G(\mathbf{R})=-\frac{\mathrm{e}^{-\operatorname{m} k}}{4 \pi R} \) where \( \mathbf{R}=\mathbf{x}-\mathbf{x}' \) and \( R=|\mathbf{R}| \).
03

- Solve for Phi

We finally solve for Phi using the given point source \( \rho(\mathbf{r})=q \delta^{3}(r) \). Substituting this into our equation gives us \( \phi(r) = \frac{q}{4\pi r} e^{-mr} \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the Green ' function for the one-dimensional diffusion equation, $$ \left.\frac{\partial^{2} G(x, t)}{\partial x^{2}}-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\varepsilon\left(x-x^{\prime}\right)\right\\}\left(t-t^{\prime}\right) $$ is given by $$ G\left(x-x^{\prime}, t-t^{\prime}\right)=-\theta\left(t-t^{\prime}\right) \sqrt{\frac{K}{4 \pi\left(t-t^{\prime}\right)}} e^{-\left(x-r^{\prime}\right)^{2} / 4(t-h)} $$ and write out the corresponding solution of the inhomogencous equation $$ \frac{\partial^{2} \psi(x, t)}{\partial x^{2}}-\frac{1}{x^{\prime}} \frac{\partial \psi(x, t)}{\partial t}=F(x, t) $$ Do the same for the two- and thrce-dimensional diffusion equations $$ \nabla^{2} G(x, t)-\frac{1}{\kappa} \frac{\partial G(x, t)}{\partial t}=\delta^{n}\left(x-x^{\prime}\right) \delta\left(t-t^{\prime}\right) \quad(n=2,3) $$

Evaluate Fourier transforms of the following distributional functions: (a) \(\delta(x-a)\). (b) \(\delta^{\prime}(x-a)\). (c) \(\delta^{(n)}(x-a)\). (d) \(\delta\left(x^{2}-a^{2}\right)\) (c) \(\delta^{\prime}\left(x^{2}-a^{2}\right)\)

We say a scquence of distributions \(T_{n}\) converges to a distribution \(T\), written \(T_{n} \rightarrow T\). if \(T_{n}(\phi) \rightarrow T(\phi)\) for all test functions \(\phi \in \mathcal{D}\) (this is sometimes called weak convergence). If a scquence of continuous functions \(f_{n}\) converges uniformly to a function \(f(x)\) on every compact subsct of \(\mathbb{R}\), show that the associated regular distributions \(T_{f_{n}} \rightarrow T_{f-}\) In the distributional sense, show that we have the following convergences. $$ \begin{aligned} f_{n}(x) &=\frac{n}{\pi\left(1+n^{2} x^{2}\right)} \rightarrow \delta(x) \\ g_{n}(x) &=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-\alpha^{2} x^{2}} \rightarrow \delta(x) \end{aligned} $$

Show that for a monotone function \(f(x)\) such that \(f(\pm \infty)=\pm \infty\) with \(f(a)=0\) $$ \int_{-\infty}^{\infty} \varphi(x) \delta^{\prime}(f(x)) \mathrm{d} x=-\left.\frac{1}{f^{\prime}(x)} \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\varphi(x)}{\left|f^{\prime}(x)\right|}\right)\right|_{x=6} $$ For a general function \(f(x)\) that is monotone on a neighbourhood of all its zeros, find a general formula for the distribution \(\delta^{\prime} \circ f .\)

Find the Fourier transforms of the functions $$ f(x)= \begin{cases}1 & \text { if }-a \leq x \leq a \\ 0 & \text { otherwise }\end{cases} $$ and $$ g(x)= \begin{cases}1-\frac{\mid x}{2} & \text { if }-a \leq x \leq a \\ 0 & \text { otherwisc }\end{cases} $$

See all solutions

Recommended explanations on Combined Science Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free