Hey everybody,
Background:
I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric.
The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0):
$$ \partial_t\phi =...
I am interested in the derivation of Schrödinger’s wave equation from the Klein Gordon equation. I have looked in Penfold’s ‘The Road to Reality’, the open University’s Quantum Mechanics books, Feynman’s lectures, the internet, but not found what I want. Everyone seems to take it as a given...
1. Homework Statement
My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG
2. Homework Equations
v_g = dw/dk
3. The Attempt at a Solution
Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I...
I start by outlining the little I know about the basics of quantum field theory.
The simplest relativistic field theory is described by the Klein-Gordon equation of motion for a scalar field ##\large \phi(\vec{x},t)##:
$$\large \frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi+m^2\phi=0.$$
We...
1. Homework Statement
I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., ##\hat{\phi}(x,t)|_{boundary} = 0 ##). 1-D functions that obey the Dirichlet condition on interval ##[0,L]## are of the form below (using the discrete Fourier sine transform)
$$f(x) =...
Hi everyone,
I've been reading about the Klein Gordon equation with the Coulomb Potential. The full solution can be found here:
http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation#Klein-Gordon_equation_with_Coulomb_potential
I'm confused near the beginning of this. I understand...
If we have the normal KG scalar field expansion:
$$ \hat{\phi}(x^{\mu}) = \int \frac{d^{3}p}{(2\pi)^{3}\omega(\mathbf{p})} \big( \hat{a}(p)e^{-ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}(p)e^{ip_{\mu}x^{\mu}} \big) $$
With ## \omega(\mathbf{p}) = \sqrt{|\mathbf{p}^{2}|+m^{2}}##
Then why do we associate...
So, I am a newbie in quantum mechanics, took modern physics last fall for my physics minor.
I know that Schrodinger based his equation based on the equation K + V = E,
by using non-relativistic kinematic energy (P2/2m + V = E)
p becoming the operator p= -iħ∇ for the wave equation eigenfunction...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave equation \Box \ \phi = 0. The only problem is that I'm not being able to get the same solution as the book. In the book the author states that...
Consider the double-slit experiment done with photons from a laser. If one was interested only in computing position (vertical) probability amplitudes and did not care about spin/helicity, could the Klein-Gordon Equation (with mass set to zero) be used?
Thanks in advance.
1. Homework Statement
Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##.
2. Homework Equations
##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##
3. The Attempt at...
I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...
1. Homework Statement
Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then
##[H,F]-i \partial_0 F##
2. Homework Equations
For KG we have...
Hi All,
I've heard it said that the superluminal phase velocity of the KG eqn is not a problem for relativistic causality because signals travel at the packet/group velocity, which is the inverse of the phase velocity (c being 1). I'm a bit skeptical of this.
We can strip away all the quantum...
I've been working through a qft book by Sadovskii (while I wait for my Peskin book to come in) and I've used some later chapters of Griffith's Into to Elementary Particles as an introduction to some qft. My issue with both of these is that, where in classical mechanics we have the Lagrangian...