Recent content by acegikmoqsuwy

Never mind, I figured it out. The expansion of an eigenstate of position should instead be $$|\mathbf r\rangle = \sum\limits_{\mathbf k} e^{-i\mathbf k\cdot \mathbf r} |\mathbf k\rangle.$$

Hi, It appears that the definition of a quantum field creation operator is given by $$\Psi^{\dagger}(\mathbf r) = \sum\limits_{\mathbf k} e^{-i\mathbf k\cdot \mathbf r} a^{\dagger}_{\mathbf k}.$$ But then if we examine how this operator acts on the vacuum state, we get $$\Psi^{\dagger}(\mathbf...

Yes, this is purely classical mechanics. I posted in here because I was going to have some follow up questions regarding relativity, but I've resolved them now. Thanks.

Hi, If I have a massive particle constrained to the surface of a Riemannian manifold (the metric tensor is positive definite) with kinetic energy $$T=\dfrac 12mg_{\mu\nu} \dfrac{\text dx^{\mu}}{\text dt} \dfrac{\text dx^{\nu}}{\text dt}$$ then I believe I should be able to derive the geodesic...

Hi, I was reading a book about second quantization and there were a few things that I didn't quite understand entirely. This is what I understood so far: Given an operator ##\mathcal A## and two orthonormal bases ##|\alpha_i\rangle## and ##|\beta_i\rangle## for the Hilbert space, ##\mathcal...

Hi, For a particle in a box (so that the momentum spectrum is discrete), we can write the identity operator as a sum over all momentum eigenstates of a projection to that eigenstate: $$I=\displaystyle\sum\limits_{p} |p\rangle\langle p|.$$ I was wondering what the corresponding form of the...

OMG. Whoops. Haha, thank you, I spent the last four hours trying to figure this thing out!

Hi, I just started a book on QFT and one of the first things that was done was switch from labeling states with their individual particles and instead label states by the number of particles in each momentum eigenstate. In addition, some "algebras" (not sure if they qualify by the mathematical...

I agree one hundred percent. I am currently a high schooler and AoPS is my regular go-to for anything to do with math, whether it be studying for competitions, learning higher level math, or perhaps if I'm just bored and want someone to talk to about math. The part which really caught my eye the...

Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula \dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0 for the geodesic equation, with the metric...

Hello all. If I was to create a circuit with a 2 kV power supply, how would I reduce the current so that if I were to accidentally become part of the circuit, I would be in no danger? Another question: Does a bleeder resistor help? Thanks.

My immediate guesses: (1) Max is 10 and min is 19/10 (2) Max is 1000 and min is 1999/28

What a double integral generally represents is the volume of a solid above a region on a plane. The solid you have here is called a cylindrical surface. The equation of it is z=y^2. This is a parabola. However, since there is no x, we can let x vary from negative infinity to infinity. Doing so...

Hi, I have a very basic question about the Lagrangian that I can't seem to understand: why is it dependent on both the position function and the time derivative? I know that it is the difference between the kinetic and potential energy, but why? Is there a derivation of this, is it a definition...

Oops the domain is x^2+y^2\le 4