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Consider, two fields interact with each other and the interaction term of the action is given. Now the Lagrangian density is Fourier transformed and the interaction term of the action is expressed as an integral over the momentum space. How is the integrand related to the form factor?

Consider a Lagrangian: \begin{equation} \mathcal{L} = \mathcal{L}(q_1\, \dots\, q_n, \dot{q}_1\, \dots\, \dot{q}_n,t) \end{equation} From this Lagrangian, we get a set of ##n## equations: \begin{equation} \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}_i} - \frac{\partial...

This will transform the PDE into a wave equation. But this exercise asks to solve this problem not using this coordinate transformation. Thanks for your suggestion anyway.

Homework Statement Find out the Green's function, ##G(\vec{r}, \vec{r}')##, for the following partial differential equation: $$\left(-2\frac{\partial ^2}{\partial t \partial x} + \frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \right) F(\vec{r}) = g(\vec{r})$$ Here ##\vec{r}...

I was looking for a rigorous derivation.

Homework Statement Suppose two successive coordinate rotations through angles ##\Phi_1## and ##\Phi_2## are carried out, equivalent to a single rotation through an angle ##\Phi##. Show that ##\Phi_1##, ##\Phi_2## and ##\Phi## can be considered as the sides of a spherical triangle with the angle...

If ##\frac{d\theta}{dt} = 0## and ##\omega## is nonzero, will the piston move? What do you think? Drawing diagrams may help.

Photon has spin 1 and Higgs boson has spin 0. (Source: Wikipedia) You may find this thread on spin 0 particle helpful.

Why didn't you consider ##\omega## in your solution?

You forgot to upload the figure.

Let $$\Psi(x,t) = A(t) \psi(x)$$ Applying Schrodinger's Time dependent equation: $$\begin{equation} i\hbar\frac{\partial}{\partial t}\left(A(t)\psi(x)\right) = H\left(A(t)\psi(x)\right) \end{equation}$$ Let ##\psi(x)## is an eigenfunction of ##H## with eigenvalue ##E##. So, we get...

Isn't the Hamiltonian Operator in the Schrodinger's time dependent equation is the Hamiltonian operator defined for the particular system we are considering?

How do we experimentally apply the operator ## \exp{\left(-i\phi\frac{ S_z}{\hbar}\right)}## on a quantum mechanical system? (Here ##S_z## is the spin angular momentum operator along the z-axis) For example, on a beam of electrons?

Yes. I just wanted to show that the energy eigenkets are also eigenkets to the operator ##i\hbar \frac{\partial}{\partial t}##.

Can't ##H:=-\frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + V(x) ## act on ##\Psi (x,t)## as well?

What is the four-vector related to electric and magnetic dipole moment?

Suppose, the energy of a particle is measured, say ##E_1##. So now the state vector of the particle is the energy eigenket ##|E_1>##. Then the position of the particle is measured, say ##x##. As the Hamiltonian operator and the position operator are non-commutative, the state vector is changed...

I thought writing more equations means spending more time on physics.

How many equations does a physicist write in his/her lifetime on average? Is there any approximate statistics on this? Also how much is this correlated to his/her contribution to physics?

Consider the particle in a box problem. The number of energy eigenbasis is 'countable' infinity. But the number of position eigenbasis is 'uncountable' infinity. x can take any value from the interval [0,L] Whichever basis I choose, shouldn't the dimensionality of the vector space be the same?

Look, functions like ##f(z)=z^2##, gives you the same value for a particular ##z##, no matter you write ##z=||z||e^{i\theta}## or ##z=||z||e^{i(\theta+2\pi)}##. The problem arises when you deal with functions like ##g(z) = log (z)## or ##g(z) = z^{1/2}##. In those cases, you notice...

Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...

Thanks for your help. But still could not get it.

Why ##\mu_1, \mu_2## must be the same as ##\mu_1^*, \mu_2^*## ? What I thought is : If ##\mu_1\mu_2 = \mu_1^*\mu_2^*## and ##\mu_1+\mu_2 = \mu_1^*+\mu_2^*##, then ##\mu_1, \mu_2## are the same as ##\mu_1^*, \mu_2^*## It can be shown by taking the complex conjugate of (27.5) that $$\mu_1\mu_2 =...

I did not understand why (3) is the Fourier transform of -k instead of k. Look, I used the defination, $$ \int_{-\infty}^{\infty}f(x)e^{-ikx}\,dx=\widetilde{f}(k)\\ $$ Taking complex conjugate on both sides, $$...

I used the defination, $$\int_{-\infty}^{\infty}f(x)e^{-ikx}\,dx=\widetilde{f}(k)$$

[##f^*## represents complex conjugate of ##f##. ] [##\widetilde{f}(k)## represents Fourier transform of the function ##f(x)##.] $$\begin{align} \int_{-\infty}^{\infty}f^*(x)e^{ikx}\,dx&=\int_{-\infty}^{\infty}f^*(x)\left(e^{-ikx}\right)^*\,dx\\...

Using Fourier transformation, we have, Comparing with the equation, $$f(t)=\int_{-\infty}^{\infty}\delta(t-u)f(u)\,du$$ we have, Thus, $$\begin{align} \delta(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega t}\,d\omega\\ &=\frac{1}{2\pi}\lim_{\Omega\rightarrow...

We know, $$\delta(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}$$ And, also, $$\int_{-\infty}^{\infty}\delta(x)\,dx=1$$ Using Fourier Transformation, it can be shown that, $$\delta(x)=\lim_{\Omega \rightarrow \infty}\frac{\sin{(\Omega x)}}{\pi x}$$ Let's...

If complex numbers are allowed, then there always exists a transformation matrix between two metric tensors. So I guess complex numbers are not allowed. Could you explain me why?

Homework Statement Can the following function be represented by a Fourier series over the range indicated: $$f(x) = \cos^{-1}(\sin {2x}),~~~~-\infty<x<\infty$$ Homework Equations The Dirichlet conditions that a function must satisfy before it can be represented by a Fourier series are: (i) the...

The components of transformation matrix can be complex number, can't they? Here, I am looking for the transformation matrix between two coordinate systems while the components of metric tensor are given for the two coordinate systems.

Is that true in general and why: $$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$

Complex numbers are not allowed? Why?

So, there are ##\frac{N+N^2}{2}## equations and ##N^2## unknowns (the ##N^2## components of ##\frac{\partial \chi^i}{\partial x^j}##). As ##N^2>\frac{N+N^2}{2}##, there will be infinite number of solutions.

Aren't there ##N^2## unknowns as the transformation matrix ##\frac{\partial \chi^i}{\partial x^j}## has ##N^2## components in ##N##-dimension?

Sorry for the mistake. Is it necessary to know the transformation matrix? The metric tensor defines the geometry. Isn't it sufficient just to know the components of the metric tensor? Moreover, if I know the components of metric tensor in the two different coordinate systems, I can actually...

Is this method correct?

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...

$$\frac{1}{z}\left(\frac{1}{z-2} - \frac{1}{z-1}\right)= \frac{1}{z}\left(\frac{1}{-2}\cdot\frac{1}{1-\frac{z}{2}} - \frac{1}{z}\cdot\frac{1}{1-\frac{1}{z}}\right)$$ For ##1<|z|<2##, we have, ##\left|\frac{1}{z}\right|<1## and ##\left|\frac{z}{2}\right|<1##. So you can now use the formula for...

Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$ Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...

What are your favorite Physics jokes? One of my favorite is : Studying Entropy is a TdS PdV. My friend, Fahim Bin Selim, first told me this joke.

First, consider this problem : How many words of length ##n## contains ##2k## ##0##s? There are ##^nC_{2k}## ways to choose the ##2k## places for ##2k## ##0##s. After setting ##0##s, we have ##n-2k## places to be filled up by ##1##s and ##2##s. We can use as many ##1##s and ##2##s as we like...

Yes, you're correct.

If you know how to set mirror charges for grounded conducting plane and sphere, this picture is enough hint for you.

What's the most beautiful theory in Physics or Math you have ever read? For me, they are the Prime Number Theorem and the Theory of Relativity.

Substitute ##x=\tan{y}##.

In that case, you can break the resistance into several rectangular shaped resistances and then integrate.

The cross-sectional area is to be perpendicular to the direction of current and the length is to be measured in the direction parallel to current flow. So, in the second case, the cross-sectional area will be ##L\cdot W## and the length will be ##H##. $$R=\frac{\rho\cdot...