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}...
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...
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?
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...
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...
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)\\
$$
Taking complex conjugate on both sides,
$$...
[##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...
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.
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.
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...
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##...
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...
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...