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...
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...
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?
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 $$\mu_1\mu_2 =...
[##f^*## represents complex conjugate of ##f##. ]
[##\widetilde{f}(k)## represents fourier transform of the function ##f(x)##.]
$$\begin{align}
\int_{-\infty}^{\infty}f^*(x)e^{ikx}\,dx&=\int_{-\infty}^{\infty}f^*(x)\left(e^{-ikx}\right)^*\,dx\\...
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...
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...
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##...
Isn't the force calculated twice here? For example, the force along AB is at first calculated for the resultant force along OB, then for the resultant force along AC.
I think the compression and tensile stress should be ##\frac{F}{2a}##.
How can a permeable piston be adiabatic? If substances can go in and out of the cylinder and the substances have heat energy, heat can be exchanged through a permeable piston.
I came across this term in the book, but cannot understand.
I read in the book, "Experiments on paramagnetic materials are usually performed on samples in the form of cylinders or ellipsoids, not toroids. In these cases, the value of the magnetic field inside the material is somewhat smaller than the value of magnetic field generated by the current in...
Homework Statement
In the case of a gas obeying the equation of state
$$\begin{align}\frac{Pv}{RT}&=1+\frac{B}{v}\end{align} $$
where ##B## is a function of ##T## only, show that,
$$\begin{align}c_v&=-\frac{RT}{v}\frac{d^2}{dT^2} (BT)+\left(c_v\right)_0\end{align}$$
where ##\left(c_v\right)_0##...
We know,
$$dU=TdS-PdV$$
##\int PdV## can be calculated if the equation of state is given.
I tried to express ##S## as a function of ##P ,V## or ##T## (any two of those).
$$dS=\left(\frac{\partial S}{\partial V}\right)_T dV+\left(\frac{\partial S}{\partial T}\right)_V dT$$
$$=\left(\frac{\partial...
Homework Statement
Show that for a gas obeying the van der Waals equation ##\left(P+\frac{a}{v^2}\right)(v-b)=RT##, with ##c_v## a function of ##T## only, an equation for an adiabatic process is $$T(v-b)^{R/c_v}=constant$$
Homework Equations
##TdS=c_vdT+T\left(\frac{\partial P}{\partial...
Homework Statement
##dz=Mdx+Ndy## is an exact differential if ##(\frac{\partial M}{\partial y})_x=(\frac{\partial N}{\partial x})_y##.
By invoking the condition for an exact differential, demonstrate that the
reversible heat ##Q_R## is not a thermodynamic property.
Homework Equations...
Homework Statement
Derive the equation
##U=-T(\frac{\partial A}{\partial T})_V##
where ##U## is the internal energy, ##T## is the temperature, ##A## is the Helmholtz function.
Reference: Heat and Thermodynamics, Zemansky, Dittman, Page 272, Problem 10.4 (a)
Homework Equations
##dA=-PdV-SdT##...
Homework Statement
If ##AA^T=A^TA##, then prove that ##A## and ##A^T## have the same eigenvectors.
Homework Equations
The Attempt at a Solution
##Ax=\lambda x##
##A^TAx=\lambda A^Tx##
##AA^Tx=\lambda A^Tx##
##A(A^Tx)=\lambda (A^Tx)##
So, ##A^Tx## is also an eigenvector of ##A##.
What should...
Homework Statement
A long metal pipe of square cross-section is grounded on three sides, while the fourth (which is insulated from the rest) is maintained at constant potential ##V_0##. What is the electric field just outside the section opposite to ##V_0##?
Homework Equations
The Attempt at...