Search results

  1. arpon

    I Form factors and Interaction term of the Action

    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?
  2. arpon

    I Transformation of Lagrangian

    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...
  3. arpon

    Green's function of a PDE

    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}...
  4. arpon

    Two successive rotation (Goldstein problem 4.13)

    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...
  5. arpon

    I Experiment: Spin Rotation Operator

    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?
  6. arpon

    I Four-vector related to electric and magnetic dipole moment

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

    I Does measurement change the energy of a system?

    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...
  8. arpon

    How many equations does a physicist write in his lifetime?

    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?
  9. arpon

    I Particle in a box problem

    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?
  10. arpon

    I Is Second rank tensor always tensor product of two vectors?

    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...
  11. arpon

    I Parametric Resonance

    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 =...
  12. arpon

    I Problem -- Fourier transform

    [##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\\...
  13. arpon

    I Dirac Delta using Fourier Transformation

    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...
  14. arpon

    Fourier series

    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...
  15. arpon

    I Contraction of mixed tensor

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

    I Two metric tensors describing same geometry

    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...
  17. arpon

    Complex Integration using residue theorem

    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##...
  18. arpon

    Physics Jokes?

    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.
  19. arpon

    Most Beautiful Theory?

    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.
  20. arpon

    I Isn't the force calculated twice?

    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}##.
  21. arpon

    B Adiabatic but permeable piston

    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.
  22. arpon

    I Demagnetizing effect of toroids or long thin cylinders

    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...
  23. arpon

    Temperature dependence of Cv at very large volume

    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##...
  24. arpon

    I Can internal energy be calculated from equation of state?

    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...
  25. arpon

    Adiabatic process of real gas

    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...
  26. arpon

    Prove dQ is an inexact differential

    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...
  27. arpon

    Problem on Thermodynamics

    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##...
  28. arpon

    Problem about an Eigenvector

    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...
  29. arpon

    Electric field outside conductor

    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...
Top