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  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

    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.
  4. 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}...
  5. arpon

    Two successive rotation (Goldstein problem 4.13)

    I was looking for a rigorous derivation.
  6. 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...
  7. arpon

    Velocity of a piston in a piston-shaft mechanism

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

    I Interpretation of photons having zero spin

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

    Velocity of a piston in a piston-shaft mechanism

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

    Velocity of a piston in a piston-shaft mechanism

    You forgot to upload the figure.
  11. arpon

    I Energy operator and the Hamiltonian operator: Are they same?

    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...
  12. arpon

    I Energy operator and the Hamiltonian operator: Are they same?

    Isn't the Hamiltonian Operator in the Schrodinger's time dependent equation is the Hamiltonian operator defined for the particular system we are considering?
  13. 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?
  14. arpon

    I Energy operator and the Hamiltonian operator: Are they same?

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

    I Energy operator and the Hamiltonian operator: Are they same?

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

    I Four-vector related to electric and magnetic dipole moment

    What is the four-vector related to electric and magnetic dipole moment?
  17. 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...
  18. arpon

    How many equations does a physicist write in his lifetime?

    I thought writing more equations means spending more time on physics.
  19. 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?
  20. 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?
  21. arpon

    I Branch cut

    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...
  22. 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...
  23. arpon

    I Parametric Resonance

    Thanks for your help. But still could not get it.
  24. 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 =...
  25. arpon

    I Problem -- Fourier transform

    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, $$...
  26. arpon

    I Problem -- Fourier transform

    I used the defination, $$\int_{-\infty}^{\infty}f(x)e^{-ikx}\,dx=\widetilde{f}(k)$$
  27. 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\\...
  28. arpon

    I Dirac Delta using Fourier Transformation

    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...
  29. 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...
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