Recent content by Peeter

@Greg Bernhardt -- It's been so long since I'd attempted that problem, that I'd probably have to attempt it anew to gain any additional insight.

Homework Statement Given a coupling h \; \partial_\mu \phi^a \partial^\mu \phi^a , meant to model the first order interaction of the Higgs field h to boson fields \phi^a , compute the width \Gamma(h \rightarrow \phi^3 \phi^3) of the Higgs particle to decay to two longitudinal (say)...

That seems a bit strange in general, but in the special case of localized currents seems like a reasonable thing to calculate.

In Jackson, the following equations for the vector potential, magnetostatic force and torque are derived##\mathbf{m} = \frac{1}{{2}} \int \mathbf{x}' \times \mathbf{J}(\mathbf{x}') d^3 x'## ##\mathbf{A} = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \mathbf{x}}{\left\lvert {\mathbf{x}}...

But in this case the eigenstates were for the Hamiltonian, not for the operator itself?

Working some subsequent parts of the assignment provide a clue. We are asked to compute the following for the 3D SHO Hamiltonian: $$\left\langle \frac{\mathbf{p}^2}{m} \right\rangle - \left\langle \mathbf{x} \cdot \boldsymbol{\nabla} V \right\rangle$$ (a) for the eigenstates ##\left\langle...

The expansion you've done isn't quite valid in this case, because X_k and P_k operators are Heisenberg picture operators $$\begin{align*}X_{k,H}(t) &= U^\dagger(t) X_k U(t) \\ P_{k,H}(t) &= U^\dagger(t) P_k U(t).\end{align*}$$ Also note that the state vectors in question (say, \left\lvert \psi...

Note that I've got the sign wrong on the gradient above, but that doesn't change my question, since I want to be able to justify setting the LHS to zero.

Homework Statement Under what conditions is \left\langle{{\mathbf{x} \cdot \mathbf{p}}}\right\rangle a constant. A proof of the quantum virial theorem starts with the computation of the commutator of \left[{\mathbf{x} \cdot \mathbf{p}},{H}\right] . Using that one can show for Heisenberg...

I'm not certain how to interpret the reply of @RUber, but I resolved this after finding a hint in Griffiths, which poses a problem of relating the volume integral of \mathbf{J} to the dipole moment using by expanding \int \boldsymbol{\nabla} \cdot ( x \mathbf{J} ) d^3 x. That expansion is...

In Jackson's 'classical electrodynamics' he re-expresses a volume integral of a vector in terms of a moment like divergence: \begin{align}\int \mathbf{J} d^3 x = - \int \mathbf{x} ( \boldsymbol{\nabla} \cdot \mathbf{J} ) d^3 x\end{align} He calls this change "integration by parts". If this...

Very clever (of both you and the master;). Thanks!

Homework Statement Attempting a mechanics problem from Landau's mechanics (3rd edition) I get a different answer, as shown below. Error by me, or typo in the textbook? I can't find any errata page for the text, but since it's an older book, perhaps no such page is maintained. Chapter 1...

The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces. So, for example, in a shearing flow where there is a moving surface, the fluid velocity at the fluid/surface interface has to...

It's not clear to me how one would relate the pressure and the volume flux. We do have the pressure show up in the traction vector. In a later part of the problem, this is used to calculate the torque on the fluid. For example, the torque per unit area on the fluid from the inner cylinder...