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Given the hamiltonian: \hat{H} = \hbar \omega_{0} \hat{a}^{+}\hat{a} + \chi (\hat{a}^{+}\hat{a})^2, where ##\hat{a}^{+}##, ##\hat{a}## are creation and annihilation operators. Find evolution of the state ##|\psi(t) \rangle##, knowing that initial state ##|\psi(0)\rangle = |\alpha\rangle##...

It looks more like a computational obstacle, but here we go. Plugging all of these to the partition function: $$Q = \frac{1}{N! h^{3N}} \int -\exp(\frac{1}{2m}(p^2_{r}+p^2_{\phi}/r^2+p^2_{z})+gz)d\Gamma=$$ $$= \frac{1}{N! h^{3N}} \int \exp{(\frac{-1}{2m}p^2_{r})}dp_{r_{1}}...dp_{r_{N}}...

I remember that there's something about stability of floating bodies in "Physics of Continuous Matter Exotic and Everyday Phenomena in the Macroscopic World" by B. Lautrup. Maybe that will help?

I guess the resonator will be stable when both focal lengths of lens and curved mirror meet. The curved mirror is fixed and it's focal length: $$f_{curved} = \frac{R_{2}}{2} = 50 \, mm.$$ Then the d should be: $$d=f_{lens}+f_{curved} = 100 \, mm.$$ I think that's also the distance for which the...

Homework Statement Hello, I have a problem with my data analysis from my lab. My goal is to find drift velocity of the electron and it's diffusion coefficient. The experiment looked like this: I've measured the time difference between signals on two gaseous detectors. The source of the signal...

I think I've seen these in Callen's "Thermodynamics and Introduction to Thermostatics" in the appendix. Although, I'm not sure if (d) can be found there.

I tried to do that, but I'm afraid it doesn't get me anywhere. From definition: $$\big(r\frac{\partial^2{T}}{\partial{r}\partial{\Phi}}\big)\phi = (-1)^2T\big( r\frac{\partial^2{\phi}}{\partial{r}\partial{\Phi}} \big)$$. Evaluating the right side of this equation: $$(-1)^2T\big(...

Homework Statement Let ##T## be a distribution in ##\mathcal{D}(\mathbb{R}^2)## such that: $$T(\phi) = \int_{0}^{1}dr \int_{0}^{\pi} \phi(r, \Phi)d\Phi$$ $$\phi \in \mathcal{D}(\mathbb{R}^2)$$ calculate ##r \frac{\partial{}}{\partial{r}} \frac{\partial{}}{\partial{\Phi}}T##. Homework...

@TSny, apparently the trick is to write the potential $$\vec{A}(\rho) = \frac{\mu_{0}}{4 \pi} \int_{-\infty}^{\infty} \frac{I(t - t_{r}) \hat{e}_{z}}{\sqrt{z^2 + \rho^2}},$$ where ##t_{r} = t - \frac{1}{c}\sqrt{z^2 + \rho^2}##. The other step is to see that function under itegral is diverging...

Ok, if there's anyone else interested in this topic it's really well explained in "Introduction to electrodynamics" third edition by David Griffiths Chapter 10.

No, sorry I didn't see that in "The attempt at a solution". a) is correct.

To find a total charge you need to integrate the density ##\rho## over whole space (spherical coordinates are best here). Luckily ##\rho## gets small pretty quickly so integration gives finite result. a) is exactly like finding a mass of an infinite object with density ##\rho##.

Homework Statement There's a thin, straight, infinite wire conducting alternating current: $$I(t) = I_{0}\exp(-\kappa t^2),$$ where ##\kappa > 0##. Calculate the force exerted on point charge ##q## that is located in distance ##\rho## from the wire. Consider relativistic effects. Homework...

Homework Statement There's a very long cylinder with radius ##R## and magnetic permeability ##\mu##. The cylinder is placed in uniform magnetic field ##B_{0}## pointed perpendicularly to the axis of cylinder. Find magnetic field for ##r < R##. Assume there's a vacuum outside the cylinder...

Yes, but the problem is that logarithms are centered in points other than ##z=0## and I'm wondering if I'm trying to do it wrong, because I can't separate real and imaginary terms in order to get ##\Phi## - velocity potential and ##\Psi## - stream function,