The general formula for the magnetic moment of a charge configuration is defined as $$\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r$$ For an electron it's said that the correct equation relating it's spin and magnetic moment is is $$\vec{\mu} =g\frac{q}{2m}\vec{S}$$ It's said tha...

On page 282 of Peskin and Schroeder discussing functional quantization of scalar fields, the authors use expression 9.12, the path integral in ordinary quantum mechanics $$U(q_a,q_b;T)=\bigg(\prod_i\int\mathcal{D}q(t)\mathcal{D}p(t)\bigg)\exp\bigg[i\int_0^T\,dt\big(\sum_ip^i\dot{q}^i-H(q,p)\big)\...

I was wondering if a hologram could be made that could focus light that wasn't in the original hologram. I'm assuming that this would have to be a mathematically constructed hologram as, by my understanding, a hologram of an actual lens can only focus light of images in the actual hologram.

I have a conflict between my lecture notes on quantum mechanics, where it is stated that the probability of measuring an eigenvalue $a_i$ on a mixed state with desnsity matrix $\rho$ is $$ \operatorname{Tr}(P_i \rho P_i)\ , $$ where $P_i$ is the projector for the subspace corresponding to $a_i$. ...

If you travel on car with nearly the speed of light and turn on the car headlights: will it shine in gamma light instead of visible light?

The schrödinger equation for the harmonic oscillator is: $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{m\omega^2}{2}x^2\right)\psi(x)=E\psi(x)$$ Then often $x$ gets substitute with $x=\xi a$ where $a=\sqrt{\hbar/m\omega}$ which leads to: $$\frac{\hbar\omega}{2}\left(-\frac{\par...