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Ahh... right. Thank you for clarification. In literature, people confusingly use E(x) and E(t) and take derivatives with respect x and t which drives me crazy....

I understand that waves are function of space and time in nature, so E(x,t) will be fundamental description of a wave. I notice that often people denote a wave as E(t) for instance, an envelop function of a pulse. For this case, E is an oscillation at a fixed spatial point x? So that the point x...

I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...

This solved the mystery for me... Thank you so much.

Then either case I am doing positive work. But the above and other text says when volume expands, external work done on the system is negative and vice versa (so the signs of work I do for compression and expansion of gas are opposite). How does this work?

If I consider a piston in equilibrium with atmosphere such that it won't move unless I push it or pull it. This case, whichever way I do, won't I be doing positive work? From above formula for work.

I am stuck on like the first page of thermal physics. It seems like the signs of the work done on the system are opposite when the volume is expanded and compressed. But when I imagine myself pushing or pulling the piston, I get confused from W = \textbf{F}\cdot \textbf{d} This work will be...

I see. Thank you!

Boyd - Nonlinear Optics page 5, there says 'Here a laser beam whose electric field strength is represented as $$\widetilde{E}(t) = Ee^{-iwt} + c.c$$But why is it written like this? Is it because the strength is the real part of the complex electric field? Then why doesn't he divide it by 2 after...

Hi, I am a fourth year physics student currently enrolled in a research project course with a supervisor. It's a research in optics and I need to learn about some nonlinear optics. For the report, I want to write about what I have learned during the term, for instance, Rayleigh and Raman...

! I see. So the set of solutions we found as above is indeed the set of all the possible solutions by uniqueness. Thank you very much, now I understand what was going on!

It looks like that what you showed here is that the time evolution of linear combination of separable solutions is ##\Psi(x,t) = \sum c_n\psi_n(x)\phi_n(t)##. But this only states about evolution of separable solutions, not evolution of general solutions. Can you state the PDE theorem here...

I apologize that I forgot to mention that the Hamiltonian concerned here is time-independent, as PeroK pointed out.

I see that the eigenfunctions of the Hamiltonian indeed form a complete set such that any function that satisfies time-independent Schrodinger equation ##H\psi=E\psi## can be written as linear combination of them. I also see that the time evolution of that function corresponds to the time...

I found this on Griffiths page 25, "Moreover (as is typically the case with separation of variables) we will be able at the end to patch together the separable solutions in such a way as to construct the most general solution."

Hi, I am wondering why every general solution to Schrodinger equation can be built from separable solutions. In other words, I don't follow that why every solution to Schrodinger equation can be written as $$\Psi(x,t) = \sum c_n\Psi_n(x,t)=\sum c_n\psi_n(x)\phi_n(t)$$ I know that the right hand...

Oh the Lagrangian does not necessarily be T-U

The principle of the least action, that the particle will take the path of least Lagrangian, here given as T-U, is Hamilton's principle in classical mechanics. I am wondering if this is just an empirical, experimental observation that is not mathematically driven from elsewhere, just like...

I am only considering idealized theoretical situations. The position of harmonic oscillator, for example, according to the postulate, a measured position value q is an eigenvalue of the equation xψ = qψ. But from this equation we can deduce that σx = 0. This is not true since |ψ|2 is not a...

According to one of the postulates of quantum mechanics, every measured observable q is an eigenvalue of a corresponding linear Hermitian operator Q. Which means, that q must satisfy the equation Qψ = qψ. But according to Griffiths chapter 3, this equation can only be followed from σQ = 0. It...

Yeah I tried that function and it did the job. Thank you. I wonder what are the functions Griffiths will be dealing with and how it makes sense to throw the term away for those functions...

I am following Griffiths' intro to quantum mechanics and struggling(already) on page 16. When a particle is in state ##\Psi##, $$\frac{d<x>}{dt} = \frac{i\hbar}{2m}\int_{-\infty}^{\infty} x\frac{\partial}{\partial t}\bigg (\Psi^*\frac{\partial \Psi}{\partial x}-\frac{\partial \Psi^*}{\partial...

I get it now...!! thank you very much!

Actually ##\vec{r}## is the position vector of target charge, and ##\vec{r}'## is the position vector of source charge, so we integrate it over ##\vec{r}'##, so I think I need the divergence with respect to ##\vec{r}'##

Homework Statement For a volume charge, ##\textbf{E}(\textbf{r}) = \frac{1}{4\pi\epsilon_0}\int_{all space}\frac{\hat{\gamma}}{\gamma^2}\rho(r')d\tau'## and I am trying to get the divergence of it. Homework Equations The book says ##\nabla\cdot\textbf{E} = \frac{1}{4\pi\epsilon_0}\int_{all...

This condition should add to the last equation: ##i\neq j##

Homework Statement My text book states that for point charges ##q_1, q_2, ... ,## where distance between ##q_i## and ##q_j## is ##r_{ij},## the total potential energy U is the sum ## U = \dfrac{1}{4\pi\epsilon_0} \sum_{i<j} \dfrac{q_iq_j}{r_{ij}} ## and specifically mentions not to count them...

So why the charges on the surface of the cage move around to cancel the electric field from outside?

Homework Statement I am having trouble understanding how a Faraday cage works. Homework Equations $$\oint \vec{E}\cdot\,d\vec{A} = \dfrac{Q_{encl}}{\epsilon_{0}}$$ The Attempt at a Solution It says that Faraday cage is a hollow metallic conductor and hence, inside the cage, $$\vec{E} = 0$$ I...

I'm a 4th year student in a university in Canada, studying pure math. My GPA so far is barely about 3.0/4.0. I was young and totally screwed my undergraduate years. I have planned to go to grad school right after I graduate, but these days I feel so regretful about my GPA that even makes me want...