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For a system of interacting particles, is it possible to define single-particle phase spaces? If not, why?

Can you show a few lines of computation because I cannot figure out how are you getting that result. No, I am using $\alpha=1.$

Can you show a few lines of computation because I cannot figure out how are you getting that result. No, I am using ##\alpha=1## .

In David Tong's QFT notes (see http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf , page 131, Eq. 6.38) the expression for canonical momentum ##\pi^0## is given by ##\pi^0=-\partial_\rho A^\rho## while my calculation gives ##\pi^\rho=-\partial_0 A^\rho## so that ##\pi^0=-\partial_0 A^0##. Is it...

Can all differential equations be turned into algebraic equations by Fourier transform (FT)? If not, what kind of differential equations can be solved by the FT technique?

What's your objection to moving charges? Currents are produced by moving charges. In general, a given charge can also accelerate due to the forces experienced by the electric and magnetic field produced by the other charges.

Thanks for the help. :-)

Thanks A. Neumaier. So for problems which are not exactly solvable, and in which the 'unsolvable part' of the Hamiltonian is not "small", other nonperturbative approximation methods become useful for calculating energy eigenvalues and eigenfunctions. Do I get it right?

So nonperturbative methods are also approximation methods?

Even perturbation theory beyond a certain order will have to be dealt with numerically. Even when you don't use perturbation theory, your numerical algorithm must be based on some framework, some approximation method if not perturbation theory. The question is which framework(s) do people have...

Which method other than the perturbation theory should one apply in this example of yours? Do you have some other approximation method such as the variational principle in mind? Thanks!

Dear DrClaude, I appreciate your help but unfortunately, your answer didn't address my actual question. The question asks about an explanation for the nonperturbative approach of quantum mechanics and occasions when it becomes indispensable perhaps with an illustration. I'm aware of what is the...

No. It's not a homework. It's for my personal understanding. My exposure to quantum mechanics is that of the undergraduate level. Sorry for the advanced tag. Not a frequent user. Title changed now.

What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.

You can't travel at 2c (twice the velocity of light in vacuum)! This violates the postulate of special relativity. Massive objects cannot even travel at the speed of light $c$. So you can modify your question to make it meaningful so that someone can help.

Triple (and higher) integrals will appear quite often in physics in several context. So if you enroll in physics major or engineering you'll definitely encounter triple integrals. Any good book on mathematics and mathematical physics deals with double and triple integrals. Triple integrals arise...

Firstly, a field operator \hat{\phi}(x) is not necessarily hermitian. For example, a complex scalar field is not hermitian but the Hamiltonian is always. A more clearer statement would be \langle 0|\hat{\phi}(x_1)\hat{\phi}(x_2)...\hat{\phi}(x_n)|n\rangle is the configuration space wavefunction...

I think, your understanding, calculation and inference is correct.

Yes. Are the colours in these pictures completely unreal? I thought there must be some 'atomic physics and spectroscopy theory' behind these beautiful pictures.

Why do galaxy and galaxy clusters look so colorful?

If at any instant of time all the atoms are excited to the higher energy level i.e., a population inversion is achieved, it will eventually relax back to a Boltzmann distribution in presence of a temperature bath.

Radioactive decay is governed by the exponential law $$N(t)=N_0e^{-\lambda t}$$ where N_0 denotes the number of nuclei at t=0 , and \lambda is the decay constant which is different for different nuclei. If you use the word "probability of radioactive decay" for "rate of radioactive decay" it is...

I have just started reading about neutrino physics and recently came across two terms called double seesaw and singular seesaw. Although I’m familiar with other seesaw mechanisms (such as type-I and II) for explaining smallness of neutrino mass. I’m completely at dark about double and singular...