In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity
\sum_n |n\rangle \langle n| = 1
In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...
Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?
Hi,
For single-electron atomic systems, the electron can be specified by four quantum numbers n, l, m_l, m_s (principal, orbital, z-orbital, z-spin). The orbital quantum numbers are well defined since the problem is spherically symmetric.
However, for many-electron systems, the spherical...
The Feynman slash
\slashed{a}=\gamma^\mu a_\mu
maps a four-vector a to its Clifford algebra-representation. This is a linear combination of the gamma matrices with the components of a acting as expansion coefficients. What physical significance does this new object have?
The gamma...
What is meant by saying that the Goldstone-bosons are "eaten" by gauge bosons?
I've seen this statement all over, but can't find a good explanation of what this actually means. Anyone care to shed some light?
When an electron and a positron annihilate, they typically produce two gamma rays, each of energy mc^2 plus whatever kinetic energy available before annihilation. I was recently told that it is an experimental fact that the electrostatic energy between the electron and the positron does NOT...
Are point-particles "real"?
I've heard a couple of times the claim that the electron (among other elementary particles) is point-like, having essentially no spatial extension.
In the framework of QFT (which forms the basis of the standard model and therefore our best current understanding of...
Hi!
The Dirac delta satisfies
\int dx f(x) \delta(x-a) = f(a)
But how about
\int d^3x f(x) \delta^{(4)}(x-a)
Here, x and a are four-momenta, and the integral is over the regular 3-dim momentum.
How does the delta behave here?
Hi!
I am currently taking a first course in QFT with Peskin & Schroeder's book. I've got stuck with the equation that relates the differential decay rate of a particle A at rest into a set of final particles with the invariant matrix element M of the process. M can be found from the Feynman...
From what I understand, Einstein basically scrapped the concept of gravity being a force and instead said that energy (and thereby mass) and momentum causes spacetime to curve. Objects still travel on geodesics in spacetime (Newton's first law), but since it is curved, the geodesics near massive...
Hi!
Work is defined as
dW = F\cdot dr
so there is no work required to keep things spatially fixed in a gravitational potential. However, consider a hovering helicopter. Even though it is not moving in the gravitational field, it will eventually run out of fuel. Ofcourse there are...
Hi!
This might be a silly question, but I can't seem to figure it out and have not found any remarks on it in the literature.
When diagonalizing an NxN matrix A, we solve the characteristic equation:
Det(A - mI) = 0
which gives us the N eigenvalues m. Then, to find the eigenvectors v...
Hi,
does anybody know about some good books with tabulated series expansions for functions and algebraic expressions? I've got one entitled Mathematics Handbook, which has maybe 20-40 different expansions, but I want more!!
Consider an isolated system of two ideal, identical gases in thermal equilibrium. Gas A is occupying a volume A, separated by a wall from volume B, where gas B resides. They are in thermal contact. The volumes are the same, so are the pressures and temperatures, and it follows that the number of...
Hi all!
I've been thinking of something lately.
When an atom absorbs an incoming photon, the atom must gain some momentum in order to conserve linear momentum, right? Sort of like a totally inelastic collision? This momentum corresponds to some amount of kinetic energy and thus a raise in...
Why is it that in a room, if you're close to any of the walls the lower frequencies of a sound become louder? Ofcourse, there's an interference pattern in the room. You can hear this by walking around and you'll notice that at some points the bass is weaker and at some points it's stronger (the...
Hi everyone!
As I understand it, magnetism can be explained by special relativity when charges are moving - length contraction in different frames of reference leads to coulomb forces between the charges in these frames.
But how does magnetism in a permanent magnet arise? Are there...