Is the following true if the momentum operator changes the direction in which it acts?
\langle \phi | p_\mu | \psi \rangle = -\langle \phi |\overleftarrow{p}_\mu| \psi \rangle
My reasoning:
\langle \phi | p_\mu | \psi \rangle = -i\hbar \langle \phi | \partial_\mu | \psi \rangle
\langle...
Is it possible to use separation of variables on this equation?
au_{xx} + bu_{yy} + c u_{xy} = u + k
Where u is a function of x and y, abck are constant.
I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed.
Thank you.
It is easy to find that the equation for an ellipse is:
$$1 = x^2/a^2 + y^2/b^2$$
Then according to Kepler's equation:
$$x = a(\cos(E)-e)$$
$$y = b\sin(E)$$
where E is the eccentric anomaly and e is the eccentricity.
If you plug the Kepler's equations' x and y into the equation for the ellipse...
It is fairly trivial to do this with a circular orbit: $$(x,y) = (cos(\omega t),sin(\omega t))$$
where t is time, and $$\omega = \sqrt{GM/r^3}$$
How this parametric equation look for an elliptical orbit?
It's common knowledge that it takes about 3 days to get to the moon. With a Hohmann transfer, I get a transit time of 5 days, not 3. I see NASA used something called "trans-lunar injection". Is this distinct from a Hohmann transfer, and more time efficient? What makes this trajectory different...
In collisions that are inelastic or partially elastic, how can we predict how much of the energy lost to the surroundings becomes heat, and how much becomes sound? What determines that fraction?
Hi everyone,
I am looking at a paper on compact dimensions. Equation 65 makes sense except for the term of 4*pi*n*R in the denominator. Why is it 4*pi and not 2*pi? I cannot rationalize this. Please help. Thank you.
https://arxiv.org/ftp/hep-ph/papers/0609/0609260.pdf
So the universe is expanding, and galaxies are getting farther apart from one another on average. Does this motion count the same as ordinary motion, in that if a galaxy is being expanded away from us at 0.5c, that clocks in that galaxy would appear to tick slower at 0.866 the rate of clocks here?
In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1).
m^2 = E^2 - \vec{p}^2
This can also be written with the Minkowski metric as:
m^2 = \eta_{\mu\nu} p^\mu p^\nu
More...
If a Lagrangian has explicit time dependence due to the potential changing, or thrust being applied to the object in question, how does calculus of variations handle this?
It's easy to get the Lagrange equations from:
δL = ∂L/∂x δx + ∂L/∂ẋ δẋ
What is not clear is how this works when t is an...
I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles.
In math speek:
Ai = uniform(0,2π)
dA = Ai - Aj
dA is not uniform.
Here is a rough image of...
If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors.
For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5).
Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which...
I have only seen scenarios so far where the elements are all along the diagonal, but what are some known cases where there are off-diagonal elements?
Thank you.
Minimal surfaces are sort of the "shortest path" but in terms of surface shapes.
So I figured I could characterize the shape of a hammock by adding the influence of gravity, much like you can get the shape of a catenary cable (y=cosh(x)).
The equation of motion I get from the Lagrangian is...
I'm basically trying to understand the 2-D case of the catenary cable problem. The 1-D case is pretty straightforward, you have a functional of the shape of a cable with a constraint for length and gravity, and you get the explicit function of the shape of a cable.
But if you imagine a square...
So we know that in GR electromagnetic waves have their trajectories effected by the gravity of stars and planets. But how about gravitational waves. Are their trajectories altered by gravity? If so, would this imply that gravitons are self-interacting if they exist?
So I have heard it mentioned that there are causally separate regions in the CMB. For instance a point A and point B that we can see here on earth, but are outside of each other's light cones. My question is then, how far apart are these points A and B at minimum to be causally separate in this...
Is there a way to identify a factorial without referring to computation of a factorial?
For example, is there a way to identify 5040 as a factorial and a way to identify 5050 as not a factorial?
Is there any prime number pn, such that it has a relationship with the next prime number pn+1
p_{n+1} > p_{n}^2
If not, is there any proof saying a prime like this does not exist?
I have the exact same question about this relation:
p_{n+1} > 2p_{n}
What happens when charged particles fall into a black hole?
Say like N electrons fall in, giving the black hole a net charge of -N.
Since light cannot escape the event horizon, I imagine electric fields cannot either, since they are mediated by photons.
So is that charge effectively lost until...
If so, what causes it to rip? Does it "heal"? What would a rip look like to us? What is the tension threshold at which spacetime rips? What are the units of this "tension"?
Is there a general solution explicitly worked out for how a monopole and electic charge would interact? Of course the electro-static solution is that there is no interaction, but the electro-dynamic solution would not be so trivial, as moving charges/monopoles would induce magnetic/electic fields.
In QFT, an ultraviolet cutoff is imposed to avoid singularities. One physical reason for why this works may be that there actually is an ultraviolet cutoff from spacetime being quantized. Since in GR spacetime is responsible for the force of gravity, and gravitons are the hypothesized quanta of...
Has anyone attempted build a magnetic monopole?
If you took a bunch of wedge-shaped permanent magnets and assembled them into a ball such that the same pole for each magnet was pointing out radially, then you would have a magnetic monopole. (see attached).
This is a 3 part question.
1. I've come to understand that the mass values in the mass terms (pole mass) of the standard model don't represent what we actually measure. That there are loop corrections. (I understand the concept: there's screening either adding or subtracting from the true...
Just wondering how much validity there is to this derivation, or if it's just a convenient coincidence that this works.
We have a Lagrangian dependent on position and velocity: \mathcal{L} (x, \dot{x})
Let's say now that we've perturbed the system a bit so we now have:
\mathcal{L} (x +...
Are there any theories that have mixing between fields and space?
For instance, a theory with mixing between two fields might look like:
L_{int} = k \phi(x) \psi(x)
Where mixing between a field and space might look like:
L_{int} = k \phi(x) x
What are the consequences of something like this?