Ah, I see your point. Okay suppose we interchange the variables s and t in the second integral, and then exchange the order of integration (not sure if this is justified). Then we can combine the integrals and factor out the ##\eta(s)## such that
$$
0=\int\int\left( \frac{\partial}{\partial...
Well if you're really interested in getting an analytical solution, you could try expanding the ##(7.625+.275 \cos(4x))^{1.5}## as a taylor series. At that point you'd have a sum of integrals that are just powers of trig functions, which should be integrable (though ugly). You could then...
So I've been thinking about this problem some more (sorry I didn't reply earlier). Following your suggestion, let's assume only ##\phi## is varied for simplicity. In that case we want to minimize a functional of the form...
Okay, so I've run into a rather weird functional that I am trying to optimize using calculus of variations. It is a functional of three functions of a single variable, with a constraint, but I can't figure out how to set up the Euler-Lagrange equation. The functional in question is (sorry it's...
So I recently watched the new movie Interstellar, and I've been inspired to do some more general relativity. At one point in the movie they mention that 1 hour on a planet orbiting a black hole is 7 years back on Earth, and so I decided my first project would be to figure out exactly how close...
Simply put, can you find the function which extremizes the integral
J[f]=\iint L\left(x,y,f(x),f(y),f'(x),f'(y)\right) \,dx \,dy
Where ##f## is the function to be extremized, and ##x## and ##y## are independent variables? A result seems possible by using the usual calculus of variation...
Okay, so I guess I'm still confused by the whole choice of Gauge thing. You say:
Does this mean that there is a degenerate energy level for each choice of Gauge? If not, how does the same state correspond to several different wavefunctions?
I was wondering about what the wavefunction of a particle in a magnetic field would look like, so after some quick work and a little research, I found the the Hamiltonian is
\hat{H} = \frac{(\hat{p}+qA)^{2}}{2m}
where A is the vector potential such that B=∇×A. I thought that the vector...
Homework Statement
Is it possible given a wavefunction ψ(x,t) to find the probability that the particle is at a particular location over an interval of time?
Homework Equations
The Attempt at a Solution
Intuitively, given that the probability of finding the particle in a region...