There are certain approximations - things you neglected - that a more careful treatment will not neglect. Consider: the electron is actually a relativistic particle, yet you used the nonrelatavistic schrodinger equation to derive those energy levels.
Look up the fine structure of the...
Part iv is actually pretty easy. Find the total CM energy-momentum 4-vector, and calculate the invariant mass (<E,E> = \frac{1}{c^2}\sqrt{E^2 - (pc)^2}). You can also do part iii using this information as well as some Lorentz transformations... might save you a bit of tedious algebra, although...
Well, you know what the pattern from TWO slits looks like, so you're most of the way there. Three slits is the same thing as three pairs of slits, three double-slit patterns superposed on each other. The rest is just some moderately complicated algebra.
It problably doesn't matter since this thread is several months old, but anyway, the plot posted above is incorrect, all values should be multiplied by 4/3. Franz101010 did not normalize the energies correctly. See the attached mathematica notebook.
Because the unit vectors are actually functions of position in cylindrical coordinates. This means all the derivative in the gradient operator act not only on the components of a particular vector, but also the unit vectors themselves.
Homework Statement
Solve Laplace's equation inside a circular annulus (a<r<b) subject to the boundary conditions \frac{\partial{u}}{\partial{r}}(a,\theta) = f(\theta)\text{, }\frac{\partial{u}}{\partial{r}}(b,\theta) = g(\theta)
Homework Equations
Assume solutions of the form u(r,\theta)...
Homework Statement
Solve Laplace's equation inside a circular annulus (a<r<b) subject to the boundary conditions \frac{\partial{u}}{\partial{r}}(a,\theta) = f(\theta)\text{, }\frac{\partial{u}}{\partial{r}}(b,\theta) = g(\theta)
Homework Equations
Assume solutions of the form u(r,\theta)...
What you've written isn't exactly right, it isn't
I=\int_{V}\frac{1}{\left|\texbf{x}\right|}d^{3}\texbf{x}
It's,
I=\int_{V}\frac{1}{\texbf{r}}d^{3}\texbf{x}
with r = \sqrt{x^2 + y^2 + z^2} . Try doing that integral, it'll converge.
Homework Statement
Solve Laplace's equation inside the rectangle 0 \le x \le L, 0 \le y \le H with the following boundary conditions
u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0
Homework Equations
The Attempt at a Solution
I know that with...
Homework Statement
Identify the zeroes, poles and essential singularities of f(z) = z^{2/3}
Homework Equations
f(z) = e^{\frac{2}{3}\log{z}}
Which I'm not sure will be useful...
The Attempt at a Solution
I know that f is 0 at z=0, but what is the order of this zero? Is there such a...