Homework Statement
Trying to solve this integral
\int_0^T \frac{dT'}{T'}\frac{d}{dT'}U(T',V)
where the temperature dependent part of U is
\Sigma \frac{h\omega}{\exp(\beta\omega)-1}
The Attempt at a Solution
using x = hw/T I find that I need to integrate
\frac{x^3...
I made a typo in my boundary conditions
Boundary conditions (in volts):
V(a,\phi) = 2 \cos \phi
V(b,\phi) = 12 \sin \phiTaking V(r,\phi)_{k=1} gives,
V(r,\phi) = r(a_1 \cos \phi + b_1 \sin \phi)+\frac{1}{r}(c_1\cos \phi + d_1 \sin \phi)
I will take a look at this...
Homework Statement
Two coaxial cylinders, radii {a,b} where b>a. Find the potential between the two cylinder surfaces.
Boundary conditions:
V(a,\phi) = 2 \cos \phi
V(b,\phi) = 12 \sin \phiHomework Equations
Solution by separation of variables:
V(r,\phi) = a_0 + b_0 \ln s + \sum_k \left[...
So because U has nothing to do with the magnetic field, I need to eliminate the vector potential from,
\Pi_n \int \exp \left [-\frac{\beta}{2m_n} (\vec{p}_n - q \vec{A})^2 \right] d^3 p_n
and change variable,
\vec{u}_n = \vec{p_n}-q\vec{A}
thus eliminating vector potential?
Are the end...
Homework Statement
Show that the free energy of classical particles with no internal magnetic moment is always independent of magnetic field. Hint: Write down Z for N classical particles. Let the particles interact by U which depends only on the positions of the interacting particles. Show...
Homework Statement
Point electric dipole \vec{p}=p_0 \hat{z} is a distance d above an infinite metal plane of surface normal \hat{n}=\hat{z}. What is the force on the dipole. Is the dipole attracted to, or repelled from the surface?Homework Equations
V(r) = \frac{\hat{n} \cdot \hat{p}}{4 \pi...
Homework Statement
H = -D(S^z)^2 for cases D>0, D<0, where D<0 should remove the degeneracy in the ground state.Homework Equations
H = -D(S^z)^2 = -D\hbar^2 (1 0 0; 0 0 0; 0 0 1)
(`;' separates rows)
det(H-1E)= 0, or by inspection...
The Attempt at a Solution
I get, for D>0,
E_1 = -D...
Homework Statement
A hydrogen atom is placed in a time-dependent homogeneous electric field given by \epsilon = \epsilon_0 (t^2 + \tau^2)^{-1} where \epsilon_0,\tau are constants. If the atom is in the ground state at t=-\inf, obtain the probability that it ill be found in a 2p state at...
Homework Statement
The time-averaged potential of a neutral hydrogen atom is given by
V = \frac{q}{4 \pi \epsilon_0} \frac{e^{-\alpha r}}{r} \left ( 1 + \frac{\alpha r}{2} \right )
[tex]\alpha = 2/a_0[/itex], where a_0 is the Bohr radius.
Find the charge distribution (continuous and...
Homework Statement
Calculate the total internal energy per electron at zero temperature of a free noninteracting gas of electrons of density n, in the following two cases.
a) First assume that states with both spin directions are populated equally.
b) Now assume that the gas is fully...