I understand the mean square end-to-end distance of a polymer of stepsize b and number of units N is given by
\left\langle R^{2}\right\rangle = \sum_{i,j=1}^{N} \left\langle r_{i} r_{j}\right\rangle + \left\langle \sum_{i \neq j}^{N} r_{i} r_{j}\right\rangle
And that the cross terms drop...
The molecular flux (number of collisions per unit area per unit time) in Kinetic theory is given by
F=\frac{1}{4} n \bar{c}
where c bar is the average molecular speed and n is the density of molcules in the gas phase (molecules/m^3)
I was wondering about the origin of the 1/4 term ...
Homework Statement
- Initial conformational potential energy = 2500 J
- NVT Metropolis Monte Carlo simulation, T = 300 K
- New conformational potential energy = 5000 J
What's the probability this will be accepted within the Metropolis scheme?
Homework Equations
Acceptance...
LOL, ok probably talking to myself here. Still playing around with this, taken it further, although I'm pretty sure my final answer here is wrong...
http://voy.trekcore.com/working.jpg [Broken]
OK, I've got a little bit further (this seems brutal!)
I've been able to see that the separations of the charges mentioned above are:
q1 & -q2: \sqrt{(d+rcos\theta)^{2}+(rsin\theta)^{2}}
-q1 & q2: \sqrt{(-d+rcos\theta)^{2}+(rsin\theta)^{2}}
Simplifying:
q1 & -q2...
Homework Statement
Consider two dipoles with moments u1 and u2 arranged as in the following diagram. Each dipole is depicted as two charges of equal magnitude separated by a distance d. The centre-to-centre separation of the two dipoles is the distance r. The line joining the two dipole...
I'm trying to clarify for myself the difference between a Unit Cell, a Primitive Unit Cell and a Conventional Unit Cell.
As far as I know,
Primitive Unit Cells contain only one lattice point and are the smallest possible unit cell
Unit Cells are the crystal's smallest building block, and...
I'm having difficulty with this problem:
Consider a two state system consisting of N distinguishable and indeppendent particles where each particle can occupy one of two states separated by an energy E. What is the entropy of the system at:
(A) T=0
(B) T=infinity
I'm assuming this...
Homework Statement
A crystal has a basis of one atom per lattice point and a set of primitive translation vectors of
a = 3i, b = 3j, c = 1.5(i+j+k)
where i,j,k are unit vectors in the x,y,z directions of a Cartesian coordinate system. What is the Bravais lattice type of this crystal...
Homework Statement
I'm trying to follow through a derivation involving the equation of motion for the displacement x(t) of a damped driven harmonic oscillator.
m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}cos(\omega t)
Where
cos(\omega t) = \frac{1}{2}\left( e^{i \omega...
Homework Statement
(Lead in to question:
"Assume that charges +q, -3q, 2q lie at positions -2a, 0, +2a along the x-axis respectively.")
I've calculated dipole/quadrupole moments about the origin as well as the exact potential at x=+10a, however I'm confused by this next part to the...
hmm, if that was the case, wouldn't the equation take the form
\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_0}{L^2}\frac{L^{3}}{(3\pi)^{3}}\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi
(Note the L^3/... prefactor) ??
[SOLVED] Integration change of variables
Homework Statement
An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation:
V(x) =...
Homework Statement
I'm trying to calculate the energy shift given an electron in a 1D harmonic potential has a wavefunction
\Psi_{0}(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}exp\left(\frac{-m\omega x^{2}}{2\hbar}\right)
The shift in E_{0} = \frac{\hbar\omega}{2} = 2eV
due to...
I'm trying to bridge the gap between several expressions describing the insertion of a constant perturbation:
a_{f}(t) = \frac{1}{i\hbar} V_{fi} \int^{t}_{0} e^{i(E_{f}-E_{i})t'/\hbar}dt' = \frac{1}{i\hbar}V_{fi}\frac{e^{i(E_{f}-E_{i})t/\hbar} - 1}{i(E_{f}-E_{i})/\hbar}
so...
Yep, another quick question on the Dirac Equation!
I've become slightly more clued about the use of the DE now in illustrating the negative energy problem in relativistic QM as well as the existence of spin, however one thing is still puzzling me.
I've read this excerpt in a text:
I'm...
Could anybody explain to me the difference between a Landau Gauge and Symmetric Gauge?
I know the Landau Gauge is given by A = (0,Bx,0) producing a constant magnetic field in the z direction. I am *assuming* (process of elimination!) that A = ½B × r = (-yB/2,xB/2,0) is an example of a...
Ah, I understand now why the commutator arises, thanks Dick. However, I'm still confused about the middle term business. If I understand you right, you're saying that:
(-2im\gamma^{\mu}\partial_{\mu})\Psi
is analagous to:
(i\gamma^{\mu}\partial_{\mu} - m)\Psi = 0
and hence should...
[SOLVED] The Dirac Equation
I'm trying to understand the following property of the Dirac equation:
(i \gamma^{\mu}\partial_{\mu} - m)\Psi(x) = 0
Acting twice with (i \gamma^{\mu}\partial_{\mu} - m):
(i \gamma^{\mu}\partial_{\mu} - m)^{2} \Psi(x) = 0
= [ -...
Sorry, I should have probably just transcribed the question as it's written here:
I thought that maybe I'd need to use this relation:
a_{n}(t) = \int \psi^{*}_{m}(r) \Psi(r,t) dV
But that gives me a sin² integral which seems very involved for the question...
I'm trying to get my head around the idea of expansion coefficients when describing a wavefunction as
\Psi(\textbf{r}, t) = \sum a_{n}(t)\psi_{n}(\textbf{r})
As I understand it, the expansion coefficients are the a_{n} s which include a time dependence and also dictate the probability of...
I'm working through a proof of the orthormality condition for a complete set of states and am struggling with one element of it:
Consider the eigenstates of the Hamiltonian in the following way:
1: \int\Psi^{*}_{m}H\Psi_{n}dV = E_{n}\int\Psi^{*}_{m}H\Psi_{n}dV
and
2...
I'm probably missing something obvious here, but I'm trying to integrate the following expression;
\intsin{^2}(kx)dx
I've tried doing it by part but with no luck. Is there some specific method I need to follow, or is it one of those I can only get by looking it up?