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...
I have the following equation:
\frac{1}{2N^{2}}\int_{s} \int_{p} \left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle
which describes the radius of gyration of a polymer. (the term being integrated is the average position between beads p and s)
This is shown to be...
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
I have the following expression:
cos\theta = \frac{\vec{a}\cdot \vec{b}}{ab}
(this is simply taken from a dot product rule for two vectors)
However I need to find \nabla_{\vec{r}i} \theta
Is there a way I can do it without involving differentials of arccos and...
Homework Statement
Find \frac{dU_{ave}}{d\beta}
where
U_{ave}=\sum_{k}\left(\frac{U_{k}exp(-U_{k}\beta)}{exp(-U_{k}\beta)}\right)
Homework Equations
The Attempt at a Solution
My answer is supposed to be -(U_{ave}^{2})+(U_{ave})^{2}
However I keep getting zero. I can...
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...
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...
I'm trying to derive a formula but can't seem to work the algebra.
I need to combine these two:
V_{1}p_{1} + V_{2}p_{2} = N
V_{1} + V_{2} = V
to get this:
\frac{V_{1}}{V} = \frac{p-p_{1}}{p_{2}-p_{1}}
where p = N/V
If anyone could show me the steps that would be a huge help...
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...
Hey all, I'm looking for some suggestions on topics concerning Molecular Simulation which I can research. I've been tasked with giving a talk and writing a report on one specific topic of this field but I'm unsure what to specialise in. Any and all advise and suggestions would be appreciated!!
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...
[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...
[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
= [ -...
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?
Homework Statement
I'm trying to determine a normalization value, A, for the following wavefunction:
\Psi = Ax{^2}exp(-\alpha x)}, x>0
\Psi = 0, x<0
In the past, I've had an i term in my exponential, so when applying the Normalization Condition:
\int|\Psi(x)|^2 dx = \int\Psi{^*}(x)...
Simple question, and pretty sure I already know the answer - I just wanted confirmation,
Considering the Hermitian Conjugate of a matrix, I understand that
A^{+} = A where A^{+} = (A^{T})^{*}
Explicitly,
(A_{nm})^{*} = A_{mn}
Would this mean that for a matrix of A, where A is
a...
Homework Statement
http://tng.trekcore.com/1.JPG [Broken]
I'm trying to prove that the circle is symmetrical by showing that x² + y² = a² holds when the circle rotates.
I know that this is proved given the following two formulae:
x = x'cosθ - y'sinθ
y = x'sinθ + y'cosθ
but I don't...
Einstein Summation Convention / Lorentz "Boost"
Homework Statement
I'm struggling to understand the Einstein Summation Convention - it's the first time I've used it. Would someone be able to explain it in the following context?
Lorentz transformations and rotations can be expressed in...
Homework Statement
Homework Equations
Using the Carnahan and Starling equation, estimate the coefficient of volumetric expansion, α, and the coefficient of compressibility, β, defined as
α ≡ 1/V * (δV/δT) [holding P constant] and
β ≡ -1/V * (δV/δP) [holding T constant]
(I've used δ...