Homework Statement
Suppose an initial number density of ink particles (i.e. number per unit length) is
given by:
f(x) = 2Nx ; for 0 < x < 1
f(x) = 0 ; otherwise.
Suppose also add a point source containing N molecules at the point x = − 1.
(a) Showing that the initial total...
Homework Statement
Considering the surface of a sphere of radius 1 with its centre at coordinates
(0,0,0).
For the function:
f(x,y,z) = x^{3} + y^{3} + z^{3}
Need to find the following:
(i) All the extrema on the surface which have x, y and z all non-zero simultaneously...
Homework Statement
A rectangle is placed symmetrically inside an ellipse (i.e. with all four corners
touching the ellipse) which is defined by:
x^{2} + 4y^{2} = 1
Find the length of the longest perimeter possible for such a rectangle.
Homework Equations
Within the problem...
Homework Statement
Finding the stationary point(s) of the function:
f(x,y) = xy - \frac{y^{3}}{3}
.. on the line defined by x+y = -1.
For each point, state whether it is a minimum or maximum.
Homework Equations
.. within the problem statement and solutions.
The Attempt at a Solution...
Homework Statement
The hamiltonian of a spin in a magnetic field is given by:
\hat{H} = \alpha\left( B_{x}\hat{S_{x}} + B_{y}\hat{S_{y}} + B_{z}\hat{S_{z}}\right)
where \alpha and the three components of B all are constants.
Question: Compute the energies and eigenstates of the...
Homework Statement
The spin of an electron is described by a vector: \psi = \left(\frac{\uparrow}{\downarrow}\right) and the spin operator:
\hat{S} = \hat{S_{x}}i + \hat{S_{y}}j + \hat{S_{z}}k
with components:
\hat{S_{x}} = \frac{\hbar}{2} \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0...
Homework Statement
The Yukawa potential is given by:
V_{\gamma}(r) = -\frac{q^{2}}{4\pi \epsilon_{0}r}e^{-\gamma r}
Where \gamma is a constant. This describes a screened Coulomb potential.
I. Sketch the radial dependance of this potential.
II. State the radial Schodinger...
Homework Statement
The solution of Schrodinger’s equation for a free particle can be written in the form:
\psi(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(k)e^{i(kx-wt)}dk
[Q1]: Explain why the function \phi(k) is given by:
\phi(k) =...
Homework Statement
A differential equation is given by:
\frac{\partial^{3}u}{\partial x^{3}} + 2 \left( \frac{\partial u}{\partial x} \right) = \frac{\partial u}{\partial t}
By first Fourier transforming the equation (*) with respect to x, show by substitution that:
u(k,t) =...
Homework Statement
I(x) is the intensity of an image after passing through a material which
blurs each point according to a point spread function given by:
S\left(x'-x\right)=e^{-a\left|x'-x\right|}
The Fourier transform of I(x) is given by:
I(k) = \frac{A}{\left( a^{2}+k^{2}...
Homework Statement
The Fourier transform of a function f(x) is given by the product of the Fourier transforms of cos(\alpha x) and e^{-|x|} ;
f^{~} = F^{~}\left[cos[\alpha x]\right]F^{~}\left[e^{-|x|}\right]
Find f(x) and show that it can be written as a real function.
Note: Do...
Homework Statement
Suppose I have 'N' distinguishable objects.
1. In how many different ordered sequences can they be arranged? And why?
2. In how many ways can they be split up into two piles?
(ordering within the piles being unimportant)
The first pile to contain 'n' objects...
Homework Statement
To state the differential form of the Helmholtz thermodynamic potential and
derive the corresponding Maxwell's relation.
Homework Equations
Stated within the solution attempt.
The Attempt at a Solution
Helmholtz function: F = U - TS
Calculating the...
Homework Statement
To win a lottery, must pick 5 different numbers from the 45 available.
The order in which the numbers are chosen does not matter.
With only one ticket, what is the probability of winning (i.e. matching all 5 numbers drawn with all 5 chosen) ?
Homework Equations...
Homework Statement
Assume that the particle in the box is perturbed by a potential V_{1}(x) = x .
Calculate the energy shift of the ground state and the first excited state in first-order
perturbation theory.
Homework Equations
Unperturbed wave functions for the particle given by...
Homework Statement
i. Confirming the wavefunction is normalised
ii. Calculating the expectation values: <\hat{x}> , <\hat{x^{2}}> , <\hat{p}> , <\hat{p^{2}}> as a function of \sigma
iii. Interpreting the results in regards to Heisenberg's uncertainty relation.
Homework Equations...
Homework Statement
I. Finding the partition function Z.
II. If the middle level (only) is degenerate, i.e. there are two states with the same energy, show that the partition function is:
Z = (1+exp(\frac{-\epsilon}{k_{B}T}))^{2}
III. State the Helmholtz free energy F of the...
Homework Statement
A column of water contains fine metal particles of radius 20nm, which are in thermal
equilibrium at 25°C.
The density of the metal is 2\times10^{4} kg m^{-3}.
If there are 1000 particles per unit volume at given height, what will the particle density per unit...
Homework Statement
Determing the constant c such that \psi_{c}(x,y,z) = x^{2}+cy^{2} is an eigenfunction of \hat{L_{z}}
Homework Equations
\hat{L_{z}} = -i \hbar (x\frac{\partial \psi}{\partial y} - y\frac{\partial \psi}{\partial x}
The Attempt at a Solution
x\frac{\partial...
Homework Statement
V(x) = \frac{1}{2}mw^{2}x^{2} + \lambdax^{4}
Using first-order perturbation theory to calculate the energy shift of:
1. The ground state:
\psi_{0}(x) = (2\pi\sigma)^{\frac{-1}{4}}\exp(\frac{-x^{2}}{4\sigma})
of the harmonic oscillator, where...
Homework Statement
Homework Equations
Stated in the question.
The Attempt at a Solution
It is a eigenfunction of L_z as it has no dependance on Z? Not sure if I can just state this, I do need to actually prove it but I can't get the calculations to work.
I managed a similar...