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    Initial Number Density of Ink Particles

    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...
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    Calculating Extrema on Surface of Sphere

    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...
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    Rectangle Inside An Ellipse

    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...
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    Calculating Stationary Points of a Function

    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...
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    Hamiltonian of a Spin in a Magnetic Field

    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...
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    Quantum Mechanics - Spin.

    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...
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    Yukawa Potential

    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...
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    QM: Angular Momentum Matrices (Rotating Molecule)

    Homework Statement For l=1 the angular momentum components can be represented by the matrices: \hat{L_{x}} = \hbar \left[ \begin{array}{ccc} 0 & \sqrt{\frac{1}{2}} & 0 \\ \sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \\ 0 & \sqrt{\frac{1}{2}} & 0 \end{array} \right] \hat{L_{y}} =...
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    Use of Fourier Transform in Quantum Mechanics

    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) =...
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    Fourier Transform of Differential Equation

    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) =...
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    Convolution - Image Processing

    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}...
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    Fourier Transform Calculations

    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...
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    Ordering of 'N' Distinguishable Objects

    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...
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    Deriving Helmholtz Thermodynamic Potential & Corresponding Maxwell Relation

    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...
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    Probability Calculation - Lottery

    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...
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    Perturbation of Potential (Particle in a Box)

    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...
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    Wave Function - Normalisation & Calculation of Expectation Values

    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...
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    Partition Function for Thermodynamic System

    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...
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    Thermodynamics - Change in Density due to Change in Height

    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...
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    Determining Eigenfunction of Operator

    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...
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    Anharmonic Oscillator - Energy Shift Calculation Using 1st Order Perturbation

    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...
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    Proving an Eigenfunction of Momentum Operators

    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...
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