Chem 475 Fall 2012. HW 1 Classical Mechanics, Hamiltonian Formalism, Probabilities and Uncertainties...

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Chem 475 Fall 2012. HW 1 Classical Mechanics, Hamiltonian Formalism, Probabilities and Uncertainties ? Due : Friday Sept, XXXXXXXXXXnoon) via pdf submission ? 1. Classical 1D Harmonic Oscillator ? A. Consider a particle with mass, m, constrained to move in the ± x direction (equilibrium position x = 0). The force in the 'x' direction is - k·x, where k is the force constant in (SI) units of N/m. What is the potential energy function? ? B. Write the classical Hamiltonian function for the 1D harmonic oscillator, and set up the differential equations of motion for x • (t), p • (t). ? C. Use DSolve (or other available tools) to solve the differential equations in part B to get the x(t), p(t) solutions; your answer should be in the form of periodic functions with characteristic frequency ? = k m ? D. (Optional) Write a "Manipulate" statement that lets you visualize the x(t), p(t) trajectories as a function of total energy. Let k = 1 N/m, and m = 0.001 kg. You will need to specify initial conditions. Hint: there are multiple ways of doing this; the most straightforward way is to make a "phase-space" plot, showing the momentum (at time t) as a function of position (at time, t) ? 2. Classical Uncertainties and Probabilities ? A. Suppose we have an UNnormalized probability distribution function (PDF) of the variable "x" given by: P HxL = A × SinB 2 ? x L F 2 where P(x) = 0 for x  L. Find the value of A that normalizes this PDF. ? B. Show that the average value of  is L/2. ? C. Compute the variance and uncertainty in the variable x. ? 3. Davisson-Germer Experiment ? A. Electrons leaving a “gun” with voltage V have a mean kinetic energy given in electron-volt units as q·V (q is the charge in coulombs of the electron). In the Davisson-Germer experiment, electrons are scattered off steps in a Ni crystal separated by d = 0.091 nm. Bragg’s law of diffraction says that constructive interference occurs for specific angles T that satisfy: n ? = 2 d SinB 1 2 H? - TLF where n is an integer (the diffraction “order”). If the electron detector is tuned to an angle of T = 50°, find the voltage of the electron gun needed to match the diffraction condition. That is, use the DeBroglie hypothesis to find the electron ‘wavelength’ as a function of tip voltage

Robert · Accepted Answer

1. Classical 1D Harmonic Oscillator 
 
When a particle oscillates about its equilibrium position under the action of a linear force which is directed 
towards its equilibrium position and is proportional to the displacement at any instant (i.e. the force acts to 
oppose its displacement), the motion of the particle is said to be simple harmonic and the oscillating particle 
is called as a Simple Harmonic Oscillator (SHO)  
 
A particle of mass m on a spring having spring constant or force constant k is acted upon by a restoring force 
F =-kx. According to Newton’s law
 
where k = mω2 , ω = The angular frequency of the oscillator
 
The general solution of the equation is
    x(t) = Acos(ωt) + Bsin(ωt)  
 
Constants A and B can be determined from the initial conditions. The particle performs  
simple harmonic oscillations with frequency ν where ν is related to ω by
A. The potential energy is defined as 
   ∫     

    ∫       

 
 
 

B. The Hamiltonian H is defined to be the sum of the kinetic and potential energies: 
 
 H= K+ V 
 
Here the Hamiltonian should be expressed as a function of position x and momentum p , so that H = H(x, p).

Chem 475 Fall 2012. HW 1 Classical Mechanics, Hamiltonian Formalism, Probabilities and Uncertainties ? Due : Friday Sept, XXXXXXXXXXnoon) via pdf submission ? 1. Classical 1D Harmonic Oscillator ? A....

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