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Using time-independent perturbation theory find the first-order correction to the ground-state energy




Rutgers Camden - Department of Chemistry;Physical Chemistry I, 50:160:345;Fall 2014;Problem Set #3, Atkins Chapter 8;Distributed: 09-23-14.;Deadline: 09-30-14.;5 pt. max per Problem Set.;Equations and other relations are not given for the problem sets unless necessary, these can;be found in the textbook or through other resources.;1) An electron is confined to a square well of length L, and the walls of that well are infinitely;high. What would be the length of the box L such that the zero-point energy (ZPE) of the;electron located inside this well is equal to twice its rest mass energy, 2mec2? The ZPE is;defined as the minimal energy that corresponds to the smallest quantum number n.;2) a) Determine the energy of a photon emitted when an electron relaxes from the excited state;=3 to ground state =1 of a harmonic oscillator if the force constant is 285 N m1.;1;b) What is the wavelength of that emitted photon?;3) Calculate the energies E of the first four rotational levels (those giving the minimal energy);of 1H35Cl molecule that is free to rotate in three dimensions. Use for its moment of inertia I =;effR2, with eff = mH * mCl / (mH + mCl) and bond length R = 127 pm.;2;4) Eigenfunctions corresponding to different eigenvalues of an operator are orthogonal, which;means that, where i and j are the two orthogonal wavefunctions and i j.;Show that the 1D particle in a box solution follows this relationship by using its analytical;expressions. Hint: Use the identity;3;5) Using time-independent perturbation theory find the first-order correction to the ground-state;energy for a particle in a 1D well with a perturbation in the potential V(x) = 2b sin(x/L).;Potential well spans from x=0 to x=L. Hint: Use the expectation value for a particle in a box;and the integral;4


Paper#16014 | Written in 18-Jul-2015

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