Question

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energies o P Q10M.2 Consider a nitrogen (N2) molecule bouncing around a box 10 cm on a side. Pretend that the molecule can only move in one dimension. Note that the mass of an N2 mol- ecule is roughly 28 times that of a proton. (a) What is the approximate value of n for the molecules energy eigenfunction if it has an energy E0.025 eV that one would expect from random thermal motion? (b) Estimate the ratio AE/En for this value of n, where AE is the energy difference between adjacent possible energies in the neighborhood of E, (c) Would the energies available to this molecule seem con- tinuous, as classical mechanics would predict? 010M 3 Imagine that the proton and electron in a hydro-

Answer 1

Given length of box, L = 10 cm on a side, total length of box, L^1 = 2L = 20 cm = 20 times 10^-2 m mass of N_2 molecule, 28 M_P = 28 times 1.6 times 10^-27 kg. As we know, energy of particle in L - 0 infinite potential well is, = h^2 pi^2 t^2/2mL^1^2 = (t = h/2 n with bar) Here given, = 0.025ev = 0.04 times 10^-19 J = > 0.04 times 10^-19 = n^2((13.14)^2 times (1.054 times 10^-34)/2 times 28 times 1.67 times 10^-27 times (0.20)^2) = > n = 1.1 times 10^10 = 1 times 10^10 = > n = 10^10 = (n + 1)^2/2mL^12 - n^2 pi^2 t^2/2mL^12/n^2 pi^2 t^2/2mL^12 = (n + 1)^2 - n^2/n^2 = (n + 1)^2/n^2 - 1 = n^2 + 1 + 2n - n^2/n^2 = 2n + 1/n^2 (2 times 10^10) + 1/10^20 almostequalto 2 times 10^-10 \r\n However, particular one - dimensional box is discrete but here particularly for