An electron is confined to a one-dimensional infinite square well occupying with nm

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Quantum dynamics and its formulations

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Stationary states and bound states in one dimension

Problem

An electron is confined to a one-dimensional infinite square well occupying 0<x<a0 < x < a with a=0.80a = 0.80 nm, the potential being zero inside and infinite outside. Its energy eigenfunctions are ψn(x)=2/a sin⁡(nπx/a)\psi_n(x) = \sqrt{2/a}\,\sin(n\pi x/a). At t=0t = 0 the wave function isΨ(x,0)=35 ψ1(x)+45 ψ3(x).\Psi(x,0) = \tfrac{3}{5}\,\psi_1(x) + \tfrac{4}{5}\,\psi_3(x).Take ℏ2/2me=0.0381\hbar^2/2m_e = 0.0381 eV nm2^2, and recall from Chapter 4 that ⟨x⟩=∫x ∣Ψ(x,t)∣2 dx\langle x\rangle = \int x\,|\Psi(x,t)|^2\,dx. ((a)) Calculate E1E_1 and E3E_3 in eV. ((b)) Write Ψ(x,t)\Psi(x,t), and give the probability that a measurement of the energy returns E3E_3 and the expectation value ⟨H⟩\langle H\rangle in eV. ((c)) Find ⟨x⟩\langle x\rangle at time tt, in nm. ((d)) Chapter 3 estimates the least energy of a particle confined to a region of width aa by taking Δx=a\Delta x = a and Δp=ℏ/(2Δx)\Delta p = \hbar/(2\Delta x), giving Eest=(Δp)2/2mE_{\text{est}} = (\Delta p)^2/2m. Evaluate EestE_{\text{est}} in eV and the factor E1/EestE_1/E_{\text{est}}.

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