Chapter 3's Davisson-Germer electron has kinetic energy eV and de Broglie wavelength nm

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation and the Born rule

Problem

Chapter 3's Davisson-Germer electron has kinetic energy 54.054.0 eV and de Broglie wavelength λ=0.167\lambda=0.167 nm, giving wavenumber k=2π/λ=37.6 nm−1k=2\pi/\lambda=37.6\ \mathrm{nm^{-1}} in a field-free region, where its total energy is E=54.0E=54.0 eV. The electron now enters a second region where a battery holds the potential energy at a uniform, higher value V0=10.0V_0=10.0 eV; the electron's total energy EE does not change on crossing into this region. ((a)) Write the Schrödinger equation for the electron inside the second region. ((b)) Inside the second region a plane wave Ψ=ei(k′x−ωt)\Psi=e^{i(k'x-\omega t)} again solves the equation, now with E=ℏ2k′2/2me+V0E=\hbar^2k'^2/2m_e+V_0. Find k′k' in nm−1\mathrm{nm^{-1}}, using ℏ2/2me=0.0381\hbar^2/2m_e=0.0381 eV nm2^2. ((c)) Find the ratio k′/kk'/k, and say whether the electron's wavelength grows or shrinks on entering the region held at the higher potential.

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