Electron Trapped - Introduction to Relativity and Quantum Mechanics - Problem Sets, Exercises of Quantum Mechanics

Download Electron Trapped - Introduction to Relativity and Quantum Mechanics - Problem Sets and more Exercises Quantum Mechanics in PDF only on Docsity! Group Problems #24 - Solutions Wednesday, October 26 Problem 1 Electron trapped in a 1D well An electron is trapped in a 1D region of length 10−10 m (a typical atomic diameter). (a) How much energy must be supplied to excite the electron from the ground state to the first excited state? The allowed energies are: En = n2~2π2 2mL2 = n2h2 8mL2 , (1) so the ground state energy is: E1 = h2 8mL2 = (hc)2 8(mc2)L2 = (1240 eV nm)2 8(511× 103 eV)(10−2 nm2) (2) (3) = 37.6 eV. (4) In the first excited state, the energy is 4E1. The difference, which must be supplied, is 3E1 = 113 eV. (b) In the ground state, what is the probability of finding the electron in the region from x = 0.09× 10−10 m to 0.11× 10−10 m? The spatial wavefunctions (eigenfunctions) are given by: ψn(x) = √ 2 L sin nπx L . (5) For the ground state (n = 1), the probability is the integral of the probability density, dP/dx = ψ∗1(x)ψ1(x), over the appropriate interval: P = ∫ x2 x1 |ψ|2 = 2 L ∫ x2 x1 sin2 πx L dx (6) = [ x L − 1 2π sin 2πx L ]x2 x1 (7) (8) = 0.0038 = 0.38%. (9) 1