Particle in a box: You want to have an electron in an energy level where its speed is no more than 66 m/s. What is the length of the smallest box (an infinite well) in which you can do this? (h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg) A) 5.5 µm B) 11 µm C) 2.8 µm D) 1.4 µm

Question

Quizplus · Accepted Answer

The de Broglie wavelength of the electron must be less than or equal to twice the length of the box. The de Broglie wavelength is given by h/p, where h is Planck's constant and p is the momentum of the electron. The momentum of the electron is given by mv, where m is the mass of the electron and v is its speed. Substituting these expressions into the de Broglie wavelength equation, we get:λ = h/mvSetting this equal to twice the length of the box, we get:2L = h/mvSolving for L, we get:L = h/2mvSubstituting the given values, we get:L = (6.626 × 10^-34 J ∙ s)/(2 × 9.11 × 10^-31 kg × 66 m/s) = 5.5 µmTherefore, the length of the smallest box in which you can do this is 5.5 µm.

Particle in a Box: You Want to Have an Electron