An electron and a neutron have the same de Broglie wavelength. Which statement is true?
A) The electron and neutron have the same kinetic energy, but the electron has the higher speed.
B) The neutron has more kinetic energy but the two have the same speed.
C) The electron has more kinetic energy and a higher speed.
D) The electron has less kinetic energy and a lower speed.
E) The electron has less kinetic energy but a higher speed.

Question

Quizplus · Accepted Answer

The de Broglie wavelength of a particle is given by λ = h/p, where h is Planck's constant and p is the momentum of the particle. Since the de Broglie wavelength is the same for both the electron and neutron, we can write:

λ = h/mev_e = h/mnvc

where me and ve are the mass and velocity of the electron, mn and v are the mass and velocity of the neutron, and c is the speed of light. Rearranging this equation and squaring both sides, we get:

mv^2 = (h^2/(λ^2)(c^2))(1/m^2 + 1/m^2e)

Since λ is constant, the right-hand side of this equation is a constant, which means that the kinetic energy of the particles is proportional to their mass and velocity squared. Since the mass of the neutron is much greater than that of the electron, we can conclude that the electron must have a higher kinetic energy and a higher speed. Therefore, the correct answer is C.

An Electron and a Neutron Have the Same De Broglie