Question

The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which

(a) an electron, and

(b) a neutron, would have the same de Broglie wavelength.

Answer 1

Wavelength of light of a sodium line, λ = 589 nm = 589 × 10⁻⁹ m

Mass of an electron, m_e= 9.1 × 10⁻³¹ kg

Mass of a neutron, m_n= 1.66 × 10⁻²⁷ kg

Planck’s constant, h = 6.6 × 10⁻³⁴ Js

(a) For the kinetic energy K, of an electron accelerating with a velocity v, we have the relation:

We have the relation for de Broglie wavelength as:

Substituting equation (2) in equation (1), we get the relation:

Hence, the kinetic energy of the electron is 6.9 × 10⁻²⁵ J or 4.31 μeV.

(b) Using equation (3), we can write the relation for the kinetic energy of the neutron as:

Hence, the kinetic energy of the neutron is 3.78 × 10⁻²⁸ J or 2.36 neV.