An electron is in the n = 3 orbit of a hydrogen-like atom of atomic number Z. A neutron with thermal energy kᴮT has the same de Broglie wavelength as the electron. If T = Z²h²/(απ²a₀²mᴺkᴮ), find α.
Physics · Photoelectric effect and dual nature of matter · debrogllie duel nature of light
JEE Advanced 2025 Paper 1

Question

An electron is in the n = 3 orbit of a hydrogen-like atom of atomic number Z. A neutron with thermal energy kᴮT has the same de Broglie wavelength as the electron. If T = Z²h²/(απ²a₀²mᴺkᴮ), find α.

Physics · Photoelectric effect and dual nature of matter · debrogllie duel nature of light

G1 Physics · Accepted Answer

For the n-th Bohr orbit,

pₑrₙ = nh/(2π), rₙ = n²a₀/Z.

Hence pₑ = Zh/(2πna₀).

For n = 3: pₑ = Zh/(6πa₀).

Equal de Broglie wavelength means pₙ = pₑ.

Given neutron thermal energy kᴮT = pₙ²/(2mᴺ),

T = pₑ²/(2mᴺkᴮ) = Z²h²/(72π²a₀²mᴺkᴮ).

Comparing with the given form gives α = 72.