Physics - Modern Physics Question with Solution | TestHub

Consider an electron in the $n=4$ orbit of a hydrogen-like atom with atomic number $Z$. At absolute temperature $T$, a neutron having thermal energy $k_{\mathrm{B}}T$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $\mathrm{T}=\frac{{\mathrm{Z}}^2{\mathrm{h}}^2}{\alpha {\pi }^2{\mathrm{a}}_0^2{\mathrm{m}}_{\mathrm{N}}{\mathrm{k}}_{\mathrm{B}}}$, (where $h$ is the Planck's constant, $k_B$ is the Boltzmann constant, $m_{\mathrm{N}}$ is the mass of the neutron and $a_0$ is the first Bohr radius of hydrogen atom) then the value of $\alpha$ is $\_\_\_\_$.