Find the ratio of the de Broglie wavelengths associated with an alpha particle and a proton, if both (a) have the same speeds, (b) have the same…

$\displaystyle \lambda = \dfrac{h}{p}$

\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{m_\alpha v_\alpha} \times \frac{m_p v_p}{h} = \frac{1}{4}\]

$\displaystyle p = \sqrt{2m(K.E)}$

\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{\sqrt{2m_\alpha (\mathrm{K.E.})_\alpha}} \times \frac{\sqrt{2m_p (\mathrm{K.E.})_p}}{h} = \sqrt{\frac{m_p}{m_\alpha}} = \frac{1}{2}\]

$\displaystyle v = \sqrt{\dfrac{2qV}{m}}$

\[\frac{\lambda_\alpha}{\lambda_p} = \frac{h}{m_\alpha v_\alpha} \times \frac{m_p v_p}{h} = \frac{m_p}{m_\alpha}\sqrt{\frac{2q_p V}{m_p}} \times \sqrt{\frac{m_\alpha}{2q_\alpha V}}\]

\[= \frac{m_p}{m_\alpha} \times \sqrt{\frac{m_\alpha}{m_p}} \times \sqrt{\frac{q_p}{q_\alpha}} = \frac{1}{2\sqrt{2}}\]