Two identical nonrelativistic particles move at right angles to each other, possessing de Broglie wavelengths, ${\lambda }_1$ and ${\lambda }_2$. Find the de Broglie wavelength of each particle in the frame of their centre of mass.

Initial momentum of each particle is

${\overrightarrow{p}}_1=\left(\frac{h}{{\lambda }_1}\right)\hat{i}$ and 

${\overrightarrow{p}}_2=\left(\frac{h}{{\lambda }_2}\right)\hat{j}$

$\Rightarrow {\overrightarrow{v}}_1=\left(\frac{h}{m{\lambda }_1}\right)\hat{i}\text{ and }{\overrightarrow{v}}_2=\left(\frac{h}{m{\lambda }_2}\right)\hat{j}$

Velocity of center of mass is gives as

${\overrightarrow{v}}_{cm}=\frac{m_1{\overrightarrow{v}}_1+m_2{\overrightarrow{v}}_2}{m_1+m_2}=\frac{h}{2m}\left(\frac{1}{{\lambda }_1}\hat{i}+\frac{1}{{\lambda }_2}\hat{j}\right)$

Momentum of particles in frame of center of mass is

${\overrightarrow{p}}_{1c}=m\left({\overrightarrow{v}}_1-{\overrightarrow{v}}_{cm}\right)=\frac{h}{2}\left(\frac{1}{{\lambda }_1}\hat{i}-\frac{1}{{\lambda }_2}\hat{j}\right)$

and 

${\overrightarrow{p}}_{2c}=m\left({\overrightarrow{v}}_2-{\overrightarrow{v}}_{cm}\right)=\frac{h}{2}\left(\frac{1}{{\lambda }_2}\hat{j}-\frac{1}{{\lambda }_1}\hat{i}\right)$

$\left|{\overrightarrow{p}}_{lc}\right|=\left|{\overline{p}}_{2c}\right|=\frac{h}{2}\frac{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}{{\lambda }_1{\lambda }_2}$

de Broglie wavelength of particles in frame of their centre of mass is

${\lambda }_{1c}=\frac{h}{p_{1c}}=\frac{2{\lambda }_1{\lambda }_2}{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}$

and

${\lambda }_{2c}=\frac{h}{p_{2c}}=\frac{2{\lambda }_1{\lambda }_2}{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}$