Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are ${\lambda }_1$ and ${\lambda }_2$ , their de Broglie wavelength in the frame of reference attached to their centre of mass is

Momentum of two electrons are  ${\overrightarrow{p}}_1=\frac{h}{{\lambda }_1}\hat{\mathrm{i}}$  and  ${\overrightarrow{p}}_2=\frac{h}{{\lambda }_2}\hat{\mathrm{j}}$.

Velocity of centre of mass ${\overrightarrow{V}}_{CM}=\frac{{\overrightarrow{p}}_1+{\overrightarrow{p}}_2}{2m}=\frac{h}{2m{\lambda }_1}\hat{\mathrm{i}}+\frac{h}{2m{\lambda }_2}\hat{\mathrm{j}}$

Velocity of first electron about centre of mass is

$$\begin{gathered} {\overrightarrow{V}}_{1-CM}=\frac{{\overrightarrow{p}}_1}{m}-{\overrightarrow{V}}_{CM}=\frac{h}{2m{\lambda }_1}\hat{\mathrm{i}}-\frac{h}{2m{\lambda }_2}\hat{\mathrm{j}} \\ \Rightarrow {\lambda }_{1-CM}=\frac{h}{m\left|{\overrightarrow{V}}_{1-CM}\right|}=\frac{h}{\sqrt{\frac{h^2}{4{\lambda }_1^2}+\frac{h^2}{4{\lambda }_2^2}}}=\frac{2{\lambda }_1{\lambda }_2}{\sqrt{{\lambda }_1^2+{\lambda }_2^2}}={\lambda }_{2-CM} \\ \text{∴ }{\lambda }_{CM}=\frac{2{\lambda }_1{\lambda }_2}{\sqrt{{{\lambda }_1}^2+{{\lambda }_2}^2}} \end{gathered}$$