Dual Nature of Matter

JEE Main level, single correct, medium

79 of 200

Question 79 of 200

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de brogile wavelengths are λ1{\lambda }_{1} and λ2{\lambda }_{2}, their de brogile wavelength in the frame of reference attached to their centre of mass is:

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Answer: (B) λCM=2λ1λ2λ12+λ22{\lambda }_{CM}=\dfrac{2{\lambda }_{1}{\lambda }_{2}}{\sqrt{{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}}}

Momentum of each electron hλ1i^andhλ2j^\dfrac{h}{{\lambda }_{1}}\hat{i}and\dfrac{h}{{\lambda }_{2}}\hat{j}

Velocity of centre of mass

Vcm=h2mλ1i^+h2mλ2j^{V}_{cm}=\dfrac{h}{2m{\lambda }_{1}}\hat{i}+\dfrac{h}{2m{\lambda }_{2}}\hat{j}

Velocity of 1st{1}^{st} particle about centre of mass

Vcm=h2mλ1i^−h2mλ2j^{V}_{cm}=\dfrac{h}{2m{\lambda }_{1}}\hat{i}-\dfrac{h}{2m{\lambda }_{2}}\hat{j}

λcm=hh24λ12+h24λ22=2λ1λ2λ12+λ22{\lambda }_{cm}=\dfrac{h}{\sqrt{\dfrac{{h}^{2}}{4{\lambda }_{1}^{2}}+\dfrac{{h}^{2}}{4{\lambda }_{2}^{2}}}}=\dfrac{2{\lambda }_{1}{\lambda }_{2}}{\sqrt{{\lambda }_{1}^{2}+{\lambda }_{2}^{2}}}

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