Two identical non-relativistic particles move at right angles to each other and have de Broglie wavelengths λ₁ and λ₂. Find the de Broglie wavelength of each particle in the centre-of-mass frame.
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Question

Two identical non-relativistic particles move at right angles to each other and have de Broglie wavelengths λ₁ and λ₂. Find the de Broglie wavelength of each particle in the centre-of-mass frame.

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G1 Physics · Accepted Answer

Step 1: For momenta p₁=h/λ₁ and p₂=h/λ₂ at right angles, the centre-of-mass velocity removes half the vector sum from each identical particle. Step 2: The momentum magnitude of either particle in the CM frame is |p₁−p₂|/2 = (h/2)√(1/λ₁²+1/λ₂²). Step 3: Hence λCM = 2λ₁λ₂/√(λ₁²+λ₂²). The source key prints 2λ₁λ₂/(λ₁²+λ₂²); verify against the original convention if required.