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Matter Waves

Two particles...

Question

Two particles $A_{1}$ and $A_{2}$ of masses $m_{1}, m_{2} (m_{1} > m_{2})$ have the same de Broglie wavelength. Then

A

their momenta are the same.

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B

their energies are the same.

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C

momentum of

A_{1}

is less than momentum of

A_{2}

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D

energy of

A_{1}

is more than the energy of

A_{2}

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Solution

The correct option is A their momenta are the same.
As $λ = \frac{h}{p} o r p = \frac{h}{λ} o r p \propto \frac{1}{λ}$

$∴ \frac{p_{1}}{p_{2}} = \frac{λ_{2}}{λ_{1}} = \frac{λ}{λ} = 1 o r p_{1} = p_{2}$ ..............(1)

Also $E = \frac{1}{2} \frac{p^{2}}{m} = \frac{1}{2 m} \frac{h^{2}}{λ^{2}} (∴ p = \frac{h}{λ})$

or $E \propto \frac{1}{m} ∴ \frac{E_{1}}{E_{2}} = \frac{m_{2}}{m_{1}} < 1 o r E_{1} < E_{2}$ ................(2)
From above equation: Option A

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