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Question
A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values λ1, λ2 with λ1 > λ2. Which of the following statement are true?
- The particle could be moving in a circular orbit with origin as centre.
- The particle could be moving in an elliptic orbit with origin as its focus.
- When the de Broglie wavelength is λ1, the particle is nearer the origin than when its value is λ2.
- When the de Broglie wavelength is λ2, the particle is nearer the origin than when its value is λ1.
Options
b and d
a and c
b, c and d
a, c and d
Solution
b and d
Explanation:
According to the question, here given that the de-Broglie wavelength of the particle can be varying cyclically between two values λ1 and λ2, it is possible if particle is moving in an elliptical orbit with origin as its one focus.
As shown in the figure given alongside,
Let v1 and v2 be the speed of particle at A and B respectively and origin is at focus O. If λ1 and λ2 are the de-Broglie wavelengths associated with particle while moving at A and B respectively, then `λ_1 = h/(mv_1)`
And `λ_2 = h/(mv_2)`
∴ `λ_1/λ_2 = v_2/v_1`
Since `λ_1 > λ_2`
∴ `v_2 > v_1`
By the law of conservation of angular momentum, the particle moves faster when it is closer to focus.
From figure, we note that origin O is closed to P than A.
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