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?

1. The particle could be moving in a circular orbit with origin as centre.
2. The particle could be moving in an elliptic orbit with origin as its focus.
3. When the de Broglie wavelength is λ₁, the particle is nearer the origin than when its value is λ₂.
4. When the de Broglie wavelength is λ₂, the particle is nearer the origin than when its value is λ₁.

विकल्प

• b and d

• a and c

• b, c and d

• a, c and d

Answer 1

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.