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 ${\lambda }_1, {\lambda }_2$ with ${\lambda }_1>{\lambda }_2.$ Which of the following statement are true?

The de-Broglie wavelength of the particle can be varying cyclically between two values ${\lambda }_1,$ and ${\lambda }_2$ if particle is moving in an elliptical orbit with origin as its one focus.

Consider the figure,

Let v₁ , v₂ be the speed of particle at  A and B respectively and origin is at focus O. If ${\lambda }_1,$ ${\lambda }_2$ are the de- Broglie wavelengths associated with particle while moving at  A and B respectively. Then,

$\begin{array}{r}{\lambda }_1=\frac{h}{m{v}_1}\text{ and }{\lambda }_2=\frac{h}{m{v}_2}\\ \therefore \left|\frac{{\lambda }_1}{{\lambda }_2}=\frac{v_2}{v_1}\Rightarrow \right|\text{ since }{\lambda }_1>{\lambda }_2\\ \therefore v_2>v_1\end{array}$

By 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.