Explain how,according to Born's probability interpretation,a wave having a single (unique) wavelength is extended all over space.

(N/A) According to the de Broglie hypothesis,a particle with a definite momentum $p$ is associated with a matter wave of a single,unique wavelength $\lambda = h/p$.
Mathematically,this wave is represented by a plane wave function: $\psi(x, t) = A e^{i(kx - \omega t)}$.
Born's probability interpretation states that the probability density of finding a particle at any point in space is given by $|\psi(x, t)|^2$.
For a plane wave,$|\psi(x, t)|^2 = |A e^{i(kx - \omega t)}|^2 = |A|^2$.
Since $|A|^2$ is a constant value independent of the position $x$,the probability of finding the particle is the same everywhere in space.
This implies that a particle with a perfectly defined momentum (and thus a single wavelength) is completely delocalized,meaning it is spread out over all space.