The uncertainty in the position of a particle is equal to the de Broglie wavelength of the particle. Calculate the uncertainty in the velocity of the particle in term of the velocity of the de Broglie wave associated with the particle.

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Here is given,

Uncertainty in the position of a particle say ∆x = de-Broglie wavelength of the particle (λ)

Uncertainty in the velocity (∆v) = ?

We know that, from the uncertainty principle,

\(∆x ∆p \geq \frac{h}{2x}\)

or, ∆x ∆p = \(\frac{h}{2 \pi ∆v}\)

or, ∆x ∆p = \(\frac{h}{2 \pi m ∆v}\)               ——– (1)

Again, From the de-Broglie hypothesis,

\(λ = \frac{h}{mv}\)                ——– (2)

According to the question, from the equation (1) and (2)., we get

\(\frac{h}{2\pi m ∆v} = \frac{h}{mv}\)

\(∆v = \frac{v_{wave}}{2\pi}\)

Hence, Uncertainty in velocity is equal to the \(\frac{1}{2\pi}\) times velocity of the de-Broglie wave.

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