If uncertainty in position and velocity are equal, then uncertainty in momentum will be

Question

If uncertainty in position and velocity are equal, then uncertainty in momentum will be (A) (1/2) √(m h/π) (B) (1/2) √(h/π m ) (C) (h/4 π m) (D) (m h/4 π)

Tardigrade Admin · Accepted Answer

Δ x=Δ v Δ x × Δ p ≥ (h/4 π) Δ x × m ⋅ Δ v=(h/4 π) (Δ v)2=(h/4 π m)(∵ Δ x=Δ v) ∴ Δ p=m ⋅ Δ v =m √(h/4 π m)=√(m h/4 π) Δ p=(1/2) √(m h/π)

Q. If uncertainty in position and velocity are equal, then uncertainty in momentum will be

$\frac{1}{2} \frac{mh}{π}$

$\frac{1}{2} \frac{h}{πm}$

$\frac{h}{4 πm}$

$\frac{mh}{4 π}$

$Δ x = Δ v$
$Δ x \times Δ p \geq \frac{h}{4 π}$
$Δ x \times m \cdot Δ v = \frac{h}{4 π}$
$(Δ v)^{2} = \frac{h}{4 πm} (∵ Δ x = Δ v)$
$∴ Δ p = m \cdot Δ v$
$= m \frac{h}{4 πm} = \frac{mh}{4 π}$
$Δ p = \frac{1}{2} \frac{mh}{π}$