Find the uncertainty in the momentum (in kg . m/s)of an electron if the uncertainty in its position is equal to 3.4 *10 ^- ¹⁰ m,the circumference of the first Bohr orbit. A)6.2 *10 ^- ²⁵ B)1.6 *10 ^- ²⁵ C)15.5 * 10 ^- ²⁵ D)19.4 *10 ^- ²⁵ E)2.0 *10 ^- ²⁴

Question

Quizplus · Accepted Answer

The uncertainty in momentum is given by:
Δp = h/λ = h/(2πr)      (1)
where λ is the de Broglie wavelength of the electron, h is the Planck's constant and r is the radius of the Bohr orbit.
The circumference of the first Bohr orbit is given by:
2πr = 2π(0.529 × 10^-10 m) = 3.34 × 10^-10 m
The uncertainty in position is given as Δx = 3.4 × 10^-10 m
By Heisenberg's Uncertainty Principle, we have:
ΔpΔx ≥ h/4π
Δp ≥ h/4πΔx                    (2)
Substituting Eq. (1) in Eq. (2), we get:
Δp^2 ≥ (h^2/16π^2Δx^2)
Δp ≥ (h/4πΔx) = (6.63 × 10^-34 Js)/(4π × 3.4 × 10^-10 m) ≈ 1.6 × 10^-24 kg m/s
Therefore, the uncertainty in momentum is 1.6 × 10^-24 kg m/s, which corresponds to option B.

Find the Uncertainty in the Momentum (In Kg