A nonrelativistic proton is confined to a length of 2.0 pm on the x-axis.What is the kinetic energy of the proton if its speed is equal to the minimum uncertainty possible in its speed? ,h = 1.055 × 10^-34 J • s,m_proton = 1.67 × 10^-27 kg) A) 0.13 eV B) 1.3 eV C) 13 eV D) 130 eV E) 1300 eV

Question

Quizplus · Accepted Answer

The minimum uncertainty possible in the speed of the proton is given by:

Δv ≥ h/(4πmΔx)

where Δx is the uncertainty in the position (given as 2.0 pm = 2.0 × 10^-12 m), m is the mass of the proton, and h is Planck's constant. Plugging in the values, we get:

Δv ≥ (1.055 × 10^-34 J • s) / (4π × 1.67 × 10^-27 kg × 2.0 × 10^-12 m)
Δv ≥ 9.56 × 10^6 m/s

This is the minimum uncertainty possible in speed. We know that the speed of the proton is equal to this minimum uncertainty, so we can use the formula for kinetic energy:

K = (1/2)mv^2

where m is the mass of the proton and v is the minimum uncertainty in speed. Plugging in the values, we get:

K = (1/2) × 1.67 × 10^-27 kg × (9.56 × 10^6 m/s)^2 
K ≈ 1.3 eV

Therefore, the correct choice is B) 1.3 eV.

A Nonrelativistic Proton Is Confined to a Length of 2