The position and momentum of deuteron overline ( 1 H ^ 2 ) moving with velocity of 8.35 * 10 ^ s * m / s are measured simultaneously. If its position is measured within 0.5 Å, what is the percentage of uncertainty in the momentum?

To solve this problem, we need to use the Heisenberg Uncertainty Principle, which states that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa. The principle is often expressed in the form of an inequality:

Δx * Δp ≥ h/4π

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant (6.62607015 × 10^-34 m^2 kg / s).

Given that the position is measured within 0.5 Å (or 0.5 * 10^-10 m), we can calculate the minimum uncertainty in momentum (Δp) as follows:

Δp = h / (4π * Δx) = 6.62607015 × 10^-34 m^2 kg / s / (4π * 0.5 * 10^-10 m) = 1.0545718 × 10^-24 kg m / s

The momentum (p) of the deuteron can be calculated using the formula p = m * v, where m is the mass of the deuteron (approximately 3.34 * 10^-27 kg) and v is its velocity (8.35 * 10^s m/s). However, the velocity given in the problem seems to be incorrect as it contains an unspecified variable 's'. Assuming 's' is a typo and the velocity is 8.35 m/s, the momentum would be:

p = 3.34 * 10^-27 kg * 8.35 m/s = 2.7889 * 10^-26 kg m / s

The percentage of uncertainty in the momentum can then be calculated as follows:

% uncertainty = (Δp / p) * 100% = (1.0545718 × 10^-24 kg m / s / 2.7889 * 10^-26 kg m / s) * 100% = 37.8%

So, the percentage of uncertainty in the momentum of the deuteron is approximately 37.8%.