A particle’s motion is restricted to a square of side 2 cm. If its mass is 50 g, then find theminimum uncertainty in its velocity.

To solve this problem, we need to use Heisenberg's Uncertainty Principle, which states that the product of the uncertainties in the position and momentum of a particle is always greater than or equal to half of Planck's constant. The formula is:

Δx * Δp ≥ h/4π

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

Given that the particle's motion is restricted to a square of side 2 cm (or 0.02 m), we can say that the maximum uncertainty in the particle's position is the length of the diagonal of the square, which is √2 * side. So, Δx = √2 * 0.02 m.

The uncertainty in momentum, Δp, is related to the uncertainty in velocity, Δv, by the equation Δp = m * Δv, where m is the mass of the particle. So, we can rewrite the uncertainty principle as:

Δx * m * Δv ≥ h/4π

Now we can solve for Δv:

Δv ≥ h / (4π * m * Δx)

Substituting the given values:

Δv ≥ (6.62607015 × 10^-34 m^2 kg / s) / (4π * 0.05 kg * √2 * 0.02 m)

After calculating the above expression, we get the minimum uncertainty in its velocity.