X and Y are moving particles. The uncertainty in measure of momentum of Y is half than of X .if the uncertainty of position of X is +_0.05A°than that of Y is

To tackle the question about the uncertainties in the measurements of momentum and position for particles X and Y, we can apply the principles of quantum mechanics, specifically Heisenberg's Uncertainty Principle. This principle states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. The relationship can be expressed mathematically as:

Understanding the Uncertainty Principle

The Uncertainty Principle can be formulated as:

• Δx * Δp ≥ ħ/2

Where:

• Δx is the uncertainty in position.
• Δp is the uncertainty in momentum.
• ħ (h-bar) is the reduced Planck's constant, approximately 1.055 x 10^-34 Js.

Given Information

From your question, we know the following:

• The uncertainty in momentum of particle Y (Δp_Y) is half that of particle X (Δp_X): Δp_Y = 0.5 * Δp_X.
• The uncertainty in position of particle X (Δx_X) is ±0.05 Å (Angstroms).

Calculating the Uncertainty in Position for Particle Y

Using the Uncertainty Principle, we can express the uncertainties for both particles:

• For particle X: Δx_X * Δp_X ≥ ħ/2
• For particle Y: Δx_Y * Δp_Y ≥ ħ/2

Since we know that Δp_Y = 0.5 * Δp_X, we can substitute this into the equation for particle Y:

• Δx_Y * (0.5 * Δp_X) ≥ ħ/2

Rearranging this gives us:

• Δx_Y ≥ ħ / (Δp_X)

Now, we can relate Δp_X to Δx_X using the equation for particle X:

• Δx_X * Δp_X ≥ ħ/2

From this, we can express Δp_X as:

• Δp_X ≥ ħ / (2 * Δx_X)

Substituting this back into the equation for Δx_Y gives:

• Δx_Y ≥ ħ / (0.5 * (ħ / (2 * Δx_X)))

After simplifying, we find:

• Δx_Y ≥ 2 * Δx_X

Final Calculation

Now, substituting the value of Δx_X:

• Δx_Y ≥ 2 * 0.05 Å = 0.1 Å

This means that the uncertainty in the position of particle Y is at least ±0.1 Å. In summary, the relationship between the uncertainties in position and momentum for these two particles illustrates the fundamental limits imposed by quantum mechanics.