What is the uncertainty in position if the uncertainty in momentum of a particle is 1.0 x 10^-24 kg m/s?

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Q: What is the uncertainty in position if the uncertainty in momentum of a particle is 1.0 x 10^-24 kg m/s?

Solution: Using Heisenberg's uncertainty principle, ΔxΔp ≥ ħ/2, we find Δx ≥ ħ/(2Δp) = (1.055 x 10^-34)/(2 x 1.0 x 10^-24) = 5.27 x 10^-11 m.

Steps: 10

Step 1: Understand the problem. We need to find the uncertainty in position (Δx) given the uncertainty in momentum (Δp).
Step 2: Recall Heisenberg's uncertainty principle, which states that ΔxΔp ≥ ħ/2, where ħ (h-bar) is a constant.
Step 3: Identify the value of ħ. The value of ħ is approximately 1.055 x 10^-34 kg m²/s.
Step 4: Substitute the given value of Δp into the equation. We have Δp = 1.0 x 10^-24 kg m/s.
Step 5: Rearrange the uncertainty principle to solve for Δx: Δx ≥ ħ/(2Δp).
Step 6: Plug in the values: Δx ≥ (1.055 x 10^-34)/(2 x 1.0 x 10^-24).
Step 7: Calculate the denominator: 2 x 1.0 x 10^-24 = 2.0 x 10^-24.
Step 8: Now calculate Δx: Δx ≥ (1.055 x 10^-34)/(2.0 x 10^-24).
Step 9: Perform the division: Δx ≥ 5.27 x 10^-11 m.
Step 10: Conclude that the uncertainty in position is approximately 5.27 x 10^-11 m.