Use the Heisenberg uncertainty principle to calculate the uncertainty in meters in the position of a honeybee weighing 0.68 g and traveling at a velocity of 0.85 m/s. Assume that the uncertainty in the velocity is 0.1 m/s.

Verified step by step guidance

1. Convert the mass of the honeybee from grams to kilograms by dividing by 1000, since the SI unit for mass is kilograms.
2. Use the Heisenberg uncertainty principle formula: \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \).
3. Calculate the uncertainty in momentum \( \Delta p \) using the formula \( \Delta p = m \cdot \Delta v \), where \( m \) is the mass of the honeybee in kilograms and \( \Delta v \) is the uncertainty in velocity.
4. Rearrange the Heisenberg uncertainty principle formula to solve for \( \Delta x \): \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \).
5. Substitute the values for \( h \), \( \Delta p \), and \( \pi \) into the equation to find the uncertainty in position \( \Delta x \).

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle is fundamental in quantum mechanics and implies that the more precisely one property is measured, the less precisely the other can be controlled or known. Mathematically, it is expressed as Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant.

Momentum and its Uncertainty

Momentum (p) is defined as the product of an object's mass (m) and its velocity (v), expressed as p = mv. The uncertainty in momentum (Δp) can be calculated by multiplying the mass of the object by the uncertainty in its velocity (Δv). In this case, for the honeybee, the uncertainty in momentum is crucial for determining the uncertainty in its position using the Heisenberg Uncertainty Principle.

Calculating Uncertainty in Position

To find the uncertainty in position (Δx) using the Heisenberg Uncertainty Principle, one must rearrange the formula to Δx = ħ/(2Δp). After calculating the uncertainty in momentum (Δp) from the mass and the given uncertainty in velocity, this value can be substituted into the equation to find Δx. This calculation provides insight into the limits of precision when measuring the position of the honeybee.