Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50-mg mosquito moving at a speed of 1.40 m/s if the speed is known to within {0.01 m/s;

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Identify the given values: mass of the mosquito \( m = 1.50 \times 10^{-3} \text{ g} = 1.50 \times 10^{-6} \text{ kg} \), speed \( v = 1.40 \text{ m/s} \), and uncertainty in speed \( \Delta v = 0.01 \text{ m/s} \).

Recall Heisenberg's 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 \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).

Calculate the uncertainty in momentum \( \Delta p \) using \( \Delta p = m \cdot \Delta v \).

Substitute \( \Delta p \) into the uncertainty principle formula to solve for \( \Delta x \): \( \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \).

Plug in the known values and solve for \( \Delta x \) to find the uncertainty in the position of the mosquito.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle states that it is impossible to simultaneously know both the exact position and momentum of a particle. The more accurately we know one of these values, the less accurately we can know the other. This principle is fundamental in quantum mechanics and highlights the limitations of measurement at microscopic scales.

Momentum

Momentum is defined as the product of an object's mass and its velocity (p = mv). In the context of the uncertainty principle, momentum is a key variable because it relates to the motion of the particle. For the mosquito in the question, its momentum can be calculated using its mass and the given speed, which is essential for determining the uncertainty in its position.

Uncertainty Calculation

To calculate the uncertainty in position (Δx) using the uncertainty principle, the formula Δx * Δp ≥ ħ/2 is used, where Δp is the uncertainty in momentum and ħ is the reduced Planck's constant. The uncertainty in momentum can be derived from the uncertainty in velocity (Δv) and the mass of the object. This calculation allows us to quantify the limits of our knowledge about the particle's position.

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Uncertainty Principle Formula

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