Question

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a \(1.50\)-mg mosquito moving at a speed of \(1.40 \mathrm{~m} / \mathrm{s}\) if the speed is known to within \(\pm 0.01 \mathrm{~m} / \mathrm{s}\); (b) a proton moving at a speed of \((5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}\). (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

Short answer

a) For the mosquito, we have \(m_{\text{mosquito}} = 1.50 \times 10^{-6}\text{ kg}\) and \(\Delta{v}_{\text{mosquito}} = 0.01\text{ m/s}\), so the uncertainty in momentum is \[\Delta{p}_{\text{mosquito}} = 1.50 \times 10^{-6}\text{ kg} \cdot 0.01\text{ m/s} = 1.50 \times 10^{-8}\text{ kg}\cdot\text{m/s}\]. Using Heisenberg's uncertainty principle, the uncertainty in position is \[\Delta{x}_{\text{mosquito}} \geq \frac{1.05 \times 10^{-34}\text{ Js}}{2(1.50 \times 10^{-8}\text{ kg}\cdot\text{m/s})} \approx 3.5 \times 10^{-27}\text{ m}\]. b) For the proton, we have \(m_{\text{proton}} = 1.67 \times 10^{-27}\text{ kg}\) and \(\Delta{v}_{\text{proton}} = 0.01 \times 10^{4}\text{ m/s}\), so the uncertainty in momentum is \[\Delta{p}_{\text{proton}} = 1.67 \times 10^{-27}\text{ kg} \cdot 0.01 \times 10^{4}\text{ m/s} = 1.67 \times 10^{-29}\text{ kg}\cdot\text{m/s}\]. Using Heisenberg's uncertainty principle, the uncertainty in position is \[\Delta{x}_{\text{proton}} \geq \frac{1.05 \times 10^{-34}\text{ Js}}{2(1.67 \times 10^{-29}\text{ kg}\cdot\text{m/s})} \approx 3.1 \times 10^{-6}\text{ m}\].

Step-by-step solution

Calculate the uncertainty in momentum

First, let's calculate the uncertainty in momentum for a) mosquito and b) proton. The uncertainty in momentum (Δp) can be calculated as the product of the mass, m, and the uncertainty in the speed, Δv: a) For the mosquito: \[\Delta{p}_{\text{mosquito}} = m_{\text{mosquito}} \Delta{v}_{\text{mosquito}}\] b) For the proton: \[\Delta{p}_{\text{proton}} = m_{\text{proton}} \Delta{v}_{\text{proton}}\]

Calculate the uncertainty in position using Heisenberg's uncertainty principle

Now that we have calculated the uncertainty in momentum, we can calculate the uncertainty in position (Δx) using Heisenberg's uncertainty principle equation: a) For the mosquito: \[\Delta{x}_{\text{mosquito}} \geq \frac{\hbar}{2 \Delta{p}_{\text{mosquito}}}\] b) For the proton: \[\Delta{x}_{\text{proton}} \geq \frac{\hbar}{2 \Delta{p}_{\text{proton}}}\] We'll finally calculate the values of Δx for both the mosquito and proton using the formulas above and the provided constant values.

Key concepts

Momentum Uncertainty

Heisenberg's Uncertainty Principle is a fundamental concept in quantum mechanics, positing that it's impossible to know both the position and momentum of a particle with absolute precision simultaneously. The uncertainty in momentum, denoted by \( \Delta p \), plays a crucial role in this principle. For a particle, momentum is the product of its mass and velocity. When we talk about uncertainty in momentum, we're interested in how much we can trust our measurement of this momentum. Essentially, if we know the speed of a particle but there's a bit of fuzziness in how precisely we know that speed, this translates into uncertainty in momentum. Momentum uncertainty is calculated using:

• Variation in Speed: If a mosquito's speed has an uncertainty of \( \pm 0.01 \rm{m/s} \), it signifies that there's a narrow range of possibilities for its actual speed.
• Mass: The mass of the particle, known precisely for these calculations, helps define the total momentum uncertainty.

Thus, for both the mosquito and the proton in the exercise, the uncertainty in their respective momentums is determined by multiplying the mass by the uncertainty in velocity. This step is vital as it sets the stage for understanding position uncertainty through Heisenberg's principle.

Position Uncertainty

Position uncertainty \( \Delta x \) is how we describe our uncertainty in knowing precisely where a particle is located. In physics, and particularly in quantum mechanics, this is a core part of the Heisenberg Uncertainty Principle. Once we have calculated momentum uncertainty, we can leverage this principle to find position uncertainty. The principle tells us there is a minimum limit to the product of the uncertainties of position and momentum:\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]Here, \( \hbar \) is the reduced Planck's constant, a fundamental constant of nature. This equation shows that as the certainty of momentum increases (making \( \Delta p \) smaller), the uncertainty of position \( \Delta x \) must increase, and vice versa. For the mosquito and proton example:

• Calculation: We rearrange the equation to solve for \( \Delta x \) using the previously computed \( \Delta p \).
• Outcome: This provides the smallest possible value for position uncertainty given the constraints - further underscoring the interplay between momentum and position in a quantum world.

This not only helps solve textbook problems but also illustrates a fundamental aspect of quantum behaviors.

Fundamental Constants

In understanding the principles behind momentum and position uncertainties, fundamental constants like Planck's constant (\( h \)) and its reduced form (\( \hbar = \frac{h}{2\pi} \)) are pivotal. These constants represent the scales at which quantum effects become significant and non-negligible compared to classical physics predictions.Key constants frequently encountered include:

• Planck's Constant (\( h \)): With a value of approximately \( 6.626 \times 10^{-34} \text{Js} \), this determines the smallest possible scales of quantum action.
• Reduced Planck's Constant (\( \hbar \)): This is defined as \( \hbar = \frac{h}{2\pi} \) and often appears in formulas like the Heisenberg Uncertainty Equation.

These constants don't just appear out of nowhere in physics formulas; they set the limits and scales of our understanding of quantum mechanics, dictating phenomena's behavior at microscopic levels and beyond. Acknowledging these constants is crucial for correctly applying Heisenberg's principle and similar quantum theories.