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

The position uncertainty of the mosquito is at least \(3.5 \times 10^{-27} \mathrm{m}\) and the position uncertainty of the proton is at least \(3.16 \times 10^{-12} \mathrm{m}\).

Step-by-step solution

(a) Calculate the momentum uncertainty of the mosquito

First, we need to find the momentum uncertainty of the mosquito. We can do this using Δp = m * Δv, where m is the mass (1.50 mg) and Δv is the speed uncertainty (0.01 m/s). Convert the mass to kg: \(m = 1.50 \times 10^{-6} \mathrm{kg}\) Now, calculate the momentum uncertainty: \(\Delta p = (1.50 \times 10^{-6})(0.01) = 1.5 \times 10^{-8} \mathrm{kg \cdot m/s}\)

(a) Calculate the position uncertainty of the mosquito

Now, we will use Heisenberg's uncertainty principle to calculate the position uncertainty, Δx, of the mosquito: Δx * Δp ≥ ħ / 2 Here, Δp is 1.5 × 10⁻⁸ kg m/s, and ħ = h / 2π ≈ 1.0545718 × 10⁻³⁴ J s. Solve for Δx: \(\Delta x \geq \frac{ħ}{2 \Delta p} = \frac{1.0545718 \times 10^{-34}}{2 \times 1.5 \times 10^{-8}} \approx 3.5 \times 10^{-27} \mathrm{m}\) So, the position uncertainty of the mosquito is at least 3.5 × 10⁻²⁷ m.

(b) Calculate the momentum uncertainty of the proton

First, we need to find the momentum uncertainty of the proton. We can do this using Δp = m * Δv, where m is the mass (1.67 × 10⁻²⁷ kg) and Δv is the speed uncertainty (0.01 × 10⁴ m/s). Now, calculate the momentum uncertainty: \(\Delta p = (1.67 \times 10^{-27})(0.01 \times 10^4) = 1.67 \times 10^{-23}\mathrm{kg \cdot m/s}\)

(b) Calculate the position uncertainty of the proton

Now, we will use Heisenberg's uncertainty principle to calculate the position uncertainty, Δx, of the proton: Δx * Δp ≥ ħ / 2 Here, Δp is 1.67 × 10⁻²³ kg m/s, and ħ = 1.0545718 × 10⁻³⁴ J s. Solve for Δx: \(\Delta x \geq \frac{ħ}{2 \Delta p} = \frac{1.0545718 \times 10^{-34}}{2 \times 1.67 \times 10^{-23}} \approx 3.16 \times 10^{-12} \mathrm{m}\) So, the position uncertainty of the proton is at least 3.16 × 10⁻¹² m.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It challenges classical notions of determinism and certainty by introducing probability as a core aspect of physical laws.

At its heart are the wave functions that determine the probabilities of finding a particle in a particular state. For instance, electrons in an atom do not orbit the nucleus in neat, predictable paths as planets do around a star; instead, their positions are described by a probability distribution. Measurement in quantum mechanics doesn't yield a single definitive result but a range of outcomes, each with its own probability.

It's worth keeping in mind that quantum mechanics is not merely theoretical; countless technologies such as smartphones, lasers, and MRIs all rely on quantum principles. The behavior it predicts seems counterintuitive because it's vastly different from the experiences of our everyday macroscopic world.

Momentum Uncertainty

Momentum uncertainty is a fundamental concept derived from Heisenberg's uncertainty principle within quantum mechanics. It expresses the idea that the more precisely we measure the momentum of a particle, the less precisely we can know its position at that moment, and vice versa.

This isn't due to the inadequacy of our instruments but is an intrinsic property of the universe. When we talk about the 'momentum uncertainty' (noted as \(\Delta p\)), we're referring to the extent to which the momentum of a particle is unknown. In practical terms, for an object moving with a known speed (but with some uncertainty), its momentum uncertainty can be calculated by multiplying the mass of the object by the uncertainty in its velocity. An interesting point to note is that momentum and position uncertainties are not independent; they're bound together by Heisenberg's relational expression: \(\Delta x \cdot \Delta p \geq \hbar / 2\), where \(\hbar\) is the reduced Planck constant.

Position Uncertainty

Position uncertainty in quantum mechanics refers to the indeterminacy in pinpointing the exact location of a particle. This uncertainty, symbolized as \(\Delta x\), is not just a measurement problem but rather an indication of a fundamental limit to how well we can define the position of any object at a quantum level.

According to Heisenberg's uncertainty principle, attempting to reduce the uncertainty in a particle’s momentum leads to an increased uncertainty in its position, thus highlighting the unpredictable behavior of particles in the quantum realm. For everyday objects, like a car or a baseball, these uncertainties are inconsequential and go unnoticed. However, for particles like electrons or protons, position uncertainty becomes significant and has notable implications for the understanding and development of technology at the nano-scale.

Consequently, experiments in quantum mechanics are designed to accommodate this lack of precision and instead focus on the probabilities of where a particle might be found.

Physical Chemistry

Physical chemistry is a branch of chemistry that deals with the application of the techniques and theories of physics to the study of chemical systems. It provides an in-depth understanding of fundamental physical principles governing the molecular and macroscopic levels of material.

Heisenberg's uncertainty principle is one of the many concepts from quantum mechanics that is influential in physical chemistry. This principle, for example, explains why we cannot determine the path of an electron in an atom, which consequently defines how chemical bonds form or why certain reactions occur.

A core part of physical chemistry involves using quantum mechanics to explain the behavior of atoms and molecules, and the uncertainty principle plays a crucial role in understanding reaction mechanisms, spectroscopy, and the rate of chemical processes. Physical chemists rely on this principle to explain observations that cannot be reconciled with classical chemistry.