Problem 47 - Chemistry The Central Science

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.)

Step-by-Step Solution

Answer

The uncertainty in position for the mosquito (\(\Delta x_m\)) is \(\geq 7.33 \times 10^{-32}~m\) and for the proton (\(\Delta x_p\)) is \(\geq 1.58 \times 10^{-10}~m\).

Step 1: Understand Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that there is an inherent uncertainty in the measurement of position and momentum of a particle. The principle is mathematically represented as follows: \(\Delta x \times \Delta p \geq \frac{h}{4\pi}\), where \(\Delta x\) is the uncertainty in position, \(\Delta p\) is the uncertainty in momentum, and h is Planck's constant.

Step 2: Calculate the uncertainty in momentum

We know that uncertainty in momentum is defined as the product of mass and uncertainty in speed: \(\Delta p = m \times \Delta v\) a) For the mosquito: Uncertainty in momentum of mosquito: \(\Delta p_m = m_{mosquito} \times \Delta v_{mosquito}\) \(= 1.50 \times 10^{-6}~kg \times 0.01~m/s\) b) For the proton: Uncertainty in momentum of proton: \(\Delta p_p = m_{proton} \times \Delta v_{proton}\) \(= (1.67 \times 10^{-27}~kg) \times (0.01 \times 10^4~m/s)\)

Step 3: Calculate the uncertainty in position

Now we will use Heisenberg's uncertainty principle to find the uncertainty in position, \(\Delta x\): \(\Delta x \geq \frac{h}{4\pi \times \Delta p}\) a) For the mosquito: \(\Delta x_m \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (\Delta p_m)}\) b) For the proton: \(\Delta x_p \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (\Delta p_p)}\)

Step 4: Calculation and results

Now, perform the calculations for each particle to find the minimum uncertainty in the position: a) Mosquito: \(\Delta x_m \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (1.50 \times 10^{-6}~kg \times 0.01~m/s)}\) \(\Delta x_m \geq 7.33 \times 10^{-32}~m\) b) Proton: \(\Delta x_p \geq \frac{6.626 \times 10^{-34}~J\cdot s}{4\pi \times (1.67 \times 10^{-27}~kg \times 0.01 \times 10^4~m/s)}\) \(\Delta x_p \geq 1.58 \times 10^{-10}~m\) The uncertainty in position for the mosquito is \(\geq 7.33 \times 10^{-32}~m\) and for the proton is \(\geq 1.58 \times 10^{-10}~m\).

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature on an atomic scale. It contrasts with classical physics by introducing concepts such as wave-particle duality and quantization of energy, which are not observed in the macroscopic world.

In classical physics, we can measure properties like position and momentum accurately. However, quantum mechanics reveals that at a microscopic level, particles like electrons do not have definite positions and speeds simultaneously. This challenge is beautifully encapsulated in Heisenberg's Uncertainty Principle. Quantum mechanics thus provides a greater understanding of atomic and subatomic processes by embracing inherent uncertainties.

• Describes nature at small scales, like atoms and photons.
• Introduces concepts different from classical mechanics, such as wave-particle duality.
• Allows for a probabilistic rather than deterministic view of particle behavior.

Uncertainty in Position

The uncertainty in position is a concept derived from Heisenberg's Uncertainty Principle, which reveals a fundamental limit to how precisely we can know a particle's position. Mathematically, it is represented as \[\Delta x \times \Delta p \geq \frac{h}{4\pi}\]where - \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum, and
- \(h\) is Planck's constant.This formula shows that decreasing the uncertainty in momentum will increase the uncertainty in position and vice versa. For large objects, like a baseball, this effect is negligible. However, for tiny particles, the uncertainty becomes significant.

• Larger uncertainty in one, leads to larger precision in the other.
• Significant in understanding atomic-scale behaviors.
• Cannot measure both position and momentum exactly.

Uncertainty in Momentum

Uncertainty in momentum is another core aspect of Heisenberg's Uncertainty Principle. This introduces a limitation to how accurately we can measure a particle's momentum without affecting its position.

Momentum is defined as the product of mass and velocity. In the context of Heisenberg's principle, the uncertainty in momentum \(\Delta p\) is calculated by multiplying the mass of the particle by the uncertainty in its velocity \(\Delta v\). This is expressed as:\[\Delta p = m \times \Delta v\]This relationship highlights how measuring a particle’s speed more precisely increases the uncertainty in its position. It is essential in systems with lightweight particles, like electrons or protons, where measuring changes influences the system significantly.

• Key in small particles where effects are measurable.
• Measurement changes become significant at quantum scales.
• Helps us understand particle behavior in quantum experiments.