Question

What is the de Broglie wavelength (in \(\mathrm{cm}\) ) of a 12.4-g hummingbird flying at \(1.20 \times 10^{2} \mathrm{mph}\) \((1\) mile \(=1.61 \mathrm{~km}) ?\)

Short answer

The de Broglie wavelength is approximately \(9.993 \times 10^{-32}\) cm.

Step-by-step solution

Convert Mass and Velocity to SI Units

First, we need to convert the mass of the hummingbird from grams to kilograms. Since there are 1000 grams in a kilogram, the mass \( m = \frac{12.4}{1000} \text{ kg} = 0.0124 \text{ kg} \). \\Next, we convert the velocity from miles per hour to meters per second. We use the conversion factors: 1 mile = 1.61 km and 1 km = 1000 m.\( \text{Velocity in m/s} = 1.20 \times 10^{2} \text{ mph} \times \frac{1.61 \times 1000}{3600} \approx 53.644 \text{ m/s} \).

Use de Broglie's Wavelength Formula

The de Broglie wavelength \(\lambda\) is given by the formula \( \lambda = \frac{h}{mv} \), where: - \( h = 6.626 \times 10^{-34} \text{ Js} \) (Planck's constant),- \( m = 0.0124 \text{ kg} \) (mass),- \( v \approx 53.644 \text{ m/s} \) (velocity).Substitute the values in to get: \[ \lambda \approx \frac{6.626 \times 10^{-34}}{0.0124 \times 53.644} \approx 9.993 \times 10^{-34} \text{ m} \].

Convert Wavelength to Centimeters

Convert the wavelength from meters to centimeters by using the conversion factor 1 m = 100 cm.Thus, \( \lambda \approx 9.993 \times 10^{-34} \text{ m} = 9.993 \times 10^{-32} \text{ cm} \).

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory used to understand the physical properties at microscopic scales. At these small scales, objects do not behave quite like they do in everyday life. Instead of moving in a predictable path, particles like electrons exhibit wave-like behavior, shattering the traditional Newtonian physics conventions.

The idea of wave-particle duality is a core concept within quantum mechanics. It suggests that every particle or quantum entity, such as photons or even larger objects like our 12.4-g hummingbird, can be described via wave-like properties. In the world of quantum mechanics, this property is highlighted through the de Broglie wavelength.

It's fascinating because it suggests that all moving objects have an associated wavelength. The faster an object moves, or the more massive it is, the smaller its associated wavelength. For something as large as a hummingbird in motion, the wavelength is almost negligible but nonetheless exists. This concept cemented an entirely new way of thinking about motion and interaction at microscopic scales.

Planck's Constant

Planck's constant, symbolized as \( h \), is a cornerstone of quantum mechanics, introducing fundamental limits on the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. Its value is approximately \( 6.626 \times 10^{-34} \text{ Js} \). The smallness of this constant emphasizes why quantum effects are often only significant at atomic scales.

This constant serves as a bridge between the particle- and wave-nature of matter. Without it, we would not have the grasp on energy quantization that laid the foundation for modern quantum physics. In explaining phenomena such as the color and intensity of light emitted by stars, Planck's constant proved essential.

• It facilitates the calculation of a particle's de Broglie wavelength, showing that as mass or velocity increases, the wavelength decreases—highlighting wave-particle duality in physical entities.
• Its inclusion in formulas like the de Broglie wavelength marks the threshold where classical mechanics gives way to quantum concepts.

Planck’s constant is therefore pivotal in defining the scales at which classical descriptions of matter break down. It introduces a fundamentally new layer of reality, refining our understanding of nature at the smallest scales.

SI Unit Conversion

SI unit conversion is an essential skill for carrying out scientific calculations correctly. It ensures that each component of a calculation uses a consistent system of measurements, allowing for the accurate application of mathematical formulas.

To illustrate, converting the hummingbird's mass and velocity from grams to kilograms and miles per hour to meters per second aligns them with the International System of Units (SI). This conversion is crucial for using equations like the de Broglie wavelength formula, which presumes inputs in these units.

• The mass of 12.4 g was converted to approximately 0.0124 kg.
• Velocity converted from 120 mph to around 53.644 m/s. This involves a few steps, including changing miles to meters and scaling the time unit, crucial for obtaining accurate results.

Finally, even after calculating the de Broglie wavelength in meters, converting it to centimeters ensures it matches the requested unit of measurement. Understanding and making these conversions seamlessly is fundamental in scientific investigation, helping ensure that results are reliable and easily interpreted across different scientific communities.