Calculate the de Broglie wavelength of a 143-g baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?

Verified step by step guidance

1. Convert the mass of the baseball from grams to kilograms by dividing by 1000.
2. Convert the speed of the baseball from miles per hour (mph) to meters per second (m/s) using the conversion factor: 1 mph = 0.44704 m/s.
3. Use the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \), \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second.
4. Substitute the values of \( h \), \( m \), and \( v \) into the de Broglie wavelength formula to find \( \lambda \).
5. Discuss why the wave nature of matter is not important for a baseball: The de Broglie wavelength is extremely small compared to the size of the baseball, making wave properties negligible in macroscopic objects.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is calculated using the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This wavelength becomes significant for microscopic particles, such as electrons, but is negligible for macroscopic objects like a baseball.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity (p = mv). It is a vector quantity, meaning it has both magnitude and direction. In the context of the de Broglie wavelength, momentum is crucial because it directly influences the wavelength; larger momentum results in a shorter wavelength, making wave properties less observable in larger objects.

Wave-Particle Duality

Wave-particle duality is a fundamental principle of quantum mechanics stating that every particle exhibits both wave-like and particle-like properties. While this duality is significant for small particles, such as photons and electrons, it is not observable in larger objects like a baseball due to their relatively large mass and momentum, which result in extremely short de Broglie wavelengths that are practically unnoticeable.