A 0.22-caliber handgun fires a 1.9-g bullet at a velocity of 765 m/s. Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?

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Identify the formula for the de Broglie wavelength: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) 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.

Convert the mass of the bullet from grams to kilograms by dividing by 1000. So, \( m = \frac{1.9}{1000} \text{ kg} \).

Substitute the values into the de Broglie wavelength formula: \( \lambda = \frac{6.626 \times 10^{-34}}{(1.9 \times 10^{-3}) \times 765} \).

Calculate the expression in the denominator: \( (1.9 \times 10^{-3}) \times 765 \).

Evaluate the de Broglie wavelength using the calculated denominator and discuss whether the wave nature of matter is significant for bullets by comparing the wavelength to typical macroscopic scales.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. For macroscopic objects like bullets, the wavelength is typically extremely small, making wave properties negligible in practical scenarios.

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 bullet, calculating its momentum is essential for determining the de Broglie wavelength, as it directly influences the wave properties of the bullet.

Wave-Particle Duality

Wave-particle duality is a fundamental principle of quantum mechanics stating that every particle exhibits both wave and particle characteristics. While this concept is crucial for understanding the behavior of subatomic particles, for larger objects like bullets, the wave nature is insignificant due to their large mass and the resulting minuscule de Broglie wavelength, which does not affect their motion or interactions.