Calculate the de Broglie wavelength of a 5.005.00-g bullet that is moving at 340340 m/s. Will the bullet exhibit wavelike properties?

Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problema 9

Capitolo 39, Problema 9

Calculate the de Broglie wavelength of a $5.00$ -g bullet that is moving at $340$ m/s. Will the bullet exhibit wavelike properties?

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Step 1: Recall the de Broglie wavelength formula:

λ = \frac{h}{p}

, where

λ

is the wavelength,

h

is Planck's constant (

6.626 *⁢ 10

^-34 J⋅s), and

p

is the momentum of the object.

Step 2: Calculate the momentum

p

of the bullet using the formula

p = m v

, where

m

is the mass of the bullet (5.00 g = 0.00500 kg) and

v

is its velocity (340 m/s).

Step 3: Substitute the calculated momentum

p

into the de Broglie wavelength formula:

λ = \frac{h}{p}

. This will give the de Broglie wavelength of the bullet.

Step 4: Compare the calculated wavelength to the typical size of objects or wavelengths in the macroscopic world. If the wavelength is extremely small (on the order of 10^-34 m or smaller), the bullet will not exhibit noticeable wavelike properties.

Step 5: Conclude whether the bullet exhibits wavelike properties based on the comparison in Step 4. Typically, macroscopic objects like bullets have de Broglie wavelengths that are too small to observe wavelike behavior.

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de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the particle. For a moving object, momentum is calculated as the product of its mass and velocity (p = mv). This concept illustrates the wave-particle duality of matter.

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Wave-Particle Duality

Wave-particle duality is a principle in quantum mechanics that posits that every particle or quantum entity exhibits both wave-like and particle-like properties. This means that particles such as electrons and even larger objects can show interference and diffraction patterns, which are characteristic of waves. Understanding this duality is crucial for analyzing phenomena at the quantum level, including the behavior of macroscopic objects like bullets under certain conditions.

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Quantum Effects in Macroscopic Objects

Quantum effects typically dominate at the atomic and subatomic levels, but they become negligible for larger, macroscopic objects. In the case of a 5.00-g bullet, its mass and velocity result in a de Broglie wavelength that is exceedingly small, making any wavelike properties imperceptible. This concept helps to explain why everyday objects do not exhibit noticeable quantum behavior, as their wavelengths are far too small to influence their classical motion.

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