What is the de Broglie wavelength of a 145-g baseball traveling at 30.0 m / s(67.1 mph) ? Is th

step 1 All right, so we're asked to calculate the wavelength of the baseball. Here's the De Brogley equ

What is the de Broglie wavelength of a 145-g baseball...

What is the de Broglie wavelength of a $145-\mathrm{g}$ baseball traveling at $30.0 \mathrm{~m} / \mathrm{s}(67.1 \mathrm{mph}) ?$ Is the wavelength much smaller or much larger than the diameter of an atom (on the order of $100 \mathrm{pm}$ )?

Key Concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that assigns a wavelength to any particle by relating its momentum to a wave property, using the formula ? = h/p. This idea underpins the wave-particle duality of matter, indicating that particles can exhibit wave-like characteristics, although these effects are only noticeable at very small, typically atomic, scales.

Momentum

Momentum, defined as the product of mass and velocity, is a key quantity in both classical and quantum mechanics. In the context of the de Broglie hypothesis, a particle’s momentum determines the wavelength associated with it, with larger momenta leading to smaller wavelengths, making the wave-like nature less observable in macroscopic objects.

Planck's Constant

Planck's constant is a critical physical constant in quantum mechanics that sets the scale for quantum effects. It appears in the de Broglie wavelength expression, and its extremely small value means that for everyday objects with relatively large momentum, the resulting wavelength is exceedingly small, which is why quantum wave properties are not noticeable in the macroscopic world.

Scale of Quantum Effects

Quantum mechanical effects, including the wave-like behavior of particles, are significant only when the de Broglie wavelength is comparable to the dimensions of interest (such as atomic dimensions). For macroscopic objects like a baseball, the calculated wavelengths are many orders of magnitude smaller than atomic sizes, meaning quantum effects are effectively imperceptible.

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