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The concept of wave-particle duality is a cornerstone of quantum mechanics. It reveals that all particles exhibit both wave-like and particle-like properties. This duality is not only true for tiny particles like electrons but also for larger objects, although the wave characteristics become negligible.
For example, light behaves as a wave when it diffracts and refracts. At the same time, it can behave as a particle, as demonstrated by the photoelectric effect where light knocks out electrons from a metal surface.
This dual nature applies to matter too. Even a macroscopic object like a ball has an associated wavelength.

• This wavelength is often too small to detect due to the much larger mass compared to subatomic particles.
• The De Broglie hypothesis posits that every particle or object with momentum has a wave associated with it, that's why we use the De Broglie wavelength formula for calculations.

Planck's constant (\( h \)) is a fundamental physical constant crucial in quantum mechanics. It acts as the proportionality factor between the energy of a photon and the frequency of its associated electromagnetic wave.
Given as \( 6.626 \times 10^{-34} \, \mathrm{m}^2 \, \mathrm{kg/s} \), Planck's constant helps establish the scale at which quantum effects become significant.

• It is a very small number, indicating that quantum effects are typically only visible at an atomic or subatomic scale.
• In the De Broglie wavelength formula, Planck’s constant connects momentum (mass times velocity) to wavelength.

The idea is that the smaller the wavelength, the larger the momentum or vice versa. Hence, knowing Planck’s constant is pivotal to understanding the quantum realm.

Calculating the De Broglie wavelength involves understanding the relationship between mass, velocity, and Planck’s constant. The formula \( \lambda = \frac{h}{mv} \) is used to find the wavelength \( \lambda \) of particles when their mass \( m \) and velocity \( v \) are known.
This calculation shows how an object's momentum influences its wave properties. In the given exercise, the mass of the ball is \( 0.23 \, \mathrm{kg} \) and its velocity is \( 0.10 \, \mathrm{m/s} \). Using Planck's constant \( h = 6.626 \times 10^{-34} \, \mathrm{m}^2 \mathrm{kg/s} \), you can substitute these values into the formula to find the wavelength.
Steps involved in calculations:

• Substitute the values into the equation: \( \lambda = \frac{6.626 \times 10^{-34}}{0.23 \times 0.10} \).
• Solve the equation to find \( \lambda \), which results in \( 2.88 \times 10^{-32} \) meters.

For macroscopic objects like a ball, the resulting wavelength is extremely tiny, illustrating how less significant wave behaviors become as object size increases.