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Momentum is a fundamental concept in physics which is defined as the product of an object's mass and velocity, expressed by the formula \( p = mv \). Here, \( p \) is the momentum, \( m \) is the mass of the particle, and \( v \) is its velocity.

• Momentum depends directly on both mass and velocity. This means, for two objects with the same speed, the one with the larger mass will have more momentum.
• In the context of the de Broglie wavelength, momentum plays a crucial role in determining the wavelength of a particle.

When comparing an electron and a proton moving at the same speed, we realize that the proton, which is much heavier, has a higher momentum than the electron. This mass difference is key because in the calculation of the de Broglie wavelength, the particle with the higher momentum will have a shorter wavelength when compared to the particle with less momentum.

Planck's constant is a fundamental constant in quantum mechanics, symbolized by \( h \). It has a value of approximately \( 6.626 \times 10^{-34} \) m² kg / s. This constant is crucial because it sets the scale of quantum effects, determining how the particles behave at atomic and subatomic levels.

• Planck's constant is a key component of the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), linking the wave characteristics of particles to their momentum.
• Its relatively small value implies that quantum effects are significant only for very small particles like electrons and protons.

In the exercise, Planck's constant is used to translate the momentum of particles (electron and proton) into their de Broglie wavelength. This relationship highlights how particles exhibit both wave-like and particle-like properties, central to the concept of wave-particle duality.

When comparing an electron and a proton, especially when they share the same velocity, their differing masses lead to interesting outcomes in terms of their de Broglie wavelengths.

• The electron is significantly lighter than the proton. The mass of a proton (\( m_p \)) is approximately 1836 times that of an electron (\( m_e \)).
• Since both particles are moving at the same speed, the proton's momentum is much larger due to its larger mass.

This mass disparity brings about major differences in their de Broglie wavelengths. According to the formula \( \lambda = \frac{h}{p} \), a larger momentum (associated with the proton) results in a smaller wavelength. Consequently, the electron, having a much smaller momentum due to its lighter mass, exhibits a significantly longer de Broglie wavelength. This makes the concept of wave-particle duality more apparent in lighter particles like electrons. This demonstrates why, in quantum mechanics, smaller particles tend to exhibit more pronounced wave-like behaviors than larger, more massive particles.