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Momentum is a crucial concept in physics that describes the quantity of motion an object possesses. It is defined as the product of an object's mass and velocity, represented by the formula \( p = mv \). Momentum is a vector quantity, meaning it has both magnitude and direction.

When considering the de Broglie wavelength, momentum plays a direct role. The de Broglie wavelength \( \lambda \) of a particle is inversely proportional to its momentum as given by the formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant. So, by increasing momentum, the wavelength decreases and vice versa.

In the scenario where the particle's momentum is doubled \(( p' = 2p )\), the wavelength is halved. This is because the wavelength becomes \( \lambda' = \frac{h}{2p} \), highlighting how intimately connected momentum and the de Broglie wavelength are.

Kinetic energy refers to the energy that an object possesses due to its motion, calculated through the formula \( KE = \frac{1}{2} mv^2 \). For nonrelativistic particles, kinetic energy can also be expressed as a function of momentum: \( KE = \frac{p^2}{2m} \), where \( m \) is the mass of the particle. This linkage allows us to see how changes in kinetic energy affect momentum.

When kinetic energy is doubled, the momentum transforms according to \( KE' = 2KE = \frac{(p')^2}{2m} \), leading to \( (p')^2 = 2p^2 \). Consequently, the new momentum \( p' \) becomes \( p\sqrt{2} \). This change in momentum then influences the de Broglie wavelength, which decreases by a factor of \( \frac{1}{\sqrt{2}} \) when kinetic energy is doubled, showing how kinetic energy impacts particle properties.

Nonrelativistic particles are those moving at speeds significantly less than the speed of light, making relativistic effects negligible. This categorization simplifies calculations because relativistic corrections (which arise at high speeds) are not required.

In terms of the de Broglie wavelength, the assumption of nonrelativistic speeds allows us to use simple classical formulas to approximate momentum and kinetic energy. This makes the relation between these quantities and the de Broglie wavelength straightforward.

For such particles, we can confidently use the formulas \( \lambda = \frac{h}{p} \) for wavelength, \( p = mv \) for momentum, and \( KE = \frac{1}{2} mv^2 \) or \( KE = \frac{p^2}{2m} \) for kinetic energy without considering relativistic mass changes. This simplicity is why nonrelativistic assumptions are usually made in fundamental quantum mechanics calculations.