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The concept of de Broglie Wavelength links wave and particle behaviors. It is used to determine the wavelength of a particle, such as an electron, that is moving. This relationship is defined by the equation \[\lambda = \frac{h}{mv_y}\]where

• \( \lambda \) is the de Broglie wavelength,
• \( h \) is Planck’s constant (\( 6.626 \times 10^{-34} \) Js),
• \( m \) is the mass of the particle, and
• \( v_y \) is the speed of the particle in the y-direction.

The important point here is that this wavelength is typically significant at the atomic and subatomic levels, like electrons.
It implies that every moving particle or object has a wavelength associated with it. When the speed or mass of a particle changes, so does its de Broglie wavelength. This calculation is more prominent for smaller particles, as their wavelengths become appreciably measurable.
For real-world, macroscopic objects, their calculated wavelengths are negligibly small and often ignored.

In the realm of quantum physics, determining the exact speed of a particle is not always possible. This stems from the principle known as Heisenberg's Uncertainty Principle. According to this principle, there is a limit to how precisely we can know both the position and speed (or momentum) of a particle at any given time.
When the uncertainty in the position \(\Delta y\) is equal to the de Broglie wavelength, we can establish that:

• Uncertainty in momentum \(\Delta p_y = m \Delta v_y\), where \(m\) is the mass and \(\Delta v_y\) is the uncertainty in velocity.
• The Heisenberg relation \(\Delta y \Delta p_y \geq \frac{h}{4\pi}\).

Given \(\Delta y = \frac{h}{mv_y}\), we derive \(\Delta v_y \geq \frac{v_y}{4\pi}\), showing the inherent limitation imposed by quantifying both position and speed.
Applying this, we find that the minimum uncertainty in speed is roughly 7.96% of the particle's speed. This percentage arises due to the oscillations that make it impossible to pinpoint both properties precisely.

In classical physics, assumptions often do not account for quantum behaviors, yet they provide a practical framework for understanding general mechanics. The exercise assumes that relativistic effects, those considered at speeds approaching the speed of light, can be ignored. This means that classical physics principles like Newton's laws and concepts of momentum are applicable.
In this context:

• We use non-relativistic equations: Velocity changes, as well as position, are not distorted by high-speed relativity effects.
• Classical physics assumes continuity and determinism. Outcomes can be precisely forecasted if complete initial conditions are known.

When assessing the uncertainty in speed, classical assumptions allow for straightforward generalization without considering the complex relativistic effects.
This can help bridge understanding from classical to quantum mechanics, highlighting the instances where classical physics begins to fail and quantum mechanics becomes a necessity.