Let's break down your question about wavelength, velocity, and the implications of a particle being at rest. You’re touching on some fundamental concepts in physics, particularly in wave mechanics and quantum mechanics. The relationship you mentioned, λ = h/mv, connects wavelength (λ), Planck's constant (h), mass (m), and velocity (v) of a particle. It’s important to clarify how this equation behaves under different conditions.

Understanding the Equation

The equation λ = h/mv describes the de Broglie wavelength, which is a concept that applies to all matter, not just light. It suggests that every particle has a wavelength associated with it, which is particularly significant in quantum mechanics.

When Velocity is Zero

Now, if we set v = 0 in the equation, we encounter a mathematical issue. The equation becomes:

λ = h/(m * 0)

This results in an undefined situation because division by zero is not possible. However, let’s think about what this means physically. If a particle is at rest (v = 0), it does not mean it has no properties or characteristics; rather, it indicates that the particle is not moving through space.

Displacement and Wavelength

You mentioned that if a particle is at rest, it cannot have displacement, which is true in a classical sense. Displacement refers to the change in position of an object. If there’s no movement, there’s no displacement. However, in quantum mechanics, particles can still exhibit wave-like behavior even when they are not moving in the classical sense.

Quantum Perspective

In quantum mechanics, particles can exist in a state of superposition, where they can be in multiple states at once. Even if a particle is at rest, it still has an associated wave function that describes its probability distribution in space. This wave function can have a wavelength, but it may not correspond to a classical displacement.

Wavelength and Displacement

When you consider the scenario where a particle has displacement but a wavelength of zero, it’s essential to understand that a wavelength of zero would imply that the particle is not exhibiting wave-like properties at all. In practical terms, this situation does not occur because if a particle has a defined position (displacement), it inherently has a wave function that describes its behavior, even if that wave function is not easily observable.

Key Takeaways

• The equation λ = h/mv is valid under conditions where v is not zero.
• A particle at rest does not have a defined wavelength in the classical sense, but it still possesses quantum characteristics.
• In quantum mechanics, particles can exhibit wave-like behavior even when they are not moving.

In summary, while mathematically setting v to zero leads to complications, physically, it reflects the unique nature of quantum particles. They can exist in states that defy classical intuition, and their behavior is governed by the principles of quantum mechanics rather than classical physics. If you have further questions or need clarification on specific points, feel free to ask!