Free Particle in Quantum Mechanics

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In quantum mechanics, the wave function of a free particle is often described using plane wave solutions. These plane waves represent particles with definite momentum and energy, but they come with certain issues:

Non-Normalizability: Plane waves extend infinitely, meaning their total probability density cannot be normalized to one. This conflicts with the probabilistic interpretation of the wave function.

Lack of Localization: A plane wave implies the particle is spread out over all space, making it impossible to determine its exact position.

To address these issues, the concept of a wave packet is introduced. A wave packet is a combination of multiple plane waves with different momenta. This superposition of waves results in a localized wave function, allowing the particle to be confined to a specific region in space.

Wave packets provide a more accurate representation of free particles in quantum mechanics. They allow for a localized position, which aligns with the probabilistic nature of quantum mechanics, and they can spread over time, reflecting the inherent uncertainty in a particle's momentum and position.

00:00 Introduction
01:19 Schrodinger's Equation - Solution
09:25 Constant Probability Density of Plane waves
12:39 Velocity of Plane waves
15:01 Non-Normalizability of Plane waves
17:24 Concept of Wave Packet

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