Wave-Particle Duality One of the key principles of quantum mechanics is wave-particle duality, which asserts that particles exhibit both wave-like and particle-like properties. This duality challenges our classical intuition but is supported by numerous experimental observations, such as the famous double-slit experiment. When individual particles like electrons or photons pass through two slits, they create an interference pattern characteristic of waves, despite being fired one at a time. This phenomenon has been demonstrated with particles as large as molecules containing hundreds of atoms, highlighting the universal nature of this quantum principle. The mathematical framework of quantum mechanics reconciles these seemingly contradictory behaviors through wave functions and probability amplitudes. Uncertainty Principle Another cornerstone of quantum mechanics is the uncertainty principle, formulated by Werner Heisenberg. This principle states that there is a fundamental limit to the precision with which certain pairs of properties, such as position and momentum, can be simultaneously known. Thus, quantum mechanics introduces inherent randomness and unpredictability into the behavior of particles. The mathematical expression ΔxΔp ≥ ℏ/2 quantifies this limitation, where Δx represents the uncertainty in position, Δp the uncertainty in momentum, and ℏ is the reduced Planck constant. This principle is not merely a limitation of measurement equipment but represents a fundamental aspect of quantum reality. It has profound implications for our understanding of causality and determinism in the quantum realm. Quantization Furthermore, quantum mechanics introduces the concept of quantization, where certain physical quantities, such as energy levels in atoms, are restricted to discrete, quantized values. This quantization leads to phenomena such as atomic spectra and the stability of matter. In the quantum world, electrons in atoms can only occupy specific energy levels, unlike classical systems where energy can vary continuously. This explains atomic emission spectra, where atoms emit or absorb light only at specific wavelengths corresponding to transitions between these discrete energy levels. The quantum harmonic oscillator and the hydrogen atom are prime examples where quantization manifests, showing how energy states are characterized by discrete quantum numbers rather than continuous variables. Quantum Entanglement Quantum entanglement represents one of the most fascinating aspects of quantum mechanics, described by Einstein as "spooky action at a distance." When two or more particles become entangled, their quantum states become interconnected regardless of the distance separating them. A measurement on one particle instantaneously affects its entangled partners. This non-local correlation violates classical intuitions about locality and realism, as demonstrated by violations of Bell's inequalities in experimental settings. Entanglement serves as a crucial resource in quantum information science, enabling phenomena such as quantum teleportation, superdense coding, and quantum cryptography. The phenomenon has been experimentally verified with particles separated by significant distances, confirming this fundamental aspect of quantum reality. As we progress through this lesson, we will explore the mathematical formalism of quantum mechanics, including the Schrödinger equation, which describes the evolution of quantum systems over time. We will also discuss the principles of superposition and entanglement, which are central to the power and potential of quantum computing.