Quantum Mechanics Fundamentals

Quantum Mechanics

Wave-Particle Duality

• Quantum objects (e.g., electrons, photons) can exhibit both wave-like and particle-like behavior depending on observation
• Wave-like behavior: diffraction, interference, and superposition
• Particle-like behavior: discrete packets of energy, definite position and momentum

Uncertainty Principle

• It is impossible to know certain properties of a quantum object, such as position and momentum, simultaneously with infinite precision
• Heisenberg's Uncertainty Principle: Δx * Δp >= h/4π, where Δx is uncertainty in position, Δp is uncertainty in momentum, and h is Planck's constant

Superposition

• Quantum objects can exist in multiple states simultaneously, represented by a linear combination of wave functions
• Measurement collapses superposition into a single state

Entanglement

• Quantum objects can become connected in such a way that the state of one object is dependent on the state of the other, even when separated by large distances
• Entanglement is a key feature of quantum mechanics, enabling quantum computing and cryptography

Schrödinger Equation

• A mathematical equation that describes the time-evolution of a quantum system
• Written as Hψ = Eψ, where H is the Hamiltonian operator, ψ is the wave function, and E is the energy of the system

Quantum Spin

• An intrinsic property of quantum objects, similar to angular momentum
• Can have values of +1/2 or -1/2, leading to the concept of spin-up and spin-down states

Pauli's Exclusion Principle

• No two electrons in an atom can have the same set of quantum numbers, which describe the energy, spin, and spatial distribution of an electron
• Explains the periodic table and the structure of atoms

Quantum Harmonic Oscillator

• A model system that describes the behavior of a quantum object in a potential energy well
• Used to study the quantization of energy and the properties of quantum systems

Interference and Diffraction

• Quantum objects can exhibit interference patterns, similar to wave-like behavior
• Diffraction occurs when a quantum object encounters a barrier or obstacle, resulting in a change in direction

Quantum Measurement

• The act of observing a quantum system causes the wave function to collapse, resulting in a particular outcome
• Measurement is a non-deterministic process, with the outcome uncertain until observed