lagrange's transformations & quantum theory

lagrange's transformations:

- classical mechanics alternative to newton's laws

- L = T - V (T = kinetic, V = potential energy)

- generalized coords (q_i) help simplify complex systems

- lagrange eqn: d/dt(∂L/∂q̇_i) - ∂L/∂q_i = 0

- eqn of motion, use to find system's behavior

- example: pendulum, L = (1/2)mL²θ̇² - mgL(1 - cosθ), solve eqn for motion

quantum theory:

- describes behavior of small particles (atoms, subatomic)

- key ideas:

1. quantization: energy levels are discrete (not continuous)

2. wave-particle duality: light/particles act as both wave & particle

3. uncertainty principle (Heisenberg): Δx * Δp ≥ h / 4π, can’t know position & momentum at once

4. superposition: particles can exist in multiple states at once

5. entanglement: particles linked, state of one affects the other, even at distance

- schrodinger eqn: iħ ∂Ψ/∂t = HΨ (describes how wavefunction evolves over time)

- wavefunction Ψ = probability amplitude, |Ψ|² = probability density

- quantum tunneling: particles can pass through barriers (important for fusion, semiconductors)

connections:

- lagrange is for classical (macroscopic), quantum is for microscopic (atomic/subatomic)

- both involve energy, symmetries, conservation laws

- quantum field theory uses lagrange-like formalism for fields