Physics Quantum Mechanics

# Physics Quantum Mechanics

## 1. Formalism
QM describes PS via state vectors in Hilbert space. The evolution is governed by the SE: $i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle$. The WF contains all probability amplitudes.

## 2. Observables and Operators
Physical observables are represented by Hermitian operators. Measurement outcomes correspond to EV of these operators. The expectation value is $\langle A \rangle = \langle \psi | A | \psi \rangle$.

## 3. The Uncertainty Principle
Heisenberg's principle $\sigma_x \sigma_p \ge \hbar/2$ arises from non-commuting operators $[x, p] = i\hbar$. This is a fundamental limit of nature, not a measurement error.

## 4. Quantum Dynamics
- **Stationary States**: Solving $H\psi = E\psi$ yields energy levels.
- **Superposition**: Linearity of SE allows systems to exist in multiple eigenstates simultaneously.
- **Entanglement**: Non-local correlations between particles where the joint state cannot be factored into product states.

## 5. Advanced Topics
- **Spin**: Intrinsic angular momentum without classical analogue.
- **Perturbation Theory**: Approximating complex systems by adding small corrections to solvable ones.
- **Bell's Theorem**: Proving the incompatibility of local hidden variables with QM predictions.