Quantum mechanics shakes up our understanding of reality. It introduces mind-bending concepts like wave-particle duality and the uncertainty principle, challenging our classical intuitions about how the world works.

The postulates of quantum mechanics lay the groundwork for this revolutionary theory. They define how we describe quantum states, measure observables, and predict the probabilistic outcomes of experiments in the quantum realm.

Wave-Particle Duality and Schrödinger Equation

Fundamental Concepts of Quantum Mechanics

• Wave-particle duality describes the dual nature of matter and energy exhibiting both wave-like and particle-like properties
• Light demonstrates wave-particle duality through phenomena such as diffraction (wave-like) and photoelectric effect (particle-like)
• Electrons also exhibit wave-particle duality, evidenced by electron diffraction experiments
• De Broglie wavelength relates the wavelength of a particle to its momentum: $\lambda = \frac{h}{p}$
• Schrödinger equation serves as the fundamental equation of quantum mechanics, describing the behavior of quantum systems
• Time-dependent Schrödinger equation: $i\hbar\frac{\partial}{\partial t}\Psi(r,t) = \hat{H}\Psi(r,t)$
• Time-independent Schrödinger equation for stationary states: $\hat{H}\Psi(r) = E\Psi(r)$

Mathematical Representations in Quantum Mechanics

• State vector represents the quantum state of a system in Hilbert space
• Dirac notation expresses state vectors as kets: $|\Psi\rangle$
• Superposition principle allows quantum states to exist in multiple states simultaneously
• Linear combination of state vectors represents superposition: $|\Psi\rangle = c_1|\Psi_1\rangle + c_2|\Psi_2\rangle$
• Probability amplitudes determine the likelihood of measuring specific outcomes
• Normalization condition ensures total probability equals 1: $\int_{-\infty}^{\infty} |\Psi(x)|^2 dx = 1$

Observables and Measurement

Quantum Observables and Operators

• Observable represents a physical quantity that can be measured in quantum mechanics
• Hermitian operators correspond to observables in quantum mechanics
• Common observables include position, momentum, energy, and angular momentum
• Eigenvalue equation for an observable A: $\hat{A}|\Psi\rangle = a|\Psi\rangle$
• Expectation value of an observable calculated as: $\langle A \rangle = \langle\Psi|\hat{A}|\Psi\rangle$
• Commutator of two operators determines if observables can be simultaneously measured: $[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$

Quantum Measurement Process

• Measurement in quantum mechanics involves interaction between the system and measuring apparatus
• Collapse of the wave function occurs upon measurement, reducing the state to an eigenstate of the measured observable
• Probabilistic interpretation states that quantum mechanics predicts probabilities of measurement outcomes
• Born rule relates the probability of measuring a specific value to the wave function: $P(x) = |\Psi(x)|^2$
• Repeated measurements on identically prepared systems yield a distribution of results
• Quantum entanglement describes correlated quantum states that cannot be described independently

Uncertainty Principle

Heisenberg Uncertainty Principle and Its Implications

• Uncertainty principle states that certain pairs of physical properties cannot be simultaneously known with arbitrary precision
• Heisenberg uncertainty relation for position and momentum: $\Delta x \Delta p \geq \frac{\hbar}{2}$
• Energy-time uncertainty relation: $\Delta E \Delta t \geq \frac{\hbar}{2}$
• Uncertainty principle arises from wave nature of quantum objects, not measurement limitations
• Zero-point energy results from uncertainty principle, preventing particles from having zero energy in ground state
• Quantum tunneling occurs when particles penetrate potential barriers classically forbidden due to uncertainty principle
• Applications of uncertainty principle include scanning tunneling microscopy and alpha decay in radioactive nuclei