Jul 17, 2024

- At the turn of the 20th century, physics seemed complete, with advances like electricity and sophisticated theories from physicists like Laplace and Michelson suggesting near-complete knowledge.
- Michelson’s claim (1903): Future discoveries will only refine decimal precision.
- Not all was known; there were unresolved experiments related to light and matter:`
- Black Body Spectrum
- Photoelectric Effect
- Bright Line Spectrum

- Three key problematic experiments questioning classical physics:
**Black Body Spectrum**: Predicted by Rayleigh-Jeans Law (worked only for long wavelengths, ultraviolet catastrophe).**Photoelectric Effect**: Classical EM failed to predict the actual properties (variation with light intensity and frequency).**Bright Line Spectra**: Discrete light emissions from elements unexpectedly.

- Scientists like Planck, Einstein (photoelectric effect), and others formed foundational quantum mechanics contributions.

- Describes radiation emitted by hot objects.
- Classical predictions failed (infinite energy density for short wavelengths).
- Empirical fits existed but classical physics couldn't explain the full spectrum.

- Electrons emitted when light strikes material; contradicts classical EM predictions.
- Depended on light frequency rather than amplitude.

- Discrete spectral lines unique to each element, classical physics couldn’t explain.
- Quantum mechanics helped provide an explanation.

- Surrounding the famous 1927 Solvay Conference photo:
- Planck (Black Body Radiation), Einstein (Photoelectric Effect)
- Others like Niels Bohr, Heisenberg, Dirac, Schrodinger formulated complex principles.

**Classical Side**: Predictable, deterministic, all properties known with precision.**Quantum Side**: Governed by uncertainties (probabilities rather than certainties).**Planck’s Constant**: Central constant in quantum mechanics (ℏ or h-bar).**Uncertainty Principle**: Δp * Δx ~ ℏ or ΔE * Δt ~ ℏ (fundamental limits of precision).

**Wave Function (Ψ)**: Complex function representing the probabilities linked to particle’s properties, giving probabilities over certainties.**Operators**: Related to physical observables, often represented with hat notation (e.g., x̂, p̂). Includes kinetic and potential energy operators.**Schrodinger Equation**: Core equation of Quantum Mechanics (time-dependent or independent form). Solutions map physically meaningful states.

**Probability Distribution**: Ψ*Ψ gives probability density (likelihood of finding a particle within a certain space region).**Normalization Requirement**: Total probability must sum to 1 (integral over all space of Ψ*Ψ should equal 1).

- Essential for describing wave functions.
- Euler’s formula and complex exponential related to sinusoidal terms (important for wave equations).
- Algebraic manipulations include complex conjugation, magnitude, and division.

- Demonstrated where classical physics failed; applying quantum mechanics (discretized interactions) succeeded.

- Observable properties linked with operators, e.g., momentum (p̂ = -iℏ d/dx).
**Commutators**: Reveal relationships and uncertainties (e.g., [x̂, p̂] = iℏ).**Hermitian Operators**: Ensure physically observable quantities remain real.

- Solving operator equations like ĤΨ = EΨ provides energy (E) for possible states (Ψ).
**Hermitian Operator Properties**: Real eigenvalues, orthonormal eigenstates.

- Demonstrating raising and lowering operators to solve energy levels without known wave functions.
- Special functions: Laguerre and Hermite polynomials for solution forms.

- Derived from commutator relationships (providing bounds on precision simultaneous measurements of different observables).
**Heisenberg's Uncertainty Principle**: Most famous example involving position and momentum.

- Planck’s law describing spectral distribution helping deal with ultraviolet catastrophe.
- Basis for foundational quantum principles in thermodynamics and statistical physics.

- Solution of Schrodinger equation in hydrogen atom explaining spectral lines (quantized orbits and emissions).

- Quantum mechanics in describing semiconductors, superconductivity, quantum gates and computing. Contributions from foundational ideas continue to modern technology enhancements.

- Covered basics of operators, wave functions, solving Schrodinger equation, formalism involving probability distributions.
- Explored key concepts through applications in different physical systems.
- Further studies ahead delving deeper into specific system properties and modern quantum phenomena.