Quantum Mechanics

Quantum mechanics explores matter and energy at atomic and subatomic scales, where classical physics fails. At MathMultiverse, we simplify its core concepts—superposition, entanglement, and wave-particle duality—through clear formulas, examples, and visualizations.

Key Formulas

Planck’s Equation

Photon energy:

\[ E = h f \]

de Broglie Relation

Wave-particle duality:

\[ \lambda = \frac{h}{p} \]

Schrödinger Equation (1D, Time-Independent)

Wave function behavior:

\[ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V\psi = E\psi \]

Heisenberg Uncertainty Principle

Position-momentum uncertainty:

\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]

Where \( h = 6.626 \times 10^{-34} \, \text{J·s} \), \( \hbar = \frac{h}{2\pi} \approx 1.055 \times 10^{-34} \, \text{J·s} \).

Examples

Photon Energy

\( f = 5 \times 10^{14} \, \text{Hz} \):

\[ E = (6.626 \times 10^{-34}) (5 \times 10^{14}) \approx 3.313 \times 10^{-19} \, \text{J} \]

de Broglie Wavelength

Electron, \( m = 9.11 \times 10^{-31} \, \text{kg} \), \( v = 1 \times 10^6 \, \text{m/s} \):

\[ \lambda = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31}) (1 \times 10^6)} \approx 7.27 \times 10^{-10} \, \text{m} \]

Particle in a Box

\( L = 1 \times 10^{-9} \, \text{m} \), \( n = 1 \):

\[ E = \frac{(1)^2 (6.626 \times 10^{-34})^2}{8 (9.11 \times 10^{-31}) (1 \times 10^{-9})^2} \approx 6.02 \times 10^{-20} \, \text{J} \]

Uncertainty Principle

\( \Delta p = 1 \times 10^{-25} \, \text{kg·m/s} \):

\[ \Delta x \geq \frac{1.055 \times 10^{-34}}{2 \cdot 1 \times 10^{-25}} \approx 5.275 \times 10^{-10} \, \text{m} \]

Visualizations

Particle in a Box Wave Function (\( n=1 \))

Applications

  • Semiconductor: Electron in quantum well, \( E \approx 9.64 \times 10^{-21} \, \text{J} \).
  • Laser: Photon energy, \( E \approx 3.14 \times 10^{-19} \, \text{J} \).
  • Electron Diffraction: Wavelength, \( \lambda \approx 3.64 \times 10^{-10} \, \text{m} \).
  • Quantum Computing: Position uncertainty, \( \Delta x \approx 1.055 \times 10^{-9} \, \text{m} \).
  • Atomic Transition: Photon energy, \( E \approx 3.976 \times 10^{-19} \, \text{J} \).
  • Particle in a Box: Energy (\( n=3 \)), \( E \approx 1.36 \times 10^{-19} \, \text{J} \).