Quantum Mechanics Intro

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy on microscopic scales, such as atoms and subatomic particles. Unlike classical mechanics, it relies on probability and wave functions rather than deterministic paths, introducing concepts like superposition and entanglement. This guide provides an introduction to quantum mechanics, covering all basic formulas (Planck’s equation, de Broglie relation, Schrödinger equation, Heisenberg uncertainty principle), detailed examples, and practical applications to illustrate its revolutionary impact on modern science and technology.

Planck’s Equation (with de Broglie, Schrödinger, Uncertainty)

Planck’s equation relates the energy of a photon to its frequency, marking the birth of quantum theory:

E=hf

Where:

  • E: Energy (J)
  • h=6.626×10−34J·s: Planck’s constant
  • f: Frequency (Hz)

Related Formulas:

  • de Broglie Relation: λ=hp, where p=mv (momentum)
  • Schrödinger Equation (Time-Independent, 1D): −ℏ22md2ψdx2+Vψ=Eψ (simplified, ℏ=h2π)
  • Heisenberg Uncertainty Principle: Δx⋅Δp≥ℏ2

Examples

Example 1: Planck’s Equation

Energy of a photon with f=5×1014Hz:

E=hf =(6.626×10−34)(5×1014) =3.313×10−19J

Example 2: de Broglie Wavelength

Wavelength of an electron with mass m=9.11×10−31kg, velocity v=1×106m/s:

p=mv =(9.11×10−31)(1×106) =9.11×10−25kg·m/s λ=hp =6.626×10−349.11×10−25 ≈7.27×10−10m

Example 3: Schrödinger Equation (Energy Estimate)

For a particle in a box (V=0, m=9.11×10−31kg, box length L=1×10−9m, n=1):

E=n2h28mL2 ℏ=h2π =6.626×10−342⋅3.1416 ≈1.055×10−34J·s E=(1)2(6.626×10−34)28(9.11×10−31)(1×10−9)2 =(6.626×10−34)28⋅9.11×10−31⋅1×10−18 ≈4.39×10−677.288×10−48 ≈6.02×10−20J

Example 4: Heisenberg Uncertainty Principle

Minimum uncertainty in position if momentum uncertainty Δp=1×10−25kg·m/s:

Δx⋅Δp≥ℏ2 Δx≥ℏ2Δp =1.055×10−342⋅1×10−25 ≈5.275×10−10m

Applications

Quantum mechanics drives modern technology and scientific advancements. Below are examples with calculations:

Example 1: Semiconductor Energy Levels

Energy of an electron in a quantum well (L=5×10−9m, m=9.11×10−31kg, n=2):

E=n2h28mL2 =(2)2(6.626×10−34)28(9.11×10−31)(5×10−9)2 =4⋅4.39×10−678⋅9.11×10−31⋅2.5×10−17 ≈1.756×10−661.822×10−46 ≈9.64×10−21J

Example 2: Laser Photon Energy

Energy of a photon from a laser with f=4.74×1014Hz:

E=hf =(6.626×10−34)(4.74×1014) ≈3.14×10−19J

Example 3: Electron Diffraction

Wavelength of an electron with v=2×106m/s:

p=mv =(9.11×10−31)(2×106) =1.822×10−24kg·m/s λ=hp =6.626×10−341.822×10−24 ≈3.64×10−10m

Example 4: Quantum Computing (Uncertainty)

Minimum position uncertainty if Δp=5×10−26kg·m/s:

Δx⋅Δp≥ℏ2 Δx≥1.055×10−342⋅5×10−26 ≈1.055×10−9m

Example 5: Atomic Transition Energy

Energy of a photon with f=6×1014Hz:

E=hf =(6.626×10−34)(6×1014) =3.976×10−19J

Example 6: Particle in a Box (Higher State)

Energy for n=3, L=2×10−9m:

E=n2h28mL2 =(3)2(6.626×10−34)28(9.11×10−31)(2×10−9)2 =9⋅4.39×10−678⋅9.11×10−31⋅4×10−18 ≈3.951×10−662.914×10−47 ≈1.36×10−19J