6.3 Development of Quantum Theory

Light and matter behave as both waves and particles. This wave-particle duality is a key concept in quantum mechanics, explaining phenomena like the photoelectric effect and electron diffraction. It challenges our classical understanding of physics.

Quantum mechanics describes electrons in atoms using wave functions called orbitals. These represent the probability of finding an electron in a specific region. Quantum numbers define an electron's energy, angular momentum, orbital orientation, and spin.

Wave-Particle Duality and Quantum Mechanics

Wave-particle duality in physics

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• Matter and electromagnetic radiation exhibit both wave-like and particle-like properties (electrons, photons)
• Electromagnetic radiation behaves as waves and particles called photons
  • Photons are discrete packets of energy that make up light
• Matter, such as electrons, also exhibits wave-like and particle-like behavior
  • Electrons can diffract and interfere like waves (double-slit experiment)
• Photoelectric effect provides evidence for the particle nature of light
  • Electrons are ejected from a metal surface when light of sufficient frequency shines on it (ultraviolet light)
  • Explained by Einstein using the concept of photons and their energy $E=hν$
• De Broglie wavelength is the wavelength associated with a particle
  • Calculated using the formula: $\lambda = \frac{h}{mv}$, where $h$ is Planck's constant, $m$ is the particle's mass, and $v$ is its velocity
  • Demonstrates the wave nature of matter (electrons, protons, neutrons)
• Max Planck introduced the concept of energy quantization, laying the foundation for quantum theory

Quantum model of atomic electrons

• Quantum mechanical model describes electrons in atoms using wave functions called orbitals
  • Orbitals represent the probability distribution of an electron in an atom (electron cloud)
  • Electron density is proportional to the square of the wave function $\Psi^2$
• Types of orbitals:
  • s orbitals have a spherical shape (1s, 2s, 3s)
  • p orbitals have a dumbbell shape, oriented along x, y, or z axes (2p, 3p)
  • d and f orbitals have more complex shapes (3d, 4f)
• Nodes are points or planes where the probability of finding an electron is zero
  • Number of nodes increases with increasing energy of the orbital (1s has no nodes, 2s has one node)
• Niels Bohr proposed a model of the atom with quantized electron energy levels, contributing to the development of quantum theory

Quantum numbers for electron states

• Principal quantum number $n$ represents the energy level and shell of an electron
  • Positive integer values: 1, 2, 3, ...
  • Higher values indicate higher energy and larger average distance from the nucleus (n=1 is closest to nucleus)
• Angular momentum quantum number $l$ determines the subshell and shape of the orbital
  • Integer values from 0 to $n-1$
  • Subshells: s $l=0$, p $l=1$, d $l=2$, f $l=3$
• Magnetic quantum number $m_l$ specifies the orientation of the orbital in space
  • Integer values from $-l$ to $+l$
  • Determines the number of orbitals within a subshell (s has 1, p has 3, d has 5)
• Spin quantum number $m_s$ describes the intrinsic angular momentum (spin) of an electron
  • Values of $+\frac{1}{2}$ (spin up) or $-\frac{1}{2}$ (spin down)
  • Pauli exclusion principle states no two electrons in an atom can have the same set of four quantum numbers (prevents electrons from occupying same state)

Fundamental concepts in quantum mechanics

• Heisenberg uncertainty principle states that it is impossible to simultaneously determine both the exact position and momentum of a particle
• Schrödinger equation is the fundamental equation of quantum mechanics, describing the behavior of quantum systems
• Copenhagen interpretation is a widely accepted interpretation of quantum mechanics, emphasizing the probabilistic nature of quantum phenomena
• Quantum superposition describes a quantum system existing in multiple states simultaneously until measured
• Wave function collapse occurs when a quantum system is observed, causing it to settle into a definite state