Bohr's atomic model revolutionized our understanding of atoms. It introduced fixed electron orbits, quantized energy levels, and the concept of electrons jumping between orbits by absorbing or emitting specific energy packets called photons.

The model's key feature is quantized angular momentum, where electrons can only have specific, discrete values of angular momentum. This quantization, expressed as L = nℏ, explains the stability of atoms and the discrete nature of atomic spectra.

Bohr's Model and Quantized Angular Momentum

Bohr's atomic model features

• Electrons orbit the nucleus in fixed, circular orbits at specific radii and energy levels
• Electrons transition between orbits by absorbing or emitting specific amounts of energy (photons)
• Angular momentum of an electron in an orbit is quantized, restricted to integer multiples of $\hbar = h/2\pi$ (Planck's constant)
• Electrons in an orbit do not radiate energy continuously, only when transitioning between orbits

Quantized angular momentum concept

• Classical mechanics allows angular momentum to take on any continuous value
• Quantum mechanics restricts angular momentum to discrete, quantized values due to the wave nature of matter
• Allowed values of angular momentum given by $L = n\hbar$, where $n$ is a positive integer (principal quantum number) determining the electron's energy level

Electron angular momentum calculations

• Angular momentum of an electron in the $n$th orbit calculated by $L = n\hbar$
  1. For an electron in the 3rd orbit ($n = 3$), angular momentum is $L = 3\hbar$
  2. For an electron in the 5th orbit ($n = 5$), angular momentum is $L = 5\hbar$
• $\hbar$ value approximately $1.054 \times 10^{-34}$ J⋅s, explaining why quantization effects not observed in macroscopic objects (baseballs, planets)

Wave-Particle Duality and Atomic Structure

Wave nature in atomic structure

• Matter exhibits wave-particle duality, with both wave-like and particle-like properties
  • Electrons in an atom described by wave functions representing their quantum state
• Probability of finding an electron at a specific location related to the square of the absolute value of its wave function (probability density)
• Standing waves in an atom correspond to allowed electron orbits
  • Orbit circumference must be an integer multiple of the electron's de Broglie wavelength, $\lambda = h/p$ ($p$ is electron momentum)
• Wave nature of electrons results in quantization of energy levels and angular momentum in atoms
  • Quantization gives rise to discrete emission and absorption spectra (Lyman series, Balmer series)

Quantum Mechanics and Atomic Structure

• Louis de Broglie proposed that particles can exhibit wave-like properties, leading to the concept of wave-particle duality
• The Schrödinger equation describes the behavior of quantum particles, including electrons in atoms
• Solutions to the Schrödinger equation provide information about the quantum state of a system, including its energy levels and probability distributions