de Broglie and the Bohr Model
By assuming that the electron is described by a wave and a whole number of wavelengths must fit, we derive Bohr's quantization assumption.
Learning Objective

Describe reinterpretation of Bohr's condition by de Broglie
Key Points
 Bohr's condition, that the angular momentum is an integer multiple of
$\hbar$ , was later reinterpreted in 1924 by de Broglie as a standing wave condition.  For what Bohr was forced to hypothesize as the rule for allowed orbits, de Broglie's matter wave concept explains it as the condition for constructive interference of an electron in a circular orbit.
 Bohr's model was only applicable to hydrogenlike atoms. In 1925, more general forms of description (now called quantum mechanics) emerged, thanks to Heisenberg and Schrodinger.
Terms

standing wave
A wave form which occurs in a limited, fixed medium in such a way that the reflected wave coincides with the produced wave. A common example is the vibration of the strings on a musical stringed instrument.

matter wave
A concept reflects the waveparticle duality of matter. The theory was proposed by Louis de Broglie.
Full Text
Bohr's condition, that the angular momentum is an integer multiple of
Waves on a String
(a) Waves on a string have a wavelength related to the length of the string, allowing them to interfere constructively. (b) If we imagine the string bent into a closed circle, we get a rough idea of how electrons in circular orbits can interfere constructively. (c) If the wavelength does not fit into the circumference, the electron interferes destructively; it cannot exist in such an orbit.
Allowed orbits are those in which an electron constructively interferes with itself. Not all orbits produce constructive interference and thus only certain orbits are allowed (i.e., the orbits are quantized). By assuming that the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit, we have the equation:
Substituting de Broglie's wavelength of
Rearranging terms, and noting that
As previously stated, Bohr was forced to hypothesize this rule for allowed orbits. We now realize this as the condition for constructive interference of an electron in a circular orbit.
Accordingly, a new kind of mechanics, quantum mechanics, was proposed in 1925. Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model of electron motion. The new theory was proposed by Werner Heisenberg. By different reasoning, another form of the same theory, wave mechanics, was discovered independently by Austrian physicist Erwin Schrödinger. Schrödinger employed de Broglie's matter waves, but instead sought wave solutions of a threedimensional wave equation. This described electrons that were constrained to move about the nucleus of a hydrogenlike atom by being trapped by the potential of the positive nuclear charge.
Key Term Reference
 Model
 Appears in these related concepts: B.4 Chapter 4, Introduction to The Four Fundamental Spaces, and Visual Demonstrations
 angular
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, Damped Harmonic Motion, and Constant Angular Acceleration
 angular momentum
 Appears in these related concepts: Quantum Numbers, Angular Quantities as Vectors, and Angular vs. Linear Quantities
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 circumference
 Appears in these related concepts: Simple Harmonic Motion and Uniform Circular Motion, Radians, and Eratosthenes' Experiment
 constructive interference
 Appears in these related concepts: Spherical and Plane Waves, Standing Waves and Resonance, and Young's Double Slit Experiment
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 hydrogenlike
 Appears in these related concepts: Energy of a Bohr Orbit, Multielectron Atoms, and B.9 Chapter 9
 interfere
 Appears in these related concepts: Beats, Superposition and Interference, and XRay Diffraction
 interference
 Appears in these related concepts: Interference and Diffraction, Holography, and Thin Film Interference
 matter
 Appears in these related concepts: Physical and Chemical Properties of Matter, Introduction: Physics and Matter, and The Study of Chemistry
 momentum
 Appears in these related concepts: Inelastic Collisions in Multiple Dimensions, The Second Law: Force and Acceleration, and Differentiation and Rates of Change in the Natural and Social Sciences
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 nucleus
 Appears in these related concepts: The Shielding Effect and Effective Nuclear Charge, Electric Charge in the Atom, and Clusters of Neuronal Cell Bodies
 potential
 Appears in these related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 quantization
 Appears in these related concepts: General Rules for Assigning Electrons to Atomic Orbitals, Bohr Orbits, and Wave Nature of Matter Causes Quantization
 quantum mechanics
 Appears in these related concepts: Electron Configurations, Fluorescence and Phosphorescence, and Conservation of Angular Momentum
 theory
 Appears in these related concepts: Theory and Practice, Models, Theories, and Laws, and Misconceptions of Evolution
 wave
 Appears in these related concepts: Properties of Waves and Light, Waves, and Atomic Structure
 wave equation
 Appears in these related concepts: Mathematical Represenation of a Traveling Wave, Reflection and Transmission, and B.2 Chapter 2
 wavelength
 Appears in these related concepts: Energy and Momentum, Introduction to Light Energy, and Light
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