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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 Objectives

Explain how Bohr's quantization assumption is derived

Discuss application of Bohr's model to atoms

Describe reinterpretation of Bohr's condition by de Broglie
Key Points
 Bohr's condition, that the angular momentum is an integer multiple of ħ, 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 ħ, was later reinterpreted in 1924 by de Broglie as a standing wave condition. The wavelike properties of matter were subsequently confirmed by observations of electron interference when scattered from crystals. Electrons can exist only in locations where they interfere constructively. How does this affect electrons in atomic orbits? When an electron is bound to an atom, its wavelength must fit into a small space, something like a standing wave on a string, as seen in . 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:
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.
Substituting de Broglie's wavelength of h/p reproduces Bohr's rule. Since
Rearranging terms, and noting that L=mvr for a circular orbit, we obtain the quantization of angular momentum as the condition for allowed orbits:
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.
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Key Term Reference
 Model
 Appears in these related concepts: Connected Objects, Visual Demonstrations, and Models, Theories, and Laws
 angular
 Appears in these related concepts: Wavelength, Freqency in Relation to Speed, Rotational Collisions, 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: Early Ideas about Atoms, Atomic Theory of Matter, and Overview of Atomic Structure
 circumference
 Appears in these related concepts: Eratosthenes' Experiment, Simple Harmonic Motion and Uniform Circular Motion, and Using Interference to Read CDs and DVDs
 constructive interference
 Appears in these related concepts: Standing Waves and Resonance, Young's Double Slit Experiment, and Standing Waves on a String
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 hydrogenlike
 Appears in these related concepts: Energy of a Bohr Orbit, Multielectron Atoms, and B.9 Chapter 9
 interfere
 Appears in these related concepts: XRay Diffraction, Beats, and Holography
 interference
 Appears in these related concepts: Diffraction, Diffraction Gratings: XRay, Grating, Reflection, and The Michelson Interferometer
 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: Impulse, Inelastic Collisions in Multiple Dimensions, and The Second Law: Force and Acceleration
 motion
 Appears in these related concepts: Motion with Constant Acceleration, Newton and His Laws, and Motion Diagrams
 nucleus
 Appears in these related concepts: The Thomson Model, The Shielding Effect and Effective Nuclear Charge, and Electric Charge in the Atom
 potential
 Appears in these related concepts: Maslow's Hierarchy of Needs, 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, The Heisenberg Uncertainty Principle, and Conservation of Angular Momentum
 theory
 Appears in these related concepts: Scientific Reasoning, Misconceptions of Evolution, and Psychology and the Scientific Method
 wave
 Appears in these related concepts: Particle in a Box, Waves, and Properties of Waves and Light
 wave equation
 Appears in these related concepts: Position, Velocity, and Acceleration as a Function of Time, The Wave Function, and Mathematical Represenation of a Traveling Wave
 wavelength
 Appears in these related concepts: Energy and Momentum, Introduction to Light Energy, and Light
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Source: Boundless. “de Broglie and the Bohr Model.” Boundless Physics. Boundless, 08 Jan. 2016. Retrieved 10 Feb. 2016 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/atomicphysics29/theearlyatom185/debroglieandthebohrmodel6926303/