# 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.

#### 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 hydrogen-like atoms. In 1925, more general forms of description (now called quantum mechanics) emerged, thanks to Heisenberg and Schrodinger.

#### Terms

• A concept reflects the wave-particle duality of matter. The theory was proposed by Louis de Broglie.

• 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.

#### Figures

1. ##### 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.

2. ##### de Broglie's Matter Waves Justify Bohr's Magic Electron Orbital Radii

I include a summary of the hydrogen atom's electronic structure and explain how an electron can interfere with itself in an orbit just like it can in a double-slit experiment.

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 wave-like 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 Figure 1. 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:

$nλ=2πr$

Substituting de Broglie's wavelength of h/p reproduces Bohr's rule. Since $\lambda = h/m_ev$, we now have: $\frac{nh}{m_ev} = 2\pi r_n$.

Rearranging terms, and noting that L=mvr for a circular orbit, we obtain the quantization of angular momentum as the condition for allowed orbits:

$L = m_e v r_n = n \frac{h}{2\pi}, (n=1,2,3...)$

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 three-dimensional wave equation. This described electrons that were constrained to move about the nucleus of a hydrogen-like atom by being trapped by the potential of the positive nuclear charge.

Figure 2

#### Key Term Glossary

angular
Relating to an angle or angles; having an angle or angles; forming an angle or corner; sharp-cornered; pointed; as in, an angular figure.
##### Appears in these related concepts:
angular momentum
A vector quantity describing an object in circular motion; its magnitude is equal to the momentum of the particle, and the direction is perpendicular to the plane of its circular motion.
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
circumference
The line that bounds a circle or other two-dimensional figure
##### Appears in these related concepts:
constructive interference
Occurs when waves interfere with each other crest to crest and the waves are exactly in phase with each other.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
interfere
(of waves) To be correlated with each other when overlapped or superposed.
##### Appears in these related concepts:
interference
An effect caused by the superposition of two systems of waves, such as a distortion on a broadcast signal due to atmospheric or other effects.
##### Appears in these related concepts:
matter
The basic structural component of the universe. Matter usually has mass and volume.
##### Appears in these related concepts:
matter wave
A concept reflects the wave-particle duality of matter. The theory was proposed by Louis de Broglie.
##### Appears in these related concepts:
Model
A representation of something difficult or impossible to display directly
##### Appears in these related concepts:
momentum
(of a body in motion) the product of its mass and velocity.
##### Appears in these related concepts:
motion
A change of position with respect to time.
##### Appears in these related concepts:
nucleus
the massive, positively charged central part of an atom, made up of protons and neutrons
##### Appears in these related concepts:
potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
##### Appears in these related concepts:
quantization
The process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as quantum mechanics.
##### Appears in these related concepts:
quantum
the smallest possible, and therefore indivisible, unit of a given quantity or quantifiable phenomenon
##### Appears in these related concepts:
quantum mechanics
The branch of physics that studies matter and energy at the level of atoms and other elementary particles; it substitutes probabilistic mechanisms for classical Newtonian ones.
##### Appears in these related concepts:
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.
##### Appears in these related concepts:
theory
An explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers
##### Appears in these related concepts:
wave
A moving disturbance in the energy level of a field.
##### Appears in these related concepts:
wave equation
An important second-order linear partial differential equation for the description of waves such as sound waves, light waves, and water waves.
##### Appears in these related concepts:
wavelength
The length of a single cycle of a wave, as measured by the distance between one peak or trough of a wave and the next; it is often designated in physics as λ, and corresponds to the velocity of the wave divided by its frequency.