Question

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Short answer

De Broglie's hypothesis explains quantized energies as a result of stable wave conditions for electrons in an atom.

Step-by-step solution

Understand de Broglie's Hypothesis

De Broglie's hypothesis proposes that particles, such as electrons, have wave-like properties. Specifically, he suggested that the wavelength of a particle is inversely proportional to its momentum. This means electrons can exhibit wave properties in addition to particle properties.

Relate de Broglie's Hypothesis to Orbital Quantization

According to de Broglie's hypothesis, electrons in an atom have wave properties. For these wave properties to be stable, only certain wavelengths can 'fit' into the circular orbit of an electron without destructive interference. This requirement leads to quantized wavelengths and, therefore, quantized energies for electrons in an atom.

Apply to the Hydrogen Atom

In a hydrogen atom, for an electron to remain in a stable orbit, its wavefunction must be a standing wave. This condition is mathematically expressed as the circumference of the orbit being an integer multiple of the de Broglie wavelength: $n\lambda =2\pi r$, where $n$ is an integer, $\lambda$ is the wavelength, and $r$ is the radius of the orbit. From this, the quantization of angular momentum follows as $mvr=n\hslash$, which results in quantized energy levels.

Key concepts

Wave-Particle Duality

Wave-particle duality is a concept that provides a unique perspective on how particles like electrons behave. De Broglie hypothesized that all matter exhibits both wave-like and particle-like properties. This idea was revolutionary because, traditionally, scientists viewed waves and particles as distinct. However, de Broglie suggested that electrons, which are typically considered particles, can also act as waves.

• The wavelength of a particle is given by de Broglie's equation: $\lambda =\frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum.
• As momentum increases, the wavelength decreases, showcasing the particle's wave nature.

This dual nature means that electrons can create patterns similar to those of waves when they overlap or go through slits, resulting in phenomena like diffraction and interference. Understanding wave-particle duality is crucial for decoding the behavior of electrons in atoms.

Quantized Energy Levels

Quantized energy levels are a fundamental aspect of quantum mechanics. In an atom, electrons reside in specific energy levels rather than a continuous range. This idea stems directly from de Broglie's hypothesis.

• Electrons exhibit wave-like properties, forming standing waves in certain orbits.
• Only certain wavelengths fit these orbits without causing wave interference.

This concept is significant because it implies that electrons can only exist in certain energy states. When an electron moves between these states, it must absorb or emit energy, typically in the form of a photon. Thus, de Broglie’s hypothesis explains why energy levels in atoms are quantized, which can be observed in phenomena such as atomic spectra.

Electron Wave Properties

Electrons, though classically considered particles, exhibit properties of waves. According to de Broglie's hypothesis, the wavelength of an electron is crucial for understanding its behavior in an atom.

• Electron wave properties allow electrons to form specific configurations in orbits known as standing waves.
• These configurations determine the allowed energy levels of an electron within an atom.

The wave nature of electrons explains why they are restricted to specific orbits. In essence, the wave properties ensure that only certain harmonics of the electron's path are stable. This concept is vital for understanding electron cloud formations and atomic bonding.

Hydrogen Atom

The hydrogen atom serves as a perfect example to illustrate de Broglie's hypothesis and the quantization of electron energies. In a hydrogen atom, the simplest atom, an electron orbits a single proton.

• The electron can only occupy certain standing wave patterns around the nucleus.
• This constraint results in specific orbit sizes (radii) and associated energy levels.

Mathematically, for an electron to form a stable orbit around the nucleus of a hydrogen atom, its circumference must be an integer multiple of its wavelength $(n\lambda =2\pi r)$. Thus, quantized angular momentum $(mvr=n\hslash )$ results in distinct energy levels, conforming to the Bohr model of the atom. This simple system beautifully illustrates the principles of quantum mechanics and the insightful nature of de Broglie's hypothesis.