Wave-Particle Duality, Bohr Model, and Atomic Spectra – Comprehensive Notes

Wave-Particle Duality and De Broglie Wavelength

The electron exhibits wavelength properties just as light does; this is the core idea of wave-particle duality. This concept emerged from early 20th-century quantum mechanics, challenging classical physics' view of particles and waves as distinct entities.
De Broglie proposed that matter has a wavelength: \lambda = \frac{h}{p} = \frac{h}{mv}
where p = mv is the momentum.
This equation lets us calculate the wavelength of a particle with mass (e.g., an electron, a hydrogen atom, etc.). It is specifically for massive particles and is not used for electromagnetic radiation like visible light.
The Dutch scientists (historically credited) helped establish that matter exhibits wavelength properties; de Broglie linked wavelength to momentum.
Stress: this relation is central to thinking about quantum objects and their wave-like behavior, not just classical particles.

Double-Slit Experiment and Wave-Particle Duality in Action

If an electron’s wave goes through both slits, it diffracts and interferes with itself on the far side, producing an interference pattern on an atomic crystal.
From the interference pattern, one can back-calculate where atoms were in the crystal, illustrating the wave nature of electrons.
If an observer determines which slit the electron passes through before it hits the detection screen, the superposition is perturbed and collapsed to a definite path (one slit or the other).
The act of measurement perturbs the electron, breaking the superposition and causing particle-like behavior; this is a straightforward illustration (via light interactions) of perturbation and wavefunction collapse. This perturbation is not merely an energy exchange, but a fundamental interaction that localizes the particle, consistent with the Heisenberg Uncertainty Principle, where precise knowledge of position leads to uncertainty in momentum, disrupting its wave pattern.
In practical terms: shining light on the electron to probe its path involves energy transfer. Photon energy per photon is extremely small (about 10^{-19}\ \text{J}), so ordinary light barely perturbs large objects but can perturb tiny ones like electrons.
The takeaway: electrons exhibit wavelength properties, showing wave-particle duality; quantum objects are described by both wave-like and particle-like behaviors simultaneously, not as a simple switch.
This duality underpins the ongoing development of atomic models and quantum mechanics.

Practical Evidence: Electron Microscopy and Wavelengths

One real-world application of the electron wavelength is electron microscopy. There are primarily two types: Transmission Electron Microscopes (TEM) which pass electrons through a thin sample to form an image, and Scanning Electron Microscopes (SEM) which scan the surface with a focused electron beam.
Electrons are accelerated with very high voltages (e.g., around 300\,\text{kV}), imparting large kinetic energies and producing very short de Broglie wavelengths for electrons.
The resulting electron wavelength is on the order of a picometer, i.e. \lambda \sim 10^{-12}\ \text{m}, which is about 100 times smaller than an atom (~10^{-10}\ \text{m}).
In a modern electron microscope, this allows resolving features at the atomic scale; 3D reconstructions from multiple 2D images enable visualization of structures down to individual atoms within a virus particle.
Image processing can be used to reconstruct three-dimensional structures from many 2D projections, revealing atoms and molecular arrangement.
In biology, these techniques are especially valuable for studying protein structures and viral particles.
The key point: high-energy electrons have very short wavelengths, enabling high-resolution imaging beyond visible light capabilities.

Standing Waves, Quantization, and the Bohr Model

A standing wave analogy helps explain quantization: in a circular orbit, the electron’s wave must fit an integer number of wavelengths around the circumference.
If the circumference is 2πr, the condition is 2\pi r = n\lambda\quad (n=1,2,3,…)
with \lambda = \frac{h}{mv} for the electron.
Equivalently, angular momentum is quantized: L = mvr = n\hbar = n\frac{h}{2\pi}.
In the Bohr model, electrons orbit the nucleus in circular orbits with fixed radii (quantized) and thus fixed energies; allowed orbits correspond to integer quantum numbers n = 1, 2, 3, … with no intermediate radii allowed.
The simplest visualization: the lowest energy orbit is n=1, followed by larger radii for n=2, n=3, etc. These are the allowed circular orbits in the Bohr picture.
The Bohr model connects to de Broglie wavelength by requiring the electron wave to close on itself around the orbit (standing-wave condition). This requirement ensures stable, non-radiating orbits, resolving a major issue with classical physics where orbiting electrons would continuously lose energy and spiral into the nucleus.
Bohr’s quantization resolves a long-standing problem: bulk matter shows continuous spectra, while individual atoms show line spectra due to quantized energy levels.

Bohr Model, Quantized Energies, and Atomic Spectra

Bohr postulated: electrons are particles orbiting the nucleus with fixed radii (quantized circular orbits).
The quantization condition arises from de Broglie wavelength fitting in the orbit: the circumference must accommodate an integer number of wavelengths.
The energy of each orbit is discrete. If you work through the Bohr derivation, you obtain an energy expression of the form
\ En = -\frac{hc\,R\infty}{n^2},

where R_\infty is the Rydberg constant.

The constants involved include h (Planck’s constant) and c (speed of light). The Rydberg constant R_\infty is a constant with dimensions of inverse length. Historically, the Rydberg constant was empirically derived from observed atomic spectra before Bohr's theoretical explanation.
A key connection: the discrete line spectra of atoms arise from transitions between these quantized energy levels, whereas bulk materials show continuous spectra because they host many energy states with dense spacing and interactions.
The product hcR\infty is a useful quantity in spectroscopy; a common numerical value is hcR\infty \approx 2.18\times 10^{-18}\ \text{J}.
The energy levels and transitions explain why specific wavelengths are emitted or absorbed in atomic spectra and how these lines shift with different n-values.
An important spectral relation used in hydrogen-like atoms is
\frac{1}{\lambda} = R\infty \left( \frac{1}{n1^2} - \frac{1}{n2^2} \right), with n2 > n_1.s

This equation describes the wavelengths of light associated with transitions between energy levels.

The Rydberg Constant and Important Formulas

Rydberg constant: R_\infty \approx 1.097 \times 10^7\ \text{m}^{-1}. (used in the hydrogen-like spectral formulas)
Bohr energy levels: En = -\frac{hc\,R\infty}{n^2}.
Wavelength of a transition: \frac{1}{\lambda} = R\infty \left( \frac{1}{n1^2} - \frac{1}{n2^2} \right) for integers n2 > n_1\ge 1.
If you want the energy difference explicitly: \Delta E = E{n2} - E{n1} = -\frac{hc R\infty}{n2^2} + \frac{hc R\infty}{n1^2} = hcR\infty\left( \frac{1}{n1^2} - \frac{1}{n_2^2} \right).

Connections to Quantum Models and Real-World Relevance

The Bohr model is semi-classical: it explains key features of atomic spectra and quantization, but it is not the full quantum mechanical description of atoms. It provides useful intuition and highly predictive results for simple systems (like hydrogen) and serves as a bridge to the full quantum mechanical model.
The four models of atomic structure discussed in the course are interrelated: Bohr’s model helps explain quantization and spectra; the full quantum mechanical model (Schrödinger’s equation, atomic orbitals) provides a more accurate, three-dimensional description of electron distributions and energy levels.
The passage emphasizes that while Bohr’s model is not entirely correct, it is extremely useful because it captures essential physics and makes highly accurate predictions for many observed phenomena.

Summary of Key Takeaways

Wave-particle duality: quantum objects like electrons have both particle-like and wave-like properties simultaneously.
De Broglie wavelength for particles with mass: \lambda = \frac{h}{p} = \frac{h}{mv}. This is not applicable to electromagnetic radiation in the same way.
Interference and measurement: interference arises from wave-like behavior; measurement (to determine a path) collapses the wavefunction, yielding particle-like results.
Electron microscopes exploit the short de Broglie wavelength of high-energy electrons to image structures at the atomic scale; high acceleration voltages produce wavelengths on the order of picometers.
Bohr model: electrons orbit nuclei in quantized circular orbits with radii and energies determined by a quantum number n; the standing-wave condition leads to quantized angular momentum and radii.
Energies and spectra: atomic spectra are quantized because energy levels are discrete; transitions between levels produce photons with energies given by the Bohr formula or, more generally, by spectral line equations using the Rydberg constant.
The Rydberg constant and its role in predicting hydrogen-like spectra: En = -\frac{hcR\infty}{n^2},\quad \frac{1}{\lambda} = R\infty\left(\frac{1}{n1^2}-\frac{1}{n_2^2}\right).
The constant product hcR_\infty is a useful quantity in spectroscopy (numerically, ≈ 2.18\times10^{-18}\ \text{J}).
Real-world utility: the concepts are not just theoretical; they underpin technologies like electron microscopy and the interpretation of atomic spectra in chemistry and physics.

Final Note

The instructor invites reflection: do we think these models are correct? They are remarkably predictive and useful, even if not a complete description of reality. The ongoing development of quantum models builds on these ideas and expands our ability