Atomic Spectra and Quantum Wave-Particle Duality

Atomic Emission and Absorption Spectra

• Continuous spectra are produced when light from a continuous source (such as a light bulb) passes through a prism, producing a complete spectrum of colors.

• Excited hydrogen gas inside a circuit emits discrete, specific spectral lines (such as a magenta-colored emission) rather than a full continuous spectrum.

• Dark lines in sunlight represent an absorption spectrum, occurring when photons of specific energies are absorbed by cooler atmospheric gases.

• Each element possesses a distinct absorption and emission spectrum, functioning as an atomic fingerprint used in spectroscopy to identify chemical composition.

The Bohr Model and Energy Levels

• The Bohr model depicts electrons orbiting the nucleus at specific, quantized energy levels, with the ground state defined at level $n = 1$.

• Atomic light emission occurs when excited electrons decay from higher energy levels to lower energy levels:

  • Transition $n = 3 \rightarrow n = 2$ emits red light, which has a lower frequency, longer wavelength $\lambda$, and lower energy.

  • Transition $n = 4 \rightarrow n = 2$ emits higher frequency light due to a larger energy gap.

  • Transition $n = 5 \rightarrow n = 2$ emits blue light.

  • Transitions dropping down to level $n = 1$ involve large energy gaps, emitting high-energy ultraviolet radiation.

  • Transitions involving smaller energy gaps emit low-energy infrared radiation.

• Absorption occurs in reverse: an electron absorbs a photon with an energy exactly matching the gap between its current state and a higher energy level (e.g., $n = 2 \rightarrow n = 3$).

• Higher orbital energy levels are spaced closer together, causing higher-energy spectral lines to crowd closer together.

Electron Beams and Particle-Wave Duality

• Magnetic fields exert forces on moving electric charges, bending the trajectory of electron beams.

• Cathode Ray Tubes (CRT) in older televisions utilize steered electron beams scanned across pixels to generate images.

• Electrons scattered off an annealed metallic crystal form a two-dimensional diffraction pattern of local intensity maxima and minima, proving that electrons exhibit wave-like behavior.

• Moving matter exhibits a de Broglie wavelength defined by:   $\lambda = \frac{h}{p}$   $\lambda = \frac{h}{m v}$

• Electron microscopes achieve far higher spatial resolution than optical microscopes because high-momentum electrons possess extremely short wavelengths.

Quantum Wave Functions and Measurement

• Double-slit experiments using individual electrons or photons produce single discrete impacts that aggregate over time to build a wave interference pattern.

• The quantum wave function $\psi$ dictates particle behavior, where the probability density of finding a particle at a given position is represented by:   $|\psi|^2$

• Single isolated particles demonstrate self-interference, traversing both slits simultaneously in an indefinite state prior to detection.

• Performing a "which-way" measurement to track which slit a particle passes through destroys the quantum interference pattern, forcing classical particle behavior.

• Matter wave energy $E$ and momentum $p$ relate to frequency $f$ and wavelength $\lambda$ through Planck's constant $h$:   $E = h f$   $p = \frac{h}{\lambda}$   $E = \frac{h c}{\lambda}$