Introduction to Quantum Theory and the Properties of Light

Structure of the Atom

Nuclear Model of the Atom

The modern atomic model describes the atom as consisting of a dense nucleus surrounded by electrons. This model is foundational for understanding atomic structure and quantum theory.

• Nucleus: The nucleus is positively charged and contains more than 99.95% of the atom's mass, but occupies a very small volume.

• Electrons (e-): Negatively charged, very light particles that move around the nucleus. Their mass is extremely small, but they are dispersed over a large volume.

• Size of an atom: Typically 1–5 Å (1 Å = 10-10 m).

• Nucleus composition: Made of protons (p+, positively charged) and neutrons (n, neutral).

Subatomic Particles: Mass and Charge

Atoms are composed of three fundamental subatomic particles: protons, neutrons, and electrons. Their properties are summarized below.

Particle	Charge	Mass (amu)
Proton	Positive (1+)	1.0073
Neutron	None (neutral)	1.0087
Electron	Negative (1−)	5.486 × 10-4

• 1 amu (atomic mass unit) = 1.66054 × 10-24 g

• Mass of proton: 1.67262 × 10-27 kg

• Mass of neutron: 1.67493 × 10-27 kg

• Mass of electron: 0.00091 × 10-27 kg

Electrons

Properties of Electrons

• Charge: Each electron carries a charge of $-1.602 \times 10^{-19}$ C, known as the electronic charge.

• Mass: $0.00091 \times 10^{-27}$ kg or $5.486 \times 10^{-4}$ amu.

• Electrons are among the smallest particles that compose matter.

• The arrangement and behavior of electrons determine many chemical and physical properties of atoms.

• Electrons exhibit wave-particle duality: they show both particle-like and wave-like behavior.

Additional info: The concept of wave-particle duality is central to quantum mechanics and is also observed in light.

Properties of Light

Electromagnetic Radiation

Visible light is a form of electromagnetic radiation, which is energy transmitted through space as periodic oscillations of electric and magnetic fields (waves). This energy is also called radiant energy.

• Electromagnetic radiation (EMR) consists of oscillating electric and magnetic fields that propagate through space at a constant velocity.

• The oscillations are perpendicular to each other and to the direction of wave propagation.

• From a thermodynamic perspective, EMR is a form of heat (energy transfer).

Wave Properties of Electromagnetic Radiation (EMR)

Characteristics of Waves

• Amplitude (A): The vertical height of a crest (or depth of a trough). Determines the intensity or brightness of light.

• Wavelength (\( \lambda \)): The distance between two corresponding points in a wave (e.g., crest to crest). Units: meters (m), nanometers (nm), etc.

• Frequency (\( \nu \)): The number of cycles (wave crests) passing a stationary point per unit time. Units: 1/s (Hz).

• Speed (c): In a vacuum, all EMR travels at the speed of light: $c = 3.00 \times 10^8$ m/s.

Relationships Between Wave Properties

• Waves are periodic: they repeat regularly in space and time.

• Frequency is inversely proportional to wavelength: as wavelength increases, frequency decreases.

• The speed of a wave is the product of its wavelength and frequency:

$\lambda \times \nu = c$

$\nu = \frac{c}{\lambda}$

• Amplitude and wavelength can vary independently; two waves can have the same amplitude but different wavelengths, and vice versa.

Intensity of Light

• The amplitude of the electric and magnetic field waves in light determines the light's intensity or brightness.

• Greater amplitude means greater intensity.

Electromagnetic Spectrum

Overview of the Spectrum

The electromagnetic spectrum includes all types of electromagnetic radiation, classified by wavelength and frequency.

Type of Radiation	Frequency Range (Hz)	Wavelength Range
Gamma-rays	1020 – 1024	< 10-12 m
X-rays	1017 – 1020	1 nm – 1 pm
Ultraviolet	1015 – 1017	400 nm – 1 nm
Visible	4 – 7.5 × 1014	750 nm – 400 nm
Near-infrared	1 × 1014 – 4 × 1014	2.5 μm – 750 nm
Infrared	3 × 1011 – 1 × 1014	1 mm – 2.5 μm
Microwaves	3 × 108 – 3 × 1011	1 m – 1 mm
Radio waves	< 3 × 108	> 1 mm

• Shorter wavelength corresponds to higher frequency and higher energy.

• Visible light is the only part of the spectrum detectable by the human eye (400–750 nm).

$E = h\nu$

$\nu = \frac{c}{\lambda}$

• Where $E$ is energy, $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s), $\nu$ is frequency, $c$ is the speed of light, and $\lambda$ is wavelength.

Wave Behavior: Interference and Diffraction

Interference

• Interference is the interaction of waves with each other.

• Constructive interference: When two waves of equal amplitude are in phase, their crests align, resulting in a wave with twice the amplitude.

• Destructive interference: When two waves are out of phase, the crest of one overlaps with the trough of another, canceling each other out.

Diffraction

• Diffraction occurs when a wave encounters an obstacle or slit comparable in size to its wavelength, causing the wave to bend around it.

• Diffraction patterns, especially through two slits, demonstrate the wave nature of light (Young's experiment, 1801).

Particle Properties of Electromagnetic Radiation

Blackbody Radiation and Quantization

• Classical wave theory could not explain phenomena such as blackbody radiation, the photoelectric effect, and atomic emission spectra.

• Max Planck proposed that energy is emitted or absorbed in discrete packets called quanta.

• The energy of a quantum is given by:

$E = h\nu$

• Energy can only be released or absorbed in integer multiples of $h\nu$ (i.e., $h\nu$, $2h\nu$, $3h\nu$, ...).

• This means energy is quantized—restricted to certain values.

The Photoelectric Effect

• When light shines on a clean metal surface, electrons are emitted if the light has a minimum frequency (threshold frequency) specific to the metal.

• Einstein explained this by proposing that light consists of particles called photons, each with energy $E = h\nu$.

• Electrons are emitted only if the energy of a photon exceeds the work function ($\phi$) of the metal.

• The energy balance for the photoelectric effect is:

$h\nu = \frac{1}{2}mv^2 + \phi$

• If $h\nu < \phi$, no electrons are emitted.

• If $h\nu > \phi$, electrons are emitted with kinetic energy $\frac{1}{2}mv^2$.

• The intensity of light relates to the number of photons, not the energy per photon.

Example Problems

• Calculating wavelength from frequency: For an FM radio station broadcasting at 101.1 MHz, use $\lambda = \frac{c}{\nu}$.

• Calculating energy of a photon: For yellow light with $\lambda = 583$ nm, find $\nu$ and then $E = h\nu$.

• Photoelectric effect calculation: Given the work function and electron velocity, use $h\nu = \frac{1}{2}mv^2 + \phi$ to find the wavelength of incident light.

Additional info: These calculations are essential for understanding the quantized nature of energy and the dual wave-particle behavior of light and electrons.