Modern Physics: Wave-Particle Duality, Quantum Mechanics, and Atomic Structure

A Bit of History

Classical Mechanics and Its Limitations

Classical mechanics, developed up to the 1900s, includes:

• Newtonian Laws of Motion (1600's)

• Thermodynamics (1700's)

• Electricity and Magnetism (1800's)

These theories successfully explained macroscopic phenomena involving mass, gravity, energy, and electromagnetism. However, they failed to describe micro-scale and atomic processes, leading to the development of Quantum Mechanics in the early 1900s.

Wave-Particle Duality of Light

Light as Both Wave and Particle

Light exhibits both wave-like and particle-like properties. Each photon carries energy, described by:

• Wave property: Light has frequency (f) and wavelength (λ).

• Particle property: Light consists of photons, each with energy.

Energy of a photon:

• $E = h f = rac{h c}{ u}$

• h: Planck's constant ($6.63 imes 10^{-34}$ J·s)

• c: Speed of light ($3.00 imes 10^8$ m/s)

Electromagnetic Spectrum

The electromagnetic spectrum covers a wide range of wavelengths and frequencies, from radio waves to gamma rays.

• Visible light is a small part of the spectrum.

• Frequency and energy increase as wavelength decreases.

Photon Rate Comparison

Example: Red vs. Green Laser

Given two lasers of equal power, one red and one green:

• Red light has a longer wavelength and lower frequency than green light.

• The energy per photon of red light is less than that of green light.

• To emit the same power, the red laser must emit more photons per second.

Sources of Electromagnetic Waves

Generation of EM Waves

• Oscillating voltage/current in antennas radiates radio or microwaves.

• Atoms in all objects oscillate, producing EM waves; brightness and frequency depend on temperature.

Thermal Radiation

Properties of Thermal Radiation

• Broad spectrum due to distribution of atomic velocities.

• Intensity increases with the fourth power of temperature: $I ext{ (intensity)} ightarrow T^4$

• Peak wavelength decreases as temperature increases.

Electromagnetic Waves

Nature and Speed of EM Waves

• Light is an electromagnetic wave.

• Speed of light: $c = rac{1}{ ext{sqrt}( ext{μ}_0 ext{ε}_0)} = 3.00 imes 10^8$ m/s

• EM waves are generated by oscillating electric and magnetic fields.

• Relationship between field amplitudes: $E_0 = c B_0$

Energy Carried by EM Waves

Intensity and Power

• Intensity on area A: $I = rac{P}{A} = rac{1}{2 c ext{μ}_0} E_0^2 = rac{c ext{ε}_0}{2} E_0^2$

• Intensity at distance r from a point source: $I = rac{P_{ ext{source}}}{4 ext{π} r^2}$

Example: Cellphone Radiation

• Calculating field amplitudes from a 0.6 W signal at 1.9 GHz and 10 cm distance:

• Intensity: $I = rac{0.6}{4 ext{π} (0.1)^2} = 4.8$ W/m2

• Electric field amplitude: $E_0 = ext{sqrt}(2 I / c ext{ε}_0)$

• Magnetic field amplitude: $B_0 = E_0 / c$

Photoelectric Effect

Einstein's Explanation: Light as a Particle

• Light consists of discrete packets of energy: photons.

• Energy of a photon: $E = h f$

• Matter emits/absorbs integer multiples of photon energy.

• Photoelectric effect: A photon can eject an electron if $h f ext{ (photon energy)} ext{ ≥ } E_0 ext{ (work function)}$

Experimental Observations

• Electrons are emitted immediately after light is applied.

• Emission occurs only above a threshold frequency.

• Classical theory (energy accumulation) fails; quantum theory succeeds.

Swimming Pool Analogy

• A photon (pebble) transfers all its energy to an electron (water drop), ejecting it from the metal (pool).

Wave-Particle Duality

Light and Matter as Both Waves and Particles

• Light shows wave properties (diffraction, interference) and particle properties (photon energy).

• Electrons also exhibit wave-particle duality, as shown in double-slit experiments.

De Broglie Wavelength

• Wavelength for a moving particle: $λ = rac{h}{p} = rac{h}{mv}$

• Example for electron with kinetic energy 1 eV:

  • $K = rac{mv^2}{2} = 1.6 imes 10^{-19}$ J

  • $v = ext{sqrt}(2K/m) = 5.9 imes 10^5$ m/s

  • $λ = rac{h}{mv} = 1.2 imes 10^{-9}$ m = 1.2 nm

• For macroscopic objects (e.g., baseball), wavelength is too small to observe quantum effects.

Atomic Structure and Quantum Theory

Classical vs. Quantum Model of the Atom

• Classical theory predicts electron collapse into nucleus due to energy loss.

• Quantum theory, via the uncertainty principle, prevents collapse and stabilizes atom size.

Bohr Model

• Electrons occupy only certain allowed orbits (stationary states).

• Energy levels are quantized:

  • $E_n = - rac{13.60 ext{ eV}}{n^2}$

  • $n = 1, 2, 3, ...$

• Photon emission/absorption occurs during transitions between energy levels.

Hydrogen Spectrum

• Frequency of emitted photons: $f_{ ext{photon}} = rac{ΔE_{ ext{atom}}}{h}$

• Wavelength: $λ_{ ext{photon}} = rac{c}{f_{ ext{photon}}}$

• Balmer series: visible hydrogen emission lines ($n = 2$)

Absorption vs. Emission Spectrum

• Absorption: electrons absorb photons and move to higher energy states.

• Emission: electrons drop to lower energy states, emitting photons.

Quantum Numbers and Atomic Orbitals

Quantum Numbers in Hydrogen Atom

• Principal quantum number (n): Determines energy level.

• Orbital quantum number (l): Determines angular momentum.

• Magnetic quantum number (m): Relates to orbital orientation.

• Spin quantum number (ms): $+ rac{1}{2}$ or $- rac{1}{2}$

Atomic Orbitals and Electron Clouds

• Electron probability distributions (clouds) are described by quantum numbers.

• Orbitals: s ($l=0$), p ($l=1$), d ($l=2$), f ($l=3$)

Example: Hydrogen Atom in 5p State

• Energy: $E_n = - rac{13.6 ext{ eV}}{5^2} = -0.544$ eV

• Quantum number $l = 1$

• Angular momentum: $L = ext{sqrt}(l(l+1)) imes ext{ħ}$

• Possible $m$ values: -1, 0, 1

Multi-Electron Atoms

Electron Interactions and Energy Levels

• Electrons are attracted to the nucleus and repelled by other electrons.

• Energy depends on both $n$ and $l$ due to electron-electron interactions.

Pauli Exclusion Principle

• No two electrons in an atom can have the same set of quantum numbers.

• This principle explains the structure of the periodic table.

Summary Table: Quantum Numbers for Hydrogen Atom

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level
Orbital	l	0, 1, ..., n-1	Angular momentum
Magnetic	m	-l, ..., 0, ..., l	Orbital orientation
Spin	ms	+1/2, -1/2	Electron spin

Additional info:

• Some context and terminology have been expanded for clarity and completeness.

• All equations are provided in LaTeX format for academic rigor.