Quantum Mechanics: Wave-Particle Duality, Atomic Structure, and Orbitals

Quantum Mechanics and Atomic Structure

Introduction to Quantum Mechanics

Quantum mechanics explores the behavior of electrons and light at the atomic and subatomic levels. This field explains phenomena that cannot be described by classical physics, such as the dual nature of light and electrons, and the quantization of energy in atoms.

Classical Electrons vs. Electrons as Waves (Wave-Particle Duality)

• Wave-Particle Duality: Electrons and light exhibit both wave-like and particle-like properties. This duality is fundamental to understanding atomic structure.

• Electromagnetic Waves: Light is an electromagnetic wave, characterized by its wavelength (λ), amplitude, and frequency (ν).

Key Equations:

• Relationship between wavelength, frequency, and speed of light: $c = \lambda \nu$ where $c$ is the speed of light ($3.00 \times 10^8$ m/s), $\lambda$ is wavelength, and $\nu$ is frequency.

Example: Radio waves have long wavelengths and low frequencies, while gamma rays have short wavelengths and high frequencies.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, arranged by wavelength and energy.

• High energy: Gamma rays, X-rays, Ultraviolet

• Low energy: Infrared, Microwaves, Radio waves

Type	Wavelength (m)	Energy
Gamma rays	10-12 - 10-10	Highest
X-rays	10-10 - 10-8	High
Ultraviolet	10-8 - 4x10-7	Moderate
Visible	4x10-7 - 7x10-7	Moderate
Infrared	7x10-7 - 10-3	Low
Microwaves	10-3 - 10-1	Lower
Radio waves	10-1 - 103	Lowest

Classic Wave Behavior: Waves can interfere constructively (amplitudes add) or destructively (amplitudes subtract).

Light Behaving in Non-Classic Manners

At the turn of the 20th century, experiments revealed that light sometimes behaves as particles, not just waves.

• Black Body Radiation: Atoms and molecules emit only discrete packets of energy (quanta), not a continuous range.

• Planck's Constant: $h = 6.63 \times 10^{-34}$ J·s

• Energy of a photon: $E = h\nu$

Photoelectric Effect: When light of sufficient frequency strikes a metal surface, electrons are ejected. The energy of the ejected electrons depends on the frequency, not the intensity, of the light.

• Einstein's Explanation: Light consists of photons, each with energy $E = h\nu$. Only photons with energy above a threshold can eject electrons.

Example: Shining ultraviolet light on potassium metal can eject electrons, but red light cannot, regardless of intensity.

Experimental Results of the Photoelectric Effect

• Different wavelengths (and thus energies) of light are required to eject electrons from different metals.

• The kinetic energy of ejected electrons increases with the frequency of the incident light.

Key Equation:

• $KE = h\nu - \phi$ where $\phi$ is the work function (minimum energy needed to eject an electron).

Electrons and the Double Slit Experiment

Electrons, like light, exhibit wave-particle duality. When electrons pass through two slits, they create an interference pattern, a property of waves, even though they are particles.

• Conclusion: The behavior of electrons cannot be fully explained by classical physics; quantum mechanics is required.

Emission Spectra and the Bohr Model

When atoms are excited, they emit light at specific wavelengths, producing line spectra unique to each element.

• Bohr Model: Electrons occupy quantized energy levels. When an electron transitions from a higher to a lower energy level, it emits a photon with energy equal to the difference between the levels.

Key Equation (Rydberg Formula):

• $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ where $R$ is the Rydberg constant ($1.097 \times 10^7$ m-1), $n_1$ and $n_2$ are integers with $n_2 > n_1$.

Example: The Balmer series describes visible light emissions from hydrogen when electrons fall to $n=2$ from higher levels.

Hydrogen Atom Orbitals and Electron Transitions

• Electrons in hydrogen atoms occupy discrete energy levels ($n=1,2,3,...$).

• Transitions between these levels result in absorption or emission of photons.

Energy of Levels:

• $E_n = -2.18 \times 10^{-18} \frac{Z^2}{n^2}$ J where $Z$ is the atomic number (for hydrogen, $Z=1$), $n$ is the principal quantum number.

Series:

• Lyman series: Transitions to $n=1$ (ultraviolet)

• Balmer series: Transitions to $n=2$ (visible)

• Paschen series: Transitions to $n=3$ (infrared)

Wave-Particle Duality and the Heisenberg Uncertainty Principle

• Heisenberg Uncertainty Principle: It is impossible to simultaneously know both the exact position and momentum of an electron. $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

This principle sets a fundamental limit on the precision of measurements at the quantum scale.

Schrödinger's Equation and Atomic Orbitals

Schrödinger developed a mathematical equation to describe the behavior of electrons in atoms. The solutions to this equation are called wave functions (ψ), which describe the probability distribution of an electron.

• Orbitals: Regions in space where there is a high probability of finding an electron.

Types of Orbitals:

• s-orbital: Spherical shape, holds 2 electrons.

• p-orbitals: Dumbbell-shaped, three orientations ($p_x$, $p_y$, $p_z$), each holds 2 electrons (6 total).

• d-orbitals: Cloverleaf shapes, five orientations, each holds 2 electrons (10 total).

• f-orbitals: Complex shapes, seven orientations, each holds 2 electrons (14 total).

Nodes: Regions where the probability of finding an electron is zero. The number of nodes increases with energy level.

Quantum Numbers and Electron Addresses

Each electron in an atom is described by a unique set of four quantum numbers:

• Principal quantum number ($n$): Indicates the energy level (shell).

• Angular momentum quantum number ($l$): Indicates the shape of the orbital (0 = s, 1 = p, 2 = d, 3 = f).

• Magnetic quantum number ($m_l$): Indicates the orientation of the orbital.

• Spin quantum number ($m_s$): Indicates the spin of the electron (+1/2 or -1/2).

Shells and Subshells:

Shell (n)	Types of Orbitals	Number of Orbitals
1	s	1
2	s, p	1 + 3 = 4
3	s, p, d	1 + 3 + 5 = 9
4	s, p, d, f	1 + 3 + 5 + 7 = 16

Hydrogen-Like Energy Level Diagram

Energy levels in hydrogen-like atoms fill in order of increasing energy. The 1s orbital is the lowest in energy, followed by 2s, 2p, 3s, 3p, 3d, etc.

• Aufbau Principle: Electrons fill the lowest energy orbitals first.

Example Question: Why is the 1s orbital lower in energy than any other orbital? (Answer: It is closest to the nucleus, so the electron experiences the greatest attraction.)

Summary Table: Quantum Numbers and Orbitals

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level (shell)
Angular Momentum	l	0 to n-1	Orbital shape (s, p, d, f)
Magnetic	m_l	-l to +l	Orbital orientation
Spin	m_s	+1/2, -1/2	Electron spin direction

Key Equations and Constants

• Speed of light: $c = 3.00 \times 10^8$ m/s

• Planck's constant: $h = 6.63 \times 10^{-34}$ J·s

• Energy of a photon: $E = h\nu$

• de Broglie wavelength: $\lambda = \frac{h}{mv}$

• Rydberg equation: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

• Heisenberg uncertainty: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

Additional info:

• These notes cover foundational concepts in quantum mechanics relevant to General Chemistry, including the nature of light, atomic structure, and the mathematical framework for describing electrons in atoms.