Quantum Mechanics

Introduction to Quantum Mechanics

Quantum Mechanics is the mathematical and theoretical description of matter and its electrons on the atomic scale. It explains the behavior of electrons in atoms using both wave and particle models.

• Schrödinger Wave Equation: Describes the mathematical behavior of electrons.

• Quantum Numbers: Specify the energy levels and orbitals that electrons occupy in an atom.

Wave Properties of Light

Wavelength and Frequency

Light energy travels through space as electromagnetic radiation in the form of particles or waves.

• Wavelength (λ): The distance from one crest or trough of a wave to the next, measured in meters (m).

• Frequency (ν): The number of waves that pass per second, measured in Hertz (Hz).

• Amplitude: The height of a wave measured from the origin to its crest or trough.

Relationship: At a fixed speed, the frequency of a light wave is inversely proportional to its wavelength.

Speed of Light

The speed of light (c) is a physical constant and equals m/s in a vacuum.

• Example: Calculate the frequency of red light with a wavelength of 663 nm.

The Energy of Light

Photon Energy

Light consists of packets or quanta called photons. The energy of a photon is related to its frequency and wavelength.

• Photon Energy Formula (frequency):

• Photon Energy Formula (wavelength):

• Planck's Constant (h): J·s

• Example: Calculate the energy of a photon with a wavelength of 250.7 nm.

Moles and Energy

To find the energy for a mole of photons, multiply the energy of one photon by Avogadro's number ().

Electromagnetic Spectrum

Overview

The electromagnetic spectrum is a continuum of electromagnetic radiation at all wavelengths and frequencies, from radio waves to gamma rays.

• Visible Light: The small portion of the spectrum visible to the human eye (about 400–700 nm).

• Energy and Frequency: Higher frequency radiation (e.g., gamma rays) has more energy than lower frequency radiation (e.g., radio waves).

The Photoelectric Effect

Concept

When light of sufficient energy strikes a metal surface, electrons can be ejected. This demonstrates the particle nature of light.

• Binding Energy (EBE): The minimum energy needed to eject an electron from a metal.

• Photoelectric Effect Formula:

• Kinetic Energy (EKE): The energy of the ejected electron.

• Example: Calculate the work function of a metal if electrons are emitted with a given kinetic energy and frequency.

De Broglie Wavelength

Wave-Particle Duality

All matter exhibits both wave and particle properties. The de Broglie wavelength relates the mass and velocity of a particle to its wavelength.

• h: Planck's constant

• m: mass of object (kg)

• v: velocity (m/s)

• Example: Find the wavelength of a proton with a speed of m/s.

Heisenberg Uncertainty Principle

Concept

It is impossible to simultaneously know both the exact position and momentum of a particle.

• Uncertainty in position () and momentum () are inversely related.

• Example: Calculate the uncertainty in velocity if the uncertainty in position is given.

Bohr Model of the Atom

Concept

Electrons travel around the nucleus in circular orbits called shells. Each shell corresponds to a specific energy level.

• Energy of an electron in the nth shell:

• Rydberg Constant (): J

Absorption and Emission

• Absorption: Electron moves to a higher energy level (energy absorbed).

• Emission: Electron moves to a lower energy level (energy released).

Bohr Equation for Energy Transitions

• ni: initial energy level

• nf: final energy level

Atomic Emission Spectrum

Concept

Atoms emit light at specific wavelengths, producing a line emission spectrum. Each element has a unique spectrum.

• Balmer Series: Visible lines in the hydrogen spectrum (transitions to n=2).

Quantum Numbers

Principal Quantum Number (n)

• Indicates the energy level or shell of an electron.

• n = 1, 2, 3, ...

Angular Momentum Quantum Number (l)

• Indicates the subshell or shape of the orbital.

• l = 0 to n-1

• Subshells: s (0), p (1), d (2), f (3)

Magnetic Quantum Number (ml)

• Indicates the orientation of the orbital.

• ml = -l to +l

Spin Quantum Number (ms)

• Describes the spin of the electron: +1/2 or -1/2.

• Each orbital can hold two electrons with opposite spins (Pauli Exclusion Principle).

Number of Electrons in Shells and Orbitals

• Each shell can hold a maximum of 2n2 electrons.

• Each orbital can hold 2 electrons.

Nodes

• Node: A region where the probability of finding an electron is zero.

• Total number of nodes = n - 1

• Radial nodes = n - l - 1

• Angular nodes = l

Summary Table: Quantum Numbers

Additional info: These notes include both conceptual explanations and practice problems to reinforce understanding of quantum mechanics as applied in General Chemistry.