Quantum Mechanics

Wavelength and Frequency

Light energy travels through space as electromagnetic radiation, which can behave as both particles and waves. The properties of these waves are fundamental to understanding quantum mechanics.

• Wavelength (λ, lambda): The distance from one crest or trough of a wave to the next. It is typically measured in meters (m), nanometers (nm), or other units of length.

• Frequency (ν, nu): The number of wave cycles that pass a given point per second. It is measured in Hertz (Hz), where 1 Hz = 1 s−1.

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

The relationship between wavelength and frequency is inversely proportional at a fixed speed (the speed of light): as wavelength increases, frequency decreases, and vice versa.

• High frequencies correspond to short wavelengths.

• Low frequencies correspond to long wavelengths.

Example: Of the following, the wave with the shortest wavelength (e.g., 325 nm) will have the highest frequency.

Speed of Light

The speed of light (c) in a vacuum is a physical constant, approximately m/s. The speed of light relates wavelength and frequency:

• c: Speed of light (m/s)

• λ: Wavelength (m)

• ν: Frequency (Hz or s−1)

Example: To find the wavelength of a radio wave with a frequency of 97.7 MHz, use the formula above and convert units as needed.

The Energy of Light

Light can also be described as being made up of particles called photons. The energy of a photon is related to its frequency and wavelength:

• Planck's constant (h): J·s

• Energy and frequency:

• Energy and wavelength:

Energy is directly proportional to frequency and inversely proportional to wavelength.

For a mole of photons, use Avogadro's number ( photons/mol) as a conversion factor.

Example: Calculate the energy of a photon with a wavelength of 293.7 m using .

Electromagnetic Spectrum

The electromagnetic spectrum is a continuum of all electromagnetic radiation, ranging from radio waves (longest wavelength, lowest frequency) to gamma rays (shortest wavelength, highest frequency). As you move from radio waves to gamma rays, wavelengths decrease and frequencies increase.

• Visible light is a small portion of the spectrum, typically from about 400 nm (violet) to 700 nm (red).

Example: X-rays have more energy per photon than microwaves or radio waves.

The Photoelectric Effect

When light of sufficient energy strikes a metal surface, electrons can be ejected. This is known as the photoelectric effect.

• Binding Energy (EB.E.): The minimum energy required to eject an electron from a metal (also called the work function).

• Kinetic Energy (EK.E.): The energy of the ejected electron due to its motion.

• Photoelectric Effect Formula:

If the photon's energy is less than the binding energy, no electrons are ejected. If it is greater, the excess energy becomes the kinetic energy of the ejected electron.

Example: If a photon with energy J strikes a metal with a binding energy of J, the kinetic energy of the ejected electron is the difference.

De Broglie Wavelength

Louis de Broglie proposed that all matter has wave-like properties. The wavelength of a moving object is given by:

• λ: Wavelength (m)

• h: Planck's constant (J·s)

• m: Mass of the object (kg)

• v: Velocity (m/s)

The wavelength is inversely proportional to both mass and velocity.

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

Heisenberg Uncertainty Principle

Werner Heisenberg stated that it is impossible to simultaneously know both the exact position and momentum of an electron. This is a fundamental property of quantum systems.

• Uncertainty Principle Formula:

• Δx: Uncertainty in position (m)

• Δp: Uncertainty in momentum (kg·m/s)

This principle is significant for very small particles like electrons, but negligible for macroscopic objects.

Bohr Model

The Bohr Model describes electrons as traveling in circular orbits (shells) around the nucleus. Each shell corresponds to a specific energy level.

• Absorption: Electron moves from a lower to a higher energy shell by absorbing energy.

• Emission: Electron falls from a higher to a lower energy shell, releasing energy as a photon.

The energy of an electron in a shell is given by:

• RH: Rydberg constant ( J)

• Z: Atomic number

• n: Principal quantum number (shell number)

Example: Calculate the energy of an electron in the second shell of hydrogen ().

Emission Spectrum

When electrons transition between energy levels, they emit or absorb photons of specific energies, producing a line spectrum. Each element has a unique emission spectrum.

• Balmer series: Transitions ending at (visible region).

• Lyman series: Transitions ending at (ultraviolet region).

• Pashen, Brackett, Pfund series: Transitions ending at higher values (infrared region).

Bohr Equation

The Bohr equation calculates the energy change when an electron transitions between two energy levels:

• ni: Initial energy level

• nf: Final energy level

To find the wavelength of the emitted or absorbed photon:

• R: Rydberg constant ( m−1)

Introduction to Quantum Mechanics

Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic levels. The behavior of electrons is described mathematically by the Schrödinger equation and theoretically by quantum numbers.

• Schrödinger Wave Equation: Describes the probability distribution of an electron in an atom.

• Quantum Numbers: Specify the properties and location of electrons in atoms.

Quantum Numbers

There are four quantum numbers that describe the state of an electron in an atom:

Principal Quantum Number (n)

Indicates the main energy level or shell. As n increases, the orbital becomes larger and higher in energy. n must be a positive integer (1, 2, 3, ...).

Angular Momentum Quantum Number (l)

Defines the shape of the orbital. For a given n, l ranges from 0 to n-1. Each value of l corresponds to a subshell (s, p, d, f):

Magnetic Quantum Number (ml)

Specifies the orientation of the orbital in space. For a given l, ml ranges from –l to +l (including zero).

Spin Quantum Number (ms)

Describes the spin of the electron, which can be +1/2 or –1/2. Each orbital can hold a maximum of two electrons with opposite spins (Pauli Exclusion Principle).

Number of Electrons in Shells, Subshells, and Orbitals

• Maximum electrons in a shell:

• Maximum electrons in a subshell: s = 2, p = 6, d = 10, f = 14

• Each orbital holds 2 electrons (with opposite spins)

Nodes

A node is a region in an atom where the probability of finding an electron is zero. There are two types:

• Radial nodes: Spherical regions between shells.

• Angular nodes: Flat planes or cones that dissect the orbital.

Total number of nodes = n – 1. Number of radial nodes = n – l – 1. Number of angular nodes = l.

Example: A 4d orbital (n=4, l=2) has 2 angular nodes and 1 radial node (total 3 nodes).

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