Quantum Orbitals and Quantum Numbers

Vocabulary flashcards covering core concepts from orbitals, quantum numbers, and the quantum-mechanical description of electrons.

Wave-particle duality

The concept that matter and light exhibit both wave-like and particle-like properties; electrons behave as waves with a wavelength given by De Broglie.

De Broglie wavelength

The wavelength associated with a particle of momentum p: λ = h/p (with p = mv for a mass m and velocity v).

Schrödinger equation

The fundamental equation of quantum mechanics used to determine allowed wavefunctions (ψ) for a system; its solutions describe the electron’s state and energy.

Wave function (ψ)

A mathematical function describing the state of a quantum system; its square, |ψ|^2, gives the probability density of finding the electron in space.

Probability density

The squared magnitude of the wave function, |ψ|^2, representing the likelihood of locating the electron at a point in space.

Radial probability distribution

P(r) = 4πr^2|ψ(r)|^2; shows the probability of finding the electron at a distance r from the nucleus.

Node

A region where the wave function changes sign and equals zero; the probability density is zero at nodes.

Orbital

A region or volume in space where there is a high probability of finding an electron; a solution-derived region, not a fixed path.

Principal quantum number (n)

Indicates the energy level or shell. For hydrogen-like systems, the number of nodes is n−1; higher n means larger orbitals.

Azimuthal/Angular momentum quantum number (l)

Determines subshell type with l = 0 (s), 1 (p), 2 (d), 3 (f); l ranges from 0 to n−1.

Magnetic quantum number (m)

Determines orbital orientation with m ∈ {−l, …, +l}. There are 2l+1 orbitals in a subshell.

s orbital

Orbital with l = 0; spherical in shape; there is exactly one s orbital per n level (e.g., 1s, 2s, 3s); no angular nodes.

p orbital

Orbital with l = 1; dumbbell-shaped; there are three in each n≥2 (px, py, pz); has an angular node which is a plane through the nucleus (e.g., pz has the x–y plane as its node).

d orbital

Orbital with l = 2; five orbitals (dxy, dyz, dxz, dx^2−y^2, d_z^2); more complex shapes (clover or donut-like) and can be degenerate for a given n in hydrogen.

f orbital

Orbital with l = 3; seven orbitals with highly complex shapes; important for the chemistry of lanthanides and actinides.

Degenerate energy levels

Orbitals that share the same energy in a given atom (notably in hydrogen, where all orbitals with the same n have the same energy).

Shell, subshell, and orbital terminology

Shell is defined by the principal quantum number n; subshell is defined by l (s, p, d, f); orbitals are the individual regions within a subshell (2l+1 orbitals per subshell).

Radial nodes (general)

Nodes that occur as a function of r; for an s orbital, radial nodes equal n−1 when l = 0, so 1s has 0 radial nodes, 2s has 1, 3s has 2, etc.

Phase and orbital lobes

Wavefunction lobes can have different phases (positive/negative); nodes separate regions of opposite phase in p, d, and f orbitals.

Visualization of probability (brightness/dots)

The probability density is often depicted by brightness or dot density; brighter or more dots indicate higher probability of finding the electron.

Particle in a box (simple Schrödinger problem)

A simplified one-dimensional problem used to illustrate how boundary conditions yield discrete energy levels and simple wavefunctions.

Boundary conditions in quantum systems

Physical wavefunctions must be continuous and finite; enforcing boundary conditions selects physically acceptable solutions and shapes orbital behavior.

Delocalization of electrons

Electrons behave as waves that are spread out over space rather than localized at a single point; electrons occupy orbitals describing where they are likely found.