05 Quantum Mechanics and Atomic Orbitals

Quantized Energy and Light

Atoms absorb and emit light at particular wavelengths, producing line spectra rather than a continuous range. These lines show that an electron can occupy only certain allowed energies. This restriction is called Quantization.

When an electron moves between allowed energy states, it absorbs or emits a photon whose energy matches the difference between the states:

Here, $h$ is Planck’s constant, $\nu$ is photon frequency, $c$ is the speed of light, and $\lambda$ is wavelength. Absorption raises the electron’s energy; emission lowers it. The electron does not occupy energies between the allowed states.

Takeaway: Atomic energy changes occur in discrete steps, and each transition exchanges a photon with the corresponding energy.

Wave–Particle Duality and Orbitals

Light and matter exhibit both wave-like and particle-like behavior. Electrons can produce interference patterns characteristic of waves, yet they are detected as localized particles. The de Broglie relation connects a particle’s momentum to its wavelength:

Here, $p$ is momentum and $h$ is Planck’s constant.

In quantum mechanics, an electron is described by a Wavefunction, written as $\psi$. Its squared magnitude gives the probability density of finding the electron at a location:

An Atomic orbital describes this spatial probability distribution. It is not a miniature planetary orbit or a fixed path. Orbital drawings usually show a surface enclosing a chosen fraction of the probability; they do not form solid walls, and some probability lies beyond the pictured surface.

Takeaway: Wavefunctions describe electron states, and their squared magnitudes connect those states to the probability of finding an electron.

Quantum Numbers and Subshells

Four Quantum numbers describe an electron’s state. The first three specify an orbital, while the fourth specifies the electron’s spin state.

• The principal quantum number, $n$, takes positive integer values: $1, 2, 3, \ldots$. It identifies the shell and generally relates to orbital size and energy.

• The angular momentum quantum number, $l$, ranges from $0$ to $n-1$. It identifies the subshell and relates to orbital shape. The labels are $l=0$ for $s$, $l=1$ for $p$, $l=2$ for $d$, and $l=3$ for $f$.

• The magnetic quantum number, $m_l$, is an integer from $-l$ through $+l$. It specifies an orbital’s spatial orientation.

• The spin quantum number, $m_s$, is $+\frac{1}{2}$ or $-\frac{1}{2}$, representing the electron’s intrinsic spin state.

A subshell contains $2l+1$ orbitals, and each orbital can hold up to two electrons with opposite spin. For example, when $n=3$ and $l=2$, the possible values of $m_l$ are $-2, -1, 0, +1, +2$. Therefore, the $3d$ subshell contains five orbitals.

Takeaway: The values of $n$, $l$, and $m_l$ identify an orbital’s shell, subshell, and orientation; $m_s$ distinguishes the electron’s spin state.

Orbital Shapes and Nodes

Orbital shapes depend on the angular momentum quantum number. The familiar drawings are visual representations of probability distributions.

• s orbitals are spherical. They have no angular nodes, although higher $s$ orbitals can have spherical radial nodes.

• p orbitals have two lobes separated by a nodal plane through the nucleus. Each $p$ subshell has three differently oriented orbitals.

• d orbitals generally have four lobes; one has two lobes and a ring-like region. A $d$ subshell has five orbitals.

• f orbitals have more complex shapes, and an $f$ subshell has seven orbitals.

A Node is a region where the wavefunction and therefore the probability density are zero. For an orbital described by $n$ and $l$, the number of angular nodes is $l$, the number of radial nodes is $n-l-1$, and the total number of nodes is $n-1$.

For example, a $3p$ orbital has $n=3$ and $l=1$. It has one angular node and one radial node, for a total of two nodes.

Takeaway: Orbital shape and node counts follow from the quantum numbers; nodes mark places where the electron’s probability density is zero.

Orbital Energies

For a hydrogen atom or another one-electron species, orbital energy depends only on the principal quantum number, $n$. In the nonrelativistic model:

Here, $Z$ is the nuclear charge number. As $n$ increases, the energy becomes higher, meaning closer to zero, and the orbital is larger on average. In hydrogen, orbitals with the same $n$, such as $2s$ and $2p$, have the same energy; these are Degenerate orbitals.

In atoms with multiple electrons, electron–electron interactions change the energy pattern. Orbitals with the same $n$ but different $l$ generally do not have equal energies. Their relative energies reflect electron repulsion and how strongly the electrons in each orbital are attracted to the nucleus.

Takeaway: One-electron orbital energies depend only on $n$, while interactions in multi-electron atoms generally separate the energies of subshells with the same $n$.