Quantum Mechanical Model: Schrödinger & Orbitals

The quantum mechanical model replaced fixed electron orbits with orbitals — three-dimensional regions where an electron is likely to be found. It starts from the Schrödinger equation, whose solutions (wavefunctions, ψ) do not say where the electron is, but give the probability of finding it at each point in space.

This matters because it is the model chemists actually use. Orbital shapes explain how atoms bond, why the periodic table has its structure, and why molecules have their shapes. Every modern description of chemical bonding — from simple covalent bonds to molecular orbital theory — is built on these orbitals.

Key Concepts

• Wavefunction (ψ, psi): a mathematical function describing the electron's wave; it can be positive, negative or zero.
• Probability density (|ψ|²): the square of the wavefunction, giving the probability per unit volume of finding the electron at a point. Where |ψ|² is large, the electron is likely to be.
• Orbital vs orbit: an orbit is a fixed circular path (Bohr); an orbital is a probability cloud with a characteristic shape. Electrons do not travel on tracks.
• Quantum numbers: n (size/energy: 1, 2, …), l (shape: 0 = s, 1 = p, 2 = d, 3 = f), ml (orientation: −l … +l), ms (spin: +½ or −½).
• Node: a surface where ψ = 0, so the electron is never found there. Total nodes = n − 1; radial nodes = n − l − 1; angular nodes = l.
• Orbital shapes: s orbitals are spherical, p orbitals are dumbbell-shaped (three orientations), d orbitals mostly have four lobes.

Formulas

• Schrödinger equation (time-independent): Ĥψ = Eψ, where Ĥ is the energy operator and E the allowed energy. Solving it for hydrogen gives the exact energy levels En = −13.6 eV/n².
• Node counting: total nodes = n − 1; radial nodes = n − l − 1; angular (planar) nodes = l.
• Number of orbitals in a subshell: 2l + 1 (one s, three p, five d orbitals).

The equation is solved exactly only for one-electron systems; for larger atoms, approximations are used.

Worked Example

For a 3d orbital, determine n, l, the number of orbitals in the subshell, and the node count.

1. Read the label "3d": n = 3, and d means l = 2.
2. Number of orbitals: 2l + 1 = 2 × 2 + 1 = 5.
3. Total nodes: n − 1 = 3 − 1 = 2.
4. Radial nodes: n − l − 1 = 3 − 2 − 1 = 0.
5. Angular nodes: l = 2.
6. Check: 0 radial + 2 angular = 2 total. Consistent.

A 3d subshell has 5 orbitals and each has 2 angular nodes, 0 radial nodes.

Common Misconceptions

• Electrons revolve around the nucleus inside orbitals like planets. Orbitals are not paths — they are probability maps. The electron has no defined trajectory; only the likelihood of finding it somewhere.
• |ψ|² shows where the electron's charge is smeared. It is a probability density for one electron's position, not a physical smearing of charge.
• Nodes are places the electron passes through quickly. At a node the probability is exactly zero — the electron is never there, which is why p orbitals have two separate lobes.

Real-World Applications

• Orbital overlap explains covalent bonding and molecular shapes (the basis of VSEPR and hybridisation).
• Computational chemistry designs drugs and materials by solving approximations of the Schrödinger equation.
• Semiconductor band structure, which makes transistors possible, comes from orbital theory in solids.
• Spectroscopy interprets absorption lines as transitions between orbital energy levels.
• Scanning tunnelling microscopy images surfaces by probing electron probability densities.

Solve Ĥψ = Eψ, square the wavefunction to get |ψ|², and you get orbitals — probability clouds with shapes, nodes and quantum numbers. That is the working picture of the atom behind all of modern chemistry.