Quantum Mechanical Model
The Bohr model was a giant leap forward, but it treated electrons as tiny billiard balls traveling in neat circular orbits. Modern quantum mechanics tells us something far stranger: electrons behave as both particles and waves, and we can never know exactly where an electron is and how fast it is moving at the same time.
Wave-Particle Duality
In 1924, Louis de Broglie proposed that if light can behave as both a wave and a particle (photons), then matter - including electrons - should also have wave-like properties. This was confirmed experimentally when electron beams were shown to produce diffraction patterns, just like light waves.
The Heisenberg Uncertainty Principle
Werner Heisenberg showed that there is a fundamental limit to what we can know about a particle’s position and momentum simultaneously:
It is impossible to determine both the exact position and the exact momentum of an electron at the same time.
This is not a limitation of our instruments - it is a feature of nature itself. To measure an electron’s position precisely, you must interact with it (hit it with a photon, for example), which changes its momentum. To measure its momentum precisely, you need a long wavelength photon that cannot pinpoint its position.
Orbits vs. Orbitals
This distinction is critical and frequently tested:
| Feature | Bohr Orbit | Quantum Orbital |
|---------|-----------|----------------|
| Shape | Fixed circular path | 3D region of probability |
| Electron position | Known exactly (on the path) | Probability distribution |
| Works for | Hydrogen only | All atoms |
| Based on | Classical + quantization | Full quantum mechanics |
An orbital is a region of space around the nucleus where there is a high probability (typically 90%) of finding an electron. The shape of this region depends on the quantum numbers, which we will explore in the next section.
The Electron Cloud
Instead of a planet orbiting a sun, picture a fuzzy cloud surrounding the nucleus. The cloud is denser where the probability of finding the electron is higher and thinner where the probability is lower. For an s orbital, this cloud is spherical - densest near the nucleus and fading out with distance. For a p orbital, the cloud is shaped like a dumbbell with two lobes on either side of the nucleus.
Orbital shapes and the four quantum numbers
Scroll sideways to see the whole map.
Where the capacities come fromA subshell has 2ℓ + 1 orbitals because mℓ runs from −ℓ to +ℓ, and each orbital holds two electrons because ms has two values. So s holds 2, p holds 6, d holds 10, f holds 14. Nothing there needs memorising separately.
What a node isA node is a surface where the probability of finding the electron is zero. The pinch at the centre of every p orbital is one; the two planes that separate a d cloverleaf's four lobes are two more. Higher ℓ means more nodal planes, which is most of why higher-ℓ subshells sit higher in energy.
Lobes are not pathsThe surfaces drawn here are probability contours, usually the boundary enclosing about 90 % of the electron density, not orbits. The plus and minus signs are the phase of the wavefunction, and they are what decides whether two orbitals overlap constructively into a bond.
What the Quantum Model Preserves from Bohr
Despite replacing orbits with orbitals, the quantum mechanical model keeps several key ideas from Bohr:
- Energy is still quantized - electrons can only exist at specific energy levels
- The principal quantum number (n) still describes the overall energy level
- Transitions between levels still involve absorbing or emitting photons with specific energies
- The ground state is still the lowest energy configuration
The quantum model adds more detail through additional quantum numbers (l, ml, and ms) that describe the shape, orientation, and spin within each energy level. These quantum numbers are the subject of the next section.
A Bohr orbit is a fixed, circular path at a defined distance from the nucleus. A quantum orbital is a three-dimensional region of space where there is a high probability of finding an electron. Bohr orbits give an exact location; orbitals give a probability distribution.
It is impossible to simultaneously determine both the exact position and the exact momentum of an electron. This is not a measurement limitation - it is a fundamental property of nature. The more precisely you know position, the less precisely you can know momentum, and vice versa.