The Quantum-Mechanical Model of an Atom

Position and energy are complementary properties of an electron due to its wave-particle duality. When an electron has a well-defined energy, its position becomes less precisely known. Instead of a fixed location, the electron's position is described by an electron probability density, which maps the likelihood of finding the electron at specific locations around the nucleus.

The square of the wavefunction, ψ², represents the electron probability density. The density of dots in a ψ² plot is proportional to the probability per unit volume of locating the electron at a particular position. Larger values of ψ² indicate higher probability of finding the electron at that location relative to the nucleus.

The Schrödinger Equation mathematically integrates both the wave-nature and particle-nature of electrons. It solves for the energies and probability distributions of electrons in atoms. The equation uses a Hamiltonian operator representing total energy, a wavefunction ψ, and yields the actual energy E of the electron, providing the foundation for modern quantum mechanics.

An atomic orbital is the three-dimensional region where there is the highest probability of finding an electron of specific energy. Orbitals have varying shapes depending on quantum numbers. Unlike Bohr's orbits, which represent fixed quantized energy levels, orbitals describe probability distributions as an electron cloud surrounding the nucleus rather than definite circular paths.

Electron probability density is higher closer to the nucleus and decreases with distance. A plot of ψ² versus r (distance from nucleus) shows where an electron is most likely to exist. The electron is more probable near the nucleus than far away, reflecting the attractive electrostatic force between the negatively charged electron and positively charged nucleus.

The quantum mechanical model is based on probabilities and electron probability densities rather than fixed orbits. It depicts electrons as an electron cloud surrounding the nucleus, providing a more accurate representation of atomic structure. This model successfully reproduces experimental observations like the Rydberg formula and hydrogen spectra that the Bohr model predicted.

The Born interpretation states that electrons remain particles, and the waves represented by ψ are not physical waves but complex probability amplitudes. The square of the wavefunction magnitude, |ψ|², describes the probability of finding the quantum particle near a specific location in space, allowing wavefunctions to determine electron density distribution around the nucleus.