The Quantum-Mechanical Model of an Atom

Recall that the particle-nature and wave-nature of an electron, and by extension its position and velocity, are complementary properties. Likewise, since kinetic energy is a function of velocity, position and energy are also complementary properties.

Therefore, for an electron with a well-defined energy, its position is less precisely known. Instead, the electron's position is described by an electron probability density, which maps the probability of finding an electron at a specific energy. In this representation, the electron is more likely to be closer to the atom’s nucleus than very far away from it.

The energies and probability distribution of an electron are mathematically derived by solving the Schrӧdinger Equation, which integrates both the wave-nature and particle-nature of the electron. Here, E is the actual energy of the electron, H is a mathematical operator, and ψ is a wave function.

Solving the Schrӧdinger Equation yields many possible wave functions. However, it is the square of the wave-function, ψ², that represents the electron probability density.

The density of dots is proportional to the probability per unit volume of locating the electron at a particular position. A plot of ψ² with respect to r — the distance from the atom’s nucleus — displays where the electron is most likely to exist in an atom.

The larger the value of ψ², the higher the probability of finding the electron.

This particular map represents the probability density of hydrogen, which has 1 electron. The three-dimensional region where there is the highest probability of finding an electron of specific energy is called an orbital.

Orbitals have varying shapes, ranging from spherical to more complex, depending on the values of their quantum numbers. These orbitals are not the same as the orbits that were initially described by Bohr in his atomic model. In Bohr’s model, the orbits represent quantized energy levels.

Thus, the quantum mechanical model of the atom, which is based on probabilities, is the most contemporary representation of atomic structure. It provides a more accurate representation of the atom by depicting the electron probability density as a ‘cloud’ of electrons surrounding the nucleus.