7.9:
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.
7.9:
The Quantum-Mechanical Model of an Atom
Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger described electrons as three-dimensional stationary waves, or wavefunctions, represented by the Greek letter psi, ψ.
A few years later, Max Born proposed an interpretation of the wavefunction ψ that is still accepted today: Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes. The square of the magnitude of a wavefunction ∣ψ∣2 describes the probability of the quantum particle being present near a certain location in space. This means that wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom. In the most general form, the Schrödinger equation can be written as:
where, Ĥ is the Hamiltonian operator, a set of mathematical operations representing the total energy (potential plus kinetic) of the quantum particle (such as an electron in an atom), ψ is the wavefunction of this particle that can be used to find the special distribution of the probability of finding the particle, and E is the actual value of the total energy of the particle.
Schrödinger’s work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as quantum mechanics.
The quantum mechanical model describes an orbital as a three-dimensional space around the nucleus within an atom, where the probability of finding an electron is the highest.
This text is adapted from Openstax, Chemistry 2e, Section 6.3: Development of Quantum Theory.