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In the quantum mechanical model, electrons are described by wave functions, or mathematical functions that describe the electron's location, energy, and other properties. These wave functions, denoted by the Greek letter \(\psi\), are solutions of the Schrödinger equation, the central equation of quantum mechanics.

In the quantum mechanical model, the allowed energy states of an electron are described by a set of atomic orbitals, which are derived from the wave functions. Orbitals provide information about where an electron is likely to be found around the nucleus and are often represented by shapes such as spheres, dumbbells or lobes. Each orbital has a specific energy level and shape associated with it, and can hold a maximum of two electrons.

Each electron orbiting the nucleus is characterized by four quantum numbers that define its position, energy, and spin. These quantum numbers are: 1. Principal quantum number (n): This quantum number determines the energy level of an electron and its distance from the nucleus. The value of 'n' can be an integer greater than or equal to 1 (i.e., n = 1, 2, 3, ...). 2. Angular momentum quantum number (l): This quantum number determines the shape of an orbital. The value of 'l' can range from 0 to n - 1. The different shapes of orbitals correspond to s (l = 0), p (l = 1), d (l = 2), and f (l = 3) orbitals. 3. Magnetic quantum number (m_l): This number defines the orientation of the orbital in space, and its value can range from -l to +l. 4. Spin quantum number (m_s): Electrons have an intrinsic angular momentum called "spin," and each electron in an orbital can have a spin of either +1/2 or -1/2.

The square of the wave function, denoted as \(|\psi|^2\), represents the probability density of finding an electron at a specific point in space around the nucleus. A high probability density means that an electron is more likely to be found in that region, while a low probability density means that it is less likely to be found there. This concept replaces the fixed electron paths of the Bohr model with the electron cloud concept, where electrons can be found in specific regions with defined probabilities. In conclusion, the quantum mechanical model of the atom describes electron paths using wave functions, orbitals, and probability distributions instead of fixed orbits around the nucleus. This approach provides a more accurate representation of electron behavior and the arrangement of atoms in their energy levels.