Explain electron from a quantum mechanical perspective, including a discussion of atomic radii, probabilities, and orbitals.

In quantum mechanics, the electron is described by a wavefunction (ψ), which is a mathematical function that contains all the information about the electron in an atom. The square of the wavefunction (|ψ|^2) gives the probability distribution of finding the electron in a particular region of space, known as electron cloud or electron density.

An atom consists of a positively charged nucleus and negatively charged electrons surrounding it. The distribution of electrons in the atom is described by atomic orbitals, which are mathematical functions that describe the electron's wave-like behavior in specific energy levels. There are four types of atomic orbitals (s, p, d, and f), each with particular shapes and orientations. The s-orbital is spherical, the p-orbital is dumbbell-shaped, and the d and f orbitals have more complex shapes. The orbitals are arranged in increasing energy levels, usually denoted by the principal quantum number (n). Each energy level can hold a certain number of electrons, following the formula 2n^2. For example, the first energy level (n = 1) can hold up to 2 electrons, while the second energy level (n = 2) can hold up to 8 electrons.

The atomic radius is the distance between the nucleus of an atom and its outermost electron cloud. As one moves across a row in the periodic table, from left to right, the atomic radii generally decrease. This is because, as the nuclear charge (number of protons) increases, the electrons in the same energy level are more strongly attracted to the nucleus, resulting in a smaller atomic radius. The atomic radii also increase as one moves down a column in the periodic table. This is due to the addition of energy levels with an increased number of orbitals, which increases the distance between the nucleus and outermost electron cloud.

As mentioned earlier, the square of the wavefunction (|ψ|^2) gives the probability distribution of finding the electron in a region of space. This probability distribution, often depicted as an electron cloud, displays the most probable locations for the electron to be found in the atom. The electron cloud is denser where the probability of finding an electron is higher. For example, the probability distribution for an electron in the 1s orbital is highest close to the nucleus and gradually decreases as we move away from the nucleus. The probability distribution for the 2p orbital has two lobes, one on each side of the nucleus, where the electron is most likely to be found. In summary, electrons in atoms can be understood through a quantum mechanical perspective by discussing wavefunction, probability distribution, atomic orbitals, atomic radii, and electron probability distributions. Each of these concepts provides insight into the complex behavior of electrons and their arrangement in atoms.