Phases of atomic orbitals refer to the regions in space where the probability of finding an electron is either positive or negative. In quantum mechanics, the wave function describes the behavior of an electron in an atom, and the square of the wave function gives the probability density of finding an electron at a particular location.
The positive and negative phases of atomic orbitals are assigned based on the mathematical form of the wave function. The sign convention is arbitrary and does not indicate the distance of the orbital from the nucleus. It represents the sign of the amplitude of the wave function at different points in space.
For example, in the case of the 1s orbital of hydrogen, the wave function is spherically symmetric and has a positive phase throughout its entire volume. In the case of the 2p orbitals, which have dumbbell-shaped probability distributions, there are two lobes with opposite phases. One lobe has a positive phase, while the other has a negative phase. These phases are assigned based on the mathematical solutions of the Schrödinger equation for the hydrogen atom.
It is important to note that the positive and negative signs of the phases do not represent the physical distance from the nucleus. The probability of finding an electron is determined by the square of the wave function, not the sign of the phase.