The separation of p-orbitals into px, py, and pz orbitals is a consequence of the three-dimensional nature of space and the mathematical properties of quantum mechanics.
In quantum mechanics, orbitals are described by mathematical functions called wavefunctions. These wavefunctions represent the probability distribution of finding an electron in a particular region of space. The wavefunctions for p-orbitals have different spatial orientations, corresponding to the different axes of a three-dimensional coordinate system (x, y, and z axes).
The px orbital corresponds to the x-axis and has a characteristic dumbbell shape with its lobes aligned along the x-axis. The py orbital corresponds to the y-axis and has a similar dumbbell shape with its lobes aligned along the y-axis. Similarly, the pz orbital corresponds to the z-axis and has lobes aligned along the z-axis.
These orientations arise due to the mathematical solutions to the Schrödinger equation, which governs the behavior of electrons in quantum mechanics. The solutions for the p-orbitals yield wavefunctions that have distinct nodal planes along each axis, resulting in the separation of the p-orbitals into px, py, and pz.
It's important to note that these orbital designations are arbitrary labels for convenience and don't imply any physical significance beyond their mathematical representation. The electron's behavior within an orbital is still described by a complex wavefunction that takes into account the interactions with other particles and the overall quantum state of the system.