The spherically symmetric electron orbitals in a hydrogen atom arise from the fundamental properties of the system and the solutions to the Schrödinger equation. Here are the key reasons for their spherical symmetry:
Central Coulombic Potential: In the hydrogen atom, the nucleus (proton) exerts a positive charge, creating a central Coulombic potential. This potential depends only on the distance between the electron and the nucleus, not on the direction. It implies that the electron experiences the same attractive force toward the nucleus from any direction. As a result, the electron probability density (squared wave function) becomes spherically symmetric.
Angular Momentum Quantum Number (l): The angular momentum quantum number, denoted by 'l,' determines the shape of the electron orbital. For spherically symmetric orbitals, l = 0. The solutions for l = 0 (s orbitals) have a spherical symmetry due to the mathematical form of the solutions to the Schrödinger equation.
Radial Probability Density: The radial part of the wave function, which depends only on the distance from the nucleus (r), determines the probability of finding the electron at a particular distance. The radial probability density is highest near the nucleus and decreases with increasing distance. Since the probability density is radially symmetric, the electron's position is equally likely in all directions around the nucleus.
Overall, the combination of the central Coulombic potential, the specific solutions to the Schrödinger equation for hydrogen's electron orbitals, and the resulting spherical symmetry of the wave functions lead to the spherically symmetric electron orbitals observed in the hydrogen atom.