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A quantum system may have two or more eigenstates (or eigenfunctions), each is a solution of the Schrödinger Equation - a fundamental equation in quantum mechanics. An eigenstate is the state that can actually be observed. Since all eigenstates are the solutions of the Schrödinger Equation, their superposition Ψ (called wavefunction) is also a solution. For a system with two eigenstates, Φ1 and Φ2, Ψ may be written as
Ψ = a1 Φ1 + a2 Φ2
where a1 and a2 are the probability amplitudes. That means, the probability for the system to be in Φ1 or Φ2 is |a1|2 or |a2|2, respectively.
The system consisting of an electron surrounding an atomic nucleus has numerous eigenstates. Each eigenstate is characterized by three quantum numbers: n, ℓ, and m. These eigenstates are also called orbitals. The s orbital, p orbital and d orbital refer to the eigenstates with ℓ = 0, 1 and 2 respectively (Figure 9-1). Mathematically, the eigenfunctions can be represented by spherical harmonics Yℓm(θ, φ), which have aso been used to describe the scalp electric potentials measured by EEG (Section 9.3).
Figure 9-1. Atomic orbitals.