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In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number n, the magnetic quantum number m_l, and the spin quantum number m_s). It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as l (pronounced ell). Connected with the energy states of the atom's electrons are four quantum numbers: n, l, ml, and ms. These specify the complete, unique quantum state of a single electron in an atom, and make up its wavefunction or orbital. When solving to obtain the wave function, the Schrödinger equation reduces to three equations that lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation as shown below , reliant on the spherical coordinate system, which generally works best with models having some glimpse of spherical symmetry. An atomic electron's angular momentum, L, is related to its quantum number l by the following equation: where ħ is the reduced Planck's constant, L2 is the orbital angular momentum operator and is the wavefunction of the electron. The quantum number l is always a non-negative integer: 0, 1, 2, 3, etc. L has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is better to simply use the quantum number l. Atomic orbitals have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d (a convention originating in spectroscopy) describe the shape of the atomic orbital. Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials.

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