Quantum number

A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. Each quantum number specifies the value of a conserved quantity in the dynamics of the quantum system. Since any quantum system can have one or more quantum numbers, it is a futile job to list all possible quantum numbers. This article therefore illustrates the concepts by choosing two well-known examples, after a brief introduction to the general concept of quantum numbers.

Single electron in an atom

This section is not meant to be a full description of this problem. For that, see the article on the Bohr atom.

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The most widely studied set of quantum numbers is that for a single electron in an atom: not only because it is useful in chemistry, being the basic notion behind the periodic table, valence (chemistry) and a host of other properties, but also because it is a solvable and realistic problem, and, as such, finds widespread use in textbooks.

Related Topics:
Electron - Atom - Chemistry - Periodic table - Valence (chemistry)

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In non-relativistic quantum mechanics the Hamiltonian of this system consists of the kinetic energy of the electron and the potential energy due to the Coulomb force between the nucleus and the electron. The kinetic energy can be separated into a piece which is due to angular momentum, J, of the electron around the nucleus, and the remainder. Since the potential is spherically symmetric, the full Hamiltonian commutes with J2. J2 itself commutes with any one of the components of the angular momentum vector, conventionally taken to be Jz. These are the only mutually commuting operators in this problem; hence, there are three quantum numbers.

Related Topics:
Quantum mechanics - Kinetic energy - Potential energy - Coulomb force - Nucleus

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These are conventionally known as

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• The principal quantum number (n = 1, 2, 3,...) denotes the eigenvalue of H with the J² part removed. This number therefore has a dependence only the distance between the electron and the nucleus (ie, the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells.
• The azimuthal quantum number (l = 0, 1 ... n−1) (also known as the angular quantum number or orbital quantum number) gives the angular momentum through the relation J² = l(l+1) h/2π, where h is the universal constant known as the Planck's constant. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly inflences chemical bonds and bond angles. In some contexts, l=0 is called an s orbital, l=1, a p orbital, l=2, a d orbital and l=3, an f orbital.
• The magnetic quantum number (m_l = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue, J_z=m_lh/2π.

Note that molecular orbitals require totally different quantum numbers, because the Hamiltonian and its symmetries are quite different.

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