Principal quantum number

In quantum mechanics, the principal quantum number (symbolized n) is one of four quantum numbers which are assigned to each electron in an atom to describe that electron's state. As a discrete variable, the principal quantum number is always an integer. As n increases, the number of electronic shells increases and the electron spends more time farther from the nucleus. As n increases, the electron is also at a higher potential energy and is therefore less tightly bound to the nucleus.

The principal quantum number was first created for use in the semiclassical Bohr model of the atom, distinguishing between different energy levels. With the development of modern quantum mechanics, the simple Bohr model was replaced with a more complex theory of atomic orbitals. However, modern theory still requires the principal quantum number. Apart from the principal quantum number, the other quantum numbers for bound electrons are the azimuthal quantum number, the magnetic quantum number, and the spin quantum number.

For an analogy, one could imagine a multistoried building with an elevator structure. The building has an integer number of floors, and a (well-functioning) elevator which can only stop at a particular floor. Furthermore, the elevator can only travel an integer number of levels. As with the principal quantum number, higher numbers are associated with higher potential energy.

Beyond this point the analogy breaks down; in the case of elevators the potential energy is gravitational but with the quantum number it is electromagnetic. The gains and losses in energy are approximate with the elevator, but precise with quantum state. The elevator ride from floor to floor is continuous whereas quantum transitions are discontinuous. Finally the constraints of elevator design are imposed by the requirements of architecture, but quantum behavior reflects fundamental laws of physics.

There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, ℓ, m, and s specify the complete and unique quantum state of a single electron in an atom, called its wave function or orbital. Two electrons belonging to the same atom can not have the same four quantum numbers, due to the Pauli exclusion principle. The wave function of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The principal quantum number arose in the solution of the radial part of the wave equation as shown below.

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