## Summary of Quantum Numbers For Electrons in Atoms

Solution of the Schrodinger Equation for the electron in the hydrogen atom gives rise to four quantum numbers.

1. The principal quantum number n. The allowed values of n are 1, 2, 3, 4, and so on. It may not be zero. This number along with the orbital magnetic quantum number l specifies the energy of an electron in the absence of an external field. and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell. For a hydrogen atom with n=1, the electron is in its ground state; if the electron is in the n=2 orbital, it is in an excited state. The total number of orbitals for a given n value is n^2 .

2. The angular momentum quantum number l specifies the shape of an orbital with a particular principal quantum number. This quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n:

l | 0 | 1 | 2 | 3 | 4 | 5 | ... |

Letter | s | p | d | f | g | h | ... |

The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f). The angular quantum number l can be any integer between 0 and n - 1. If n = 3, for example, l can be either 0, 1, or 2.

3. The magnetic quantum number (m-l ) can be any integer between -l and +l. If l = 2, m can be either -2, -1, 0, +1, or +2. This Specifies the orientation in space of an orbital of a given principlal quantum number n and orbital magnetic quantum number l. This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.

4. Spin Quantum Number (m_{s}):m_{s} = +½ or -½.

Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).

The allowed quantum number for each principle quantum number n are illustrated in the table.

n | l | m | Number of | Orbital | Number of |

1 | 0 | 0 | 1 | 1 | 2 |

2 | 0 | 0 | 1 | 2 | 2 |

1 | -1, 0, +1 | 3 | 2 | 6 | |

3 | 0 | 0 | 1 | 3 | 2 |

1 | -1, 0, +1 | 3 | 3 | 6 | |

2 | -2, -1, 0, +1, +2 | 5 | 3 | 10 | |

4 | 0 | 0 | 1 | 4 | 2 |

1 | -1, 0, +1 | 3 | 4 | 6 | |

2 | -2, -1, 0, +1, +2 | 5 | 4 | 10 | |

3 | -3, -2, -1, 0, +1, +2, +3 | 7 | 4 | 14 |