**Quantum
Mechanics**

The theory or subject of quantum mechanics is based on advanced physics, chemistry and high level mathematics. For nearly a century, a large number of individuals have devoted their entire careers to the theory of quantum mechanics. Quantum equations produce numbers that are used to describe subatomic particles. Chemists use them to describe the atomic orbital shapes electrons produce for individual atoms, as well as how elements and molecules will react when brought together. Four distinct quantum numbers describe each electron. No two electrons in the same atom will have the same four quantum numbers. Chemistry teachers often liken quantum numbers to apartment addresses in that no two apartments in a building will have the same address. Four quantum numbers (n, ℓ, mℓ, ms) can be assigned to any electron. The four quantum numbers can be predicted using orbital-filling diagrams, electron-configuration diagrams or the position of the element on the periodic table. Even though chemists are able to predict the four quantum numbers for any electron, a few elements have an electron that is not found in its anticipated or predicted orbital. However, the explanations for these cases are better left to theoretical physicists and chemists.

**Principal Quantum Number (n)**

The first quantum number, also known as the principal quantum number (n), tells how far the electron is from the nucleus of the atom. The larger the number, the greater the distance the electron is from the nucleus. The values for n are n = 1, 2, 3…

**Azimuthal Quantum Number (ℓ)**

An azimuthal quantum number (ℓ) or second quantum number, identifies the energy sublevel (s, p, d, f, g) in which the electron is found. The numbers used to identify the sublevel are s = 0, p = 1, d = 2 and f = 3. The second quantum number is also known as the angular quantum number or orbital quantum number. The g energy sublevel will be used starting with element 121 and larger atomic number elements.

**Magnetic Quantum Number (m**ℓ)

Orbitals are assigned a magnetic quantum number (mℓ) that tells the direction of the orbital with respect to the other orbitals in the atom's shell. The values used for identifying the orbital depends on the sublevel: s has 0; p has -1, 0, +1; d has -2, -1, 0, +1, +2; and, f has -3, -2, -1, 0, +1, +2, +3.

**Spin Quantum Number (ms) **

A spin quantum number, (ms), is assigned to distinguish between two electrons occuping the same orbital. The spin quantum number has two values: ms=+½ for an arrow up and ms=-½ for an arrow down.

**Quantum Numbers Example One: Helium's 1s² Electrons**

Helium has the electron-configuration diagram 1s². The 1 in the electron-configuration diagram identifies the principal quantum number: n=1. The s in the electron-configuration diagram identifies the azimuthal quantum number (ℓ): ℓ=0. The azimuthal quantum number for any s sublevel has just mℓ = 0. See figure 1. Since helium has one s orbital, the two electrons are in that one orbital. However, they must have opposite spin to occupy the same orbital: ms=+½ for an arrow up and ms=-½ for an arrow down. The quantum numbers (n, ℓ, mℓ, ms) for the two electrons are: 1, 0, 0, +½ and 1, 0, 0, -½.

**Quantum Numbers ****Example Two:**** Neon's 2p⁶ Electrons**

[He]2s^{2}2p^{6} is the electron-configuration
diagram for neon. The 2 in the 2p^{6} in electron-configuration diagram identifies the principal quantum number: n = 2. The p in the
electron-configuration diagram identifies the
azimuthal quantum number (ℓ): ℓ=1. The choices are s = 0,
p = 1, d = 2 and f = 3. The
magnetic quantum numbers (mℓ) for a p orbital are mℓ = -1, 0, +1. Since neon has three 2p⁶ orbitals, the six
electrons are in the three orbitals. Each
orbital will have two electrons with opposite spin: ms = +½ for an arrow up and ms = -½ for an arrow down. The
quantum numbers (n, ℓ, mℓ, ms) for
the six electrons in the 2p⁶ are: 2, 1,-1, +½;
2, 1, 0, +½; 2, 1, +1, +½; 2,
1, -1, -½; 2, 1, 0, -½; and, 2, 1, +1, -½. See figure 3.

**Quantum
Numbers ****Example Three: ****Zinc's 3d ^{10} Electrons **

The electron-configuration diagram for zinc
is [Ar]4s^{2}3d^{10}. The 3 in the 3d^{10} from
the electron-configuration diagram identifies the principal quantum number: n=3. The d in the electron-configuration
diagram identifies the azimuthal quantum number (ℓ): ℓ=2. The values used for
identifying each orbital for the d
are mℓ= -2, -1, 0, +1, +2. Since zinc has five 3d
orbitals, containing 10 electrons, each orbital will
have two electrons with opposite spin: ms = +½ for an arrow up and ms = -½ for
an arrow down. The quantum numbers (n, ℓ, mℓ, ms) for the ten
electronsorbital-filling are: 3, 2, -2, +½;
3, 2, -1, +½; 3, 2, 0, +½; 3,
2, +1, +½; 3, 2, +2, +½; 3, 2, -2, -½;
3, 2, -1, -½; 3, 2, 0, -½; 3, 2, +1, -½; and, 3, 2, +2, -½. See
figure 4.