Quantum numbers (n, ℓ, mℓ, and ms) act like an electron’s address within an atom. They come from Schrödinger’s equation and describe different aspects of an electron’s state. A loose way to think of them: n is like the country, ℓ is like the state or province, mℓ is like the city, and ms is like the house number

Principal Quantum Number (n)

This number tells us the main energy level of the electron and roughly how far it is from the nucleus. It’s a positive integer: 1, 2, 3, etc. As n increases, both energy and distance increase. For hydrogen, the energy of a level is given by the formula:

En = –13.6 eV / n²

Higher levels are more closely spaced in energy, which is a pattern seen in spectral lines like the Balmer series.

Orbital Angular Momentum Quantum Number (ℓ)

ℓ defines the subshell and the shape of the orbital. For a given n, ℓ can range from 0 to n–1. ℓ = 0: s orbitals (spherical)

• ℓ = 1: p orbitals (dumbbell-shaped)

• ℓ = 2: d orbitals (cloverleaf-shaped)

• ℓ = 3: f orbitals (more complex)

The angular momentum of the orbital is related to ℓ by the formula: 

L = √[ℓ(ℓ + 1)] ℏ

Magnetic Quantum Number (mℓ)

mℓ specifies the orientation of the orbital within a given subshell. Its values range from –ℓ to +ℓ.

• For s orbitals (ℓ = 0): mℓ = 0

• For p orbitals (ℓ = 1): mℓ = –1, 0, +1

• For d orbitals (ℓ = 2): mℓ = –2, –1, 0, +1, +2

While these values correspond to different orientations, they don’t map directly onto labels like px, py, or pz. When a magnetic field is present, the different orientations lead to energy splitting (the Zeeman effect).

Spin Quantum Number (ms)

ms describes the intrinsic spin of the electron. It can be either +½ or –½. According to the Pauli exclusion principle, no two electrons in the same atom can have all four quantum numbers identical. That’s why each orbital can hold two electrons, but only if they have opposite spins.

Atoms with unpaired electrons (with spins not cancelling) have a net magnetic moment. Atoms with all electrons paired do not.

Using Quantum Numbers to Define Orbitals

A set of n, ℓ, and mℓ defines a specific orbital:

• The 1s orbital has n = 1, ℓ = 0, mℓ = 0

• The 2p orbitals have n = 2, ℓ = 1, mℓ = –1, 0, +1

Each of these orbitals can hold two electrons, one with ms = +½, the other with ms = –½.

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