What are the four quantumn numbers and what does each number tell us?

Answer 1

1. The principal quantum number (n = 1, 2, 3, 4, …) denotes the eigenvalue of Hamiltonian (H), i.e. the energy, with the contribution due to angular momentum (the term involving J2) left out. This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., 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.
2. The azimuthal quantum number (ℓ = 0, 1, …, n − 1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular momentum through the relationL2 = ħ2 ℓ (ℓ + 1). In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences 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.
3. The magnetic quantum number (ml = −ℓ, −ℓ + 1, …, 0, …, ℓ − 1, ℓ) yields the projection of the orbital angular momentum along a specified axis. Lz = mℓħ.
4. The spin projection quantum number (ms = ±½), is the intrinsic angular momentum of the electron or nucleon. This is the projection of the spin s = ½ along the specified axis.

Answer 2

The four quantum numbers are the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s). The principal quantum number (n) indicates the energy level of an electron and its average distance from the nucleus. The azimuthal quantum number (l) determines the shape of the electron's orbital. The magnetic quantum number (m_l) specifies the orientation of the orbital within a subshell, and the spin quantum number (m_s) indicates the spin of the electron.