Quantum Numbers - Specifying the electron state

Quantum Numbers for Electron Orbital

Now it turns out that if you consider an electron trapped inside a sphere (instead of in a one-dimensional box) it will have standing waves that are very similar to an electron bound to a positively charged nucleus by electrostatic attraction. Normally we operate in 4 dimensions (3 space and 1 time dimension), so in general we will need 4 quantum numbers to fully specify an electron state (i.e., standing wave).

Let's look at the quantum numbers needed to label the possible standing waves or states of an electron trapped by its electrostatic attraction to a positively charged nucleus. For an electron trapped by its electrostatic attraction to positively charged nucleus we use the three following quantum numbers to describe the electron state (orbital).

n : the principal quantum number

The principal quantum number has integral values of n = 1, 2, 3... As n increases, the electron orbital becomes larger and the electron spends more time farther from the nucleus.

ℓ : azimuthal quantum number

The azimuthal quantum number has integral values of ℓ = 0 to ℓ = n - 1 for each value of n. This quantum number defines the shape of the orbital. There are special letters assigned to each ℓ value (see table below).

For an electron with n = 1 and ℓ = 0 we say that the electron is in the 1s state or 1s orbital

For an electron with n = 2 and ℓ = 0 we say ... 2 s orbital

For an electron with n = 2 and ℓ = 1 we say ... 2 p orbital

For an electron with n = 3 and ℓ = 2 we say ... 3 d orbital

m_ℓ : magnetic quantum number

The magnetic quantum number has integral values of m_ℓ = - ℓ to + ℓ including 0.

For example, if n = 3 and ℓ = 2 then the possible values of m_ℓ are -2, -1, 0, +1, +2

m_s, : spin quantum number

The spin quantum number has only two possible values of +1/2 or -1/2. If a beam of hydrogen atoms in their ground state (n = 1, ℓ = 0, m_ℓ = 0) or 1s is sent through a region with a spatially varying magnetic field, then the beam splits into two beams.

Clearly the three quantum numbers, n, ℓ, m_ℓ, are not enough to completely describe the state of the H-atom, Another quantum number is required to describe whether it goes up or down in a spatially varying magnetic field. This property is called spin because if electrons were balls of charge spinning about their own axes, they would behave in this way in a magnetic field. Actually, this is not a correct picture. The spin quantum number shows up when the wave function of quantum mechanics is modified to include the effects of relativity.

The electron spin is very important in understanding the electronic structure of atoms containing many electrons (not just one like Hydrogen) because of the Pauli Exclusion Principle.

Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers n, ℓ, m_ℓ, m_s.

Since there are only two values of m_s, then any orbital can only hold two electrons with opposite vales of m_s. Any more would violate Pauli's Principle.

In summary, you can think of electron orbitals as standing waves of the electron wave function when it is bound to a postively charged nucleus. Remember, however, that the electron is not just a wave. It is a particle with wave-like properties. The wave function squared gives the probability of finding the electron at any particular position.