Question

Where do quantum numbers come from and how were they determined?

Answer 1

Quantum numbers are defined when determining the eigenvalues of energy and angular momentum, arising from their quantization. Calculating them is probably beyond the scope of general chemistry. You just need to know how to use them.

• The principal quantum number is `n = 1, 2, 3, . . .`, describing most directly the atomic energy levels.

• The angular momentum quantum number is `l = 0, 1, 2, . . . , n-1` describes the shape of the orbital.

• The magnetic quantum number `m_l` is in the set `{-l, -l-1, . . . , 0, . . . , l-1, l}`.

Since the minimum `m_l` is `-l` and the maximum `m_l` is `l`, there are `2l+1` such values of `m_l`, corresponding to the number of orbitals in each subshell (given by `l`).

• The electron spin quantum number `m_s` is `pm1/2` because the spin of the electron was experimentally determined to be `pm1/2`, just like other particles that are classified as fermions.

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Nevertheless, here is a brief origin of these quantum numbers from quantum chemistry.

`ul(n - "principal quantum number")`

An example of atomic energy quantized to the quantum number `n` is for the hydrogen atom (or other atoms with one electron):

`E_color(blue)(n) = -(2pi^2me^4Z^2)/(color(blue)(n^2)h^2)`

Here, the energy `E_n` is dependent on `n`, and is proportional to `1/n^2`.

`ul(l - "angular momentum quantum number")`

The square of the orbital angular momentum, `L^2`, has the eigenvalue `l(l+1)ℏ^2`, such that:

`ul(hat(L)^2) [Y_(color(blue)(l))^(m_l)(theta,phi)] = ul(color(blue)(l(l+1))ℏ^2)" "Y_(l)^(m_l)(theta,phi)`

where `Y_(l)^(m_l)(theta,phi)` is the angular part of the wave function, which is dependent on `l` and `m_l`, and `hat(L)^2` is the operator for `vecL cdot vecL`, the dot product of the orbital angular momentum `L` with itself.

(an operator acts on a function.)

Basically, the angular momentum you observe spits back a result that is given by `sqrt(l(l+1)ℏ^2)`.

`ul(m_l - "magnetic quantum number")`

The z-component `L_z` of the orbital angular momentum `L` has the eigenvalue `m_lℏ`, such that:

`ul(hatL_z) [Y_(l)^(color(blue)(m_l))(theta,phi)] = ul(color(blue)(m_l)ℏ)" "Y_(l)^(m_l)(theta,phi)`

where `hatL_z` is the operator for the z-component of the orbital angular momentum, `L_z`.

`ul(m_s - "electron spin quantum number")`

Similarly, the z-component `S_z` of the spin angular momentum `S` has the eigenvalue `m_sℏ`, such that:

`ul(hat(S)_z) [color(blue)(vecalpha)] = ul(color(blue)(+1/2)ℏ)" "vecalpha`

`ul(hat(S)_z) [color(blue)(vecbeta)] = ul(color(blue)(-1/2)ℏ)" "vecbeta`

where `hat(S)_z` is the operator for the z-component of the spin angular momentum, `S_z`.

And it was defined that `m_s = +1/2` for the `vecalpha` wave function (which stood for an electron with an up-spin), while `m_s = -1/2` for the `vecbeta` wave function (which stood for an electron with a down-spin).