Question

Is it true that for a value of `n` (n-principal quantum number), the value of `m_l` (`m_l`-magnetic quantum number) is equal to `n^2`? If it is true, can you please explain this to me? Thanks.

Answer 1

The actual value, no. The number of values, yes.

Explanation:

The relationship between the principal quantum number, `n`, and the magnetic quantum number, `m_l`, actually goes through the angular momentum quantum number, `l`.

The angular momentum quantum number can take any value that ranges from `0` to `n-1`

`l = 0, 1, 2, ... , (n-1)`

The magnetic quantum number can take any value that ranges from `-l` to `l`

`m_l = -l, ... , -1, 0, 1, ... , l`

If you take into account the possible values of `l`, the magnetic quantum number can thus have values that range from

`m_l = -(n-1), ..., -1, 0, 1, ..., (n-1)`

So, for example, if `n=2`, `m_l` can be equal to

`m_l = 0` `->` for `l=0`

and

`m_l = {-1, 0, 1}` `->` for `l=1`

As you can see, the individual values of `m_l` don't even come close to the value of `n^2`.

However, the total number of values `m_l` can take, if you take into account the values of `l`, is indeed equal to `n^2`.

The magnetic quantum number actually tells you how many orbitals a subshell has. In the above example, if `n=2`, you get

`n=2`, `l=0`, `m_l = 0` `->` one 2s-orbital;

`{:(n=2, l=1, m_l = -1->2p_x), (n=2, l=1, m_l = 0->2p_y), (n=2, l=1, m_l = 1->2p_z) :}}` `->` three 3p-orbitals.

This means that the second energy level has a total of two subshells, the 2s-subshell and the 2p-subshell.

The number of orbitals each subshell contains is given by `m_l`. The 2s-subshell contains one orbital, since you only have one possible value for `m_l`.

The 2p-subshell contains 3 orbitals, each denoted by a different value of `m_l`.

The total number of orbitals in the second energy level is 4, which is equal to `n^2`.

If you try this with `n=3`, you'll find that you get a total number of `9` orbitals

• one 3s-orbital;
• three 3p-orbitals;
• five 3d-orbitals.