How many orbitals can have the following set of quantum numbers, n = 3, l = 1, m1 = 0 ?

The correct option is 1.

To understand why this is the correct answer, let us break down the set of quantum numbers provided in the question:
The quantum numbers given are:
- Principal quantum number, _n = 3
- Azimuthal (or orbital angular momentum) quantum number, _l = 1
- Magnetic quantum number, _ml = 0

Each unique combination of the three quantum numbers (_n, _l, and _ml) corresponds to exactly one specific atomic orbital.

Let us analyze these quantum numbers step-by-step:

1. The principal quantum number, _n = 3, specifies the third main energy level or shell.

2. The azimuthal quantum number, _l = 1, specifies the subshell. For _l = 1, this corresponds to a p-subshell (specifically, the 3p subshell since _n = 3).

3. The magnetic quantum number, _ml, describes the spatial orientation of the orbital within a subshell. For a p-subshell (_l = 1), the possible values for _ml are:
_ml = -1, 0, +1
This means there are three separate p-orbitals in the 3p subshell, corresponding to these three spatial orientations.

4. The question specifies a single, definite value for the magnetic quantum number: _ml = 0. This restricts us to one specific orbital out of the three available in the 3p subshell (often designated as the 3p_z orbital).

Since all three spatial quantum numbers (_n, _l, _ml) are fully defined, they describe a single, unique orbital. Therefore, only 1 orbital can have this specific set of quantum numbers.