# What does #m_l# represent and why does it correspond to each of the orbitals for a given subshell?

##### 1 Answer

For the **angular distribution function** *projection of the angular momentum*

**THE MAGNETIC QUANTUM NUMBER, FOR THE 4F ORBITAL**

**magnetic quantum number**, takes on the values

For the

Therefore, there are

#4f_(y(3x^2-y^2)): m_l = -3# #4f_(z(x^2-y^2)): m_l = -2# #4f_(yz^2): m_l = -1# #4f_(z^3): m_l = 0# #4f_(xz^2): m_l = +1# #4f_(xyz): m_l = +2# #4f_(x(x^2-3y^2)): m_l = +3#

But why are there *correlates* with how there are

**MORE-OR-LESS WHY THERE ARE #\mathbf(2l+1)# ORBITALS POSSIBLE**

Consider the

When you have *correlates* with three orbitals without further explanation.

You could also take the

#hatL_zY_l^(m_l)(theta,phi) = m_lℏY_l^(m_l)(theta,phi)# resembling

#hatHpsi = Epsi# .

Without getting into too many complicated details (you don't have to know the three expressions for the **angular momentum** **three different** ** trace out** the

You have **three** relevant

#m_l = -1 -> L_z => -ℏ# (forming the bottom lobe, labeled#-ℏ# )#m_l = 0 -> L_z => 0# (forming the central dot, or node, labeled#0# )#m_l = +1 -> L_z => ℏ# (forming the top lobe, labeled#ℏ# )

Naturally, the top lobe is *symmetrically-shaped* in relation to the bottom lobe.

**THE ORTHOGONALITY CONDITION**

Now, a *quantum mechanical postulate/requirement* for an orbital is that it be

**orthogonal (perpendicular) to all the other possible orbitals**.

Two mathematical ways of conveying that are the **dot product** and the **cross product**:

#hatxcdothaty = <<1,0,0>> cdot <<0,1,0>> = 0#

#hatxcdothatz = <<1,0,0>> cdot <<0,0,1>> = 0#

#hatycdothatz = <<0,1,0>> cdot <<0,0,1>> = 0#

#hatx xxhaty = <<1,0,0>> xx <<0,1,0>> = <<0,0,1>> = hatz#

#haty xxhatz = <<0,1,0>> xx <<0,0,1>> = <<1,0,0>> = hatx#

#hatz xxhatx = <<0,0,1>> xx <<1,0,0>> = <<0,1,0>> = haty#

Since the ** is** the

**axis of symmetry**, the only way the other orbitals can be

*orthogonal*is that they

**lie along the**

**and**

**axes**(as you may have recognized in the pattern from the above dot and cross products).

Clearly, the **perpendicular**.

As a result of **as well as** the *orthogonality* condition, there are **three** orbitals possible:

Similarly, because there are

There just happens to be