What Is Z component of orbital angular momentum? How can we find the Z component? What is its importance? What does it resemble?

You seem to be referring to #m_l#, which is the observed value that corresponds to the #z#-component of the total orbital angular momentum #L_z#.

Practically speaking, for general chemistry, you can simply use the value of #l# as the range of #m_l#, and express #m_l# as:

#bb(m_l = {-l,-l+1, . . . , 0, . . . , l - 1, l})#

For instance, if #l = 2# (as for a #d# orbital), then:

#m_l = {-2,-1,0,+1,+2}#

That means five #d# orbitals exist for a given principal quantum number #n#:

![http://2012books.lardbucket.org/]()

RELATION TO THE Z-COMPONENT OF THE TOTAL ORBITAL ANGULAR MOMENTUM

Recall that the Schrodinger equation is typically written as #hatHpsi = Epsi# (where #E# is the energy, #hatH# is the Hamiltonian operator, and #psi# is the wave function).

Well, it turns out that #psi#, the wave function describing the state of a quantum mechanical system, can be separated into a radial and an angular component, #R_(nl)(r)# and #Y_(l)^(m_l)(theta,phi)#:

#psi_(nlm_l)(r,theta,phi) = R_(nl)(r)Y_(l)^(m_l)(theta,phi)#

where #n#, #l#, and #m_l# are the principal, angular momentum, and magnetic quantum numbers, respectively.

Traditionally, #m_l# is defined to be the #z# component of the angular momentum #l#, and it is the eigenvalue (the quantity we expect to see over and over again), in units of #ℏ#, of the wave function, #psi#.

This eigenvalue corresponds to the operator for #L_z#, and #L_z# is the #bb(z)# component of the total orbital angular momentum.

What we just said can be expressed as:

#stackrel("Operator")overbrace(hatL_z)" "stackrel("Angular")stackrel(" Component")stackrel("of Wave Function")overbrace(Y_(l)^(m_l)(theta,phi)) = stackrel("Eigenvalue")overbrace(m_lℏ)" "stackrel("Angular")stackrel(" Component")stackrel("of Wave Function")overbrace(Y_(l)^(m_l)(theta,phi))#

If #L_z# is what you mean, then the significance of it is that it is the phenomenon we can observe that corresponds to the magnetic quantum number # m_l#.

PHYSICS PERSPECTIVE

Visually, in the presence of a magnetic field in the #z# direction, a nuclear rotation (exhibiting a total orbital angular momentum) occurs along the #z# axis, called a "Larmor precession".

This is the event described by #L_z#.

For instance, when #l = 1#, as for a #p# orbital, #m_l = {-1,0,+1}#. The "Larmor precession" that occurs looks like the following for a #2p_z# orbital:

And each #m_l# corresponds to the distance from the #z# axis in units of #ℏ#:

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For instance:

• An #m_l# of #1# corresponds to the upper half of the #2p_z# orbital.
• An #m_l# of #0# is the dot at the origin.
• An #m_l# of #-1# corresponds to the lower half.

CHEMISTRY PERSPECTIVE

From a practical point of view, what we really care about is how to use #m_l#. Each #m_l# corresponds to a unique orbital in a particular subshell. So:

• The number of #m_l# values tells you the number of orbitals in a subshell.
• The range of #m_l# is based on the chosen #l#.

For example, since #l = 2# is for a #d# subshell, then:

#m_l = {-2,-1,0,+1,+2}#

That means five #d# orbitals exist for a given principal quantum number #n#:

![http://2012books.lardbucket.org/]()