# How can we prove that #|m_l| <= l#, i.e. that the magnetic quantum number can be no bigger than #l#, and no more negative than #-l#?

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This is something I will answer myself, because I think it's actually a rather cool use of ladder operators.

This is something I will answer myself, because I think it's actually a rather cool use of ladder operators.

##### 1 Answer

By using the ladder operators, we derived:

#barul|stackrel(" ")(" "l(l+1) >= m_l (m_l pm 1)" ")|#

and from this inequality we get that

**INTRODUCTORY RELATIONS**

Following this and this page, let us introduce the **ladder operators** for orbital angular momentum

#hatL_(pm) = hatL_x pm ihatL_y# where

#hatL_i# is the angular momentum operator for the#i# th direction in 3D space.

These satisfy the **commutation relations**:

#[hatL^2, hatL_(pm)] = hatL^2hatL_(pm) - hatL_(pm)hatL^2 = 0#

#[hatL_(pm), hatL_z] = hatL_(pm)hatL_z - hatL_zhatL_(pm) = ∓ℏhatL_(pm)# where

#hatL# is the orbital angular momentum operator and#hatL_z# is its#z# component.

Now, the **eigenvalues** we get when we operate on the angular wave function

#color(green)ul(hatL^2)Y_(l)^(m_l)(theta,phi) = color(green)ul(ℏ^2l(l+1))Y_(l)^(m_l)(theta,phi)#

#color(green)ul(hatL_z)Y_(l)^(m_l)(theta,phi) = color(green)ul(m_lℏ)Y_(l)^(m_l)(theta,phi)#

**DO THESE LADDER OPERATORS CHANGE #l#?**

Now we shall ask, **what happens to the value** of

#color(red)(hatL^2hatL_(pm)Y_(l)^(m_l)(theta,phi) = ???cdotY_(l)^(m_l)(theta,phi))#

Since

#color(green)(hatL^2hatL_(pm)Y_(l)^(m_l)(theta,phi)) = hatL_(pm)hatL^2Y_(l)^(m_l)(theta,phi)#

This eigenvalue is known, so that helps...

#hatL_(pm)hatL^2Y_(l)^(m_l)(theta,phi)#

#= hatL_(pm)[ℏ^2l(l+1)Y_(l)^(m_l)(theta,phi)]#

#= color(green)(ℏ^2l(l+1)hatL_(pm)Y_(l)^(m_l)(theta,phi))#

Nothing has happened to **do not touch**

**DO THESE LADDER OPERATORS CHANGE #m_l#?**

What about

#color(red)(hatL_zhatL_(pm)Y_(l)^(m_l)(theta,phi) = ???cdotY_(l)^(m_l)(theta,phi))#

These operators do not commute, so we use the commutation relation we put earlier to note that

#color(green)(hatL_zhatL_(pm)Y_(l)^(m_l)(theta,phi)_#

#= [hatL_(pm)hatL_z pm ℏhatL_(pm)]Y_(l)^(m_l)(theta,phi)#

#= hatL_(pm)hatL_zY_(l)^(m_l)(theta,phi) pm ℏhatL_(pm)Y_(l)^(m_l)(theta,phi)#

We know the

#= hatL_(pm)m_lℏY_(l)^(m_l)(theta,phi) pm ℏhatL_(pm)Y_(l)^(m_l)(theta,phi)#

#= color(green)((m_l pm 1)ℏhatL_(pm)Y_(l)^(m_l)(theta,phi))#

We now see that

**WHAT ARE THE LIMITS OF #m_l#?**

*Now our final question is, when will* *stop decreasing, and when will it stop increasing?*

Now, the **expectation value** of the

#int_"allspace" Y_(l)^(m_l)(theta,phi)^"*"hatL_(∓)hatL_(pm)Y_(l)^(m_l)(theta,phi)d tau >= 0#

We can condense this notation down to:

#<< Y_(l)^(m_l) | hatL_(∓)hatL_(pm) | Y_(l)^(m_l) >> >= 0#

Now, it becomes physically useful to rewrite

#hatL_(∓)hatL_(pm) = hatL^2 - hatL_z^2 ∓ ℏhatL_z#

Finally, using this, we can derive **limits** on

#<< Y_(l)^(m_l) | hatL_(∓)hatL_(pm) | Y_(l)^(m_l) >>#

#= << Y_(l)^(m_l) | hatL^2 - hatL_z^2 ∓ ℏhatL_z | Y_(l)^(m_l) >>#

#= << Y_(l)^(m_l) | ℏ^2l(l+1) - m_l^2ℏ^2 ∓ m_lℏ^2 | Y_(l)^(m_l) >>#

#= << Y_(l)^(m_l) | (l(l+1) - m_l^2 ∓ m_l)ℏ^2 | Y_(l)^(m_l) >>#

The

#=> (l(l+1) - m_l^2 ∓ m_l)ℏ^2 cancel(<< Y_(l)^(m_l) | Y_(l)^(m_l) >>)^(1) >= 0#

We can divide out

#l(l+1) - m_l^2 ∓ m_l >= 0#

With further factoring and rearranging, we have the following **inequality**:

#color(blue)(barul|stackrel(" ")(" "l(l+1) >= m_l (m_l pm 1)" ")|)#

**CHECKING THE LIMITS OF #m_l#**

Testing out values of

#0(0 + 1) >= 0(0 pm 1)# #" "" "" "" "color(blue)sqrt""#

#1(1 + 1) >= 1(1 pm 1)# #" "" "" "" "color(blue)sqrt""#

#1(1 + 1) >= 0(0 pm 1)# #" "" "" "" "color(blue)sqrt""#

#1(1 + 1) >= -1(-1 pm 1)# #" "" "color(blue)sqrt""#

#2(2 + 1) >= 2(2 pm 1)# #" "" "" "" "color(blue)sqrt""#

#2(2 + 1) >= 1(1 pm 1)# #" "" "" "" "color(blue)sqrt""#

#2(2 + 1) >= 0(0 pm 1)# #" "" "" "" "color(blue)sqrt""#

#2(2 + 1) >= -1(-1 pm 1)# #" "" "color(blue)sqrt""#

#2(2 + 1) >= -2(-2 pm 1)# #" "" "color(blue)sqrt""#

So, in order to satisfy this inequality,

#bb(|m_l| <= l)# .

#"Q.E.D."#