Question

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`?

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

Answer 1

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 `|m_l| <= l`.

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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 `Y_(l)^(m_l)(theta,phi)` are given by:

`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 `l` and `m_l` when these ladder operators are applied? That is, we want to know:

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

Since `hatL^2` and `hatL_(pm)` commute, it follows that

`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 `l` here, so these ladder operators do not touch `l`.

DO THESE LADDER OPERATORS CHANGE `m_l`?

What about `m_l`? We want to know:

`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 `hatL_zhatL_(pm) = hatL_(pm)hatL_z pm ℏhatL_(pm)`:

`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_z` eigenvalue, so we proceed with that:

`= 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 `m_l` was indeed affected. So the ladder operators raise `(+)` or lower `(-)` the value of `m_l`.

WHAT ARE THE LIMITS OF `m_l`?

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

Now, the expectation value of the `hatL_(pm)` and the `hatL_(∓)` is:

`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)` in terms of components of `hatL`. It turns out to be:

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

Finally, using this, we can derive limits on `m_l`.

`<< 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 `Y_(l)^(m_l)` are normalized, so `<< Y_(l)^(m_l) | Y_(l)^(m_l) >> = 1`, and therefore:

`=> (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 `ℏ^2` to obtain the relation:

`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 `l` and `m_l`, we find:

`bbul(m_l = 0)`

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

`bbul(m_l = -1,0,+1)`

`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""`

`bbul(m_l = -2,-1,0,+1,+2)`

`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."`