Eigenstates of Angular Momentum

Let us find the simultaneous eigenstates of the angular momentum operators $L_z$ and $L^2$. Since both of these operators can be represented as purely angular differential operators, it stands to reason that their eigenstates only depend on the angular coordinates $\theta$ and $\phi$. Thus, we can write

$$L_z Y_{l,m}(\theta,\phi) = m \hbar Y_{l,m}(\theta,\phi),$$ (556)

$$L^2 Y_{l,m}(\theta,\phi) = l (l+1) \hbar^{ 2} Y_{l,m}(\theta,\phi).$$ (557)

Here, the $Y_{l,m}(\theta,\phi)$ are the eigenstates in question, whereas the dimensionless quantities $m$ and $l$ parameterize the eigenvalues of $L_z$ and $L^2$, which are $m \hbar$ and $l (l+1) \hbar^2$, respectively. Of course, we expect the $Y_{l,m}$ to be both mutually orthogonal and properly normalized (see Sect. 4.9), so that

$$\oint Y^{ \ast}_{l',m'}(\theta,\phi) Y_{l,m}(\theta,\phi) d\Omega = \delta_{ll'} \delta_{mm'},$$ (558)

where $d\Omega = \sin\theta d\theta d\phi$ is an element of solid angle, and the integral is over all solid angle.

Now,

$$L_z (L_+ Y_{l,m}) = (L_+ L_z + [L_z, L_+]) Y_{l,m}= (L_+ L_z + \hbar L_+) Y_{l,m}$$

$$= (m+1) \hbar (L_+ Y_{l,m}),$$ (559)

where use has been made of Eq. (543). We, thus, conclude that when the operator $L_+$ operates on an eigenstate of $L_z$ corresponding to the eigenvalue $m \hbar$ it converts it to an eigenstate corresponding to the eigenvalue $(m+1) \hbar$. Hence, $L_+$ is known as the raising operator (for $L_z$). It is also easily demonstrated that

$$L_z (L_- Y_{l,m}) = (m-1) \hbar (L_- Y_{l,m}).$$ (560)

In other words, when $L_-$ operates on an eigenstate of $L_z$ corresponding to the eigenvalue $m \hbar$ it converts it to an eigenstate corresponding to the eigenvalue $(m-1) \hbar$. Hence, $L_-$ is known as the lowering operator (for $L_z$).

Writing

$$L_+ Y_{l,m} = c_{l,m}^+ Y_{l,m+1},$$ (561)

$$L_- Y_{l,m} = c_{l,m}^- Y_{l,m-1},$$ (562)

we obtain

$$L_- L_+ Y_{l,m} = c^+_{l,m} c^-_{l,m+1} Y_{l,m} = [l (l+1)-m (m+1)] \hbar^2 Y_{l,m},$$ (563)

where use has been made of Eq. (541). Likewise,

$$L_+ L_- Y_{l,m} = c^+_{l,m-1} c^-_{l,m} Y_{l,m} = [l (l+1)-m (m-1)] \hbar^2 Y_{l,m},$$ (564)

where use has been made of Eq. (540). It follows that

$$c^+_{l,m} c^-_{l,m+1} = [l (l+1)-m (m+1)] \hbar^2,$$ (565)

$$c^+_{l,m-1} c^-_{l,m} = [l (l+1)-m (m-1)] \hbar^2.$$ (566)

These equations are satisfied when

$$c^\pm_{l,m} = [l (l+l) - m (m\pm 1)]^{1/2} \hbar.$$ (567)

Hence, we can write

$$L_+ Y_{l,m} = [l (l+1)-m (m+1)]^{1/2} \hbar Y_{l,m+1},$$ (568)

$$L_- Y_{l,m} = [l (l+1)-m (m-1)]^{1/2} \hbar Y_{l,m-1}.$$ (569)