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

$$L_z\,Y_{l,m}(\theta,\phi) = m\,\hbar\,Y_{l,m}(\theta,\phi),$$ (535)

$$L^2\,Y_{l,m}(\theta,\phi) = l\,(l+1)\,\hbar^{\,2}\,Y_{l,m}(\theta,\phi).$$ (536)

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

$$\oint Y^{\,\ast}_{l',m'}(\theta,\phi)\,Y_{l,m}(\theta,\phi)\,d\Omega = \delta_{ll'}\,\delta_{mm'},$$ (537)

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

Now,

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

$$= (m+1)\,\hbar\,(L_+\,Y_{l,m}),$$ (538)

where use has been made of Eq. (525). 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

$$L_z\,(L_-\,Y_{l,m}) = (m-1)\,\hbar\,(L_-\,Y_{l,m}).$$ (539)

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

$$L_+\,Y_{l,m} = c_{l,m}^+\,Y_{l,m+1},$$ (540)

$$L_-\,Y_{l,m} = c_{l,m}^-\,Y_{l,m-1},$$ (541)

we obtain

$$L_-\,L_+\,Y_{l,m} = c^+_{l,m}\,c^-_{l,m+1}\,Y_{l,m} = [l\,(l+1)-m\,(m+1)]\,\hbar^2\,Y_{l,m},$$ (542)

where use has been made of Eq. (523). Likewise,

$$L_+\,L_-\,Y_{l,m} = c^+_{l,m-1}\,c^-_{l,m}\,Y_{l,m} = [l\,(l+1)-m\,(m-1)]\,\hbar^2\,Y_{l,m},$$ (543)

where use has been made of Eq. (522). It follows that

$$c^+_{l,m}\,c^-_{l,m+1} = [l\,(l+1)-m\,(m+1)]\,\hbar^2,$$ (544)

$$c^+_{l,m-1}\,c^-_{l,m} = [l\,(l+1)-m\,(m-1)]\,\hbar^2.$$ (545)

These equations are satisfied when

$$c^\pm_{l,m} = [l\,(l+l) - m\,(m\pm 1)]^{1/2}\,\hbar.$$ (546)

Hence, we can write

$$L_+\,Y_{l,m} = [l\,(l+1)-m\,(m+1)]^{1/2}\,\hbar\,Y_{l,m+1},$$ (547)

$$L_-\,Y_{l,m} = [l\,(l+1)-m\,(m-1)]^{1/2}\,\hbar\,Y_{l,m-1}.$$ (548)