Guided reading
The position representation follows directly from \(\vec{\hat p}=-i\hbar\vec\nabla\). This connects the angular equation of central potentials to angular-momentum eigenfunctions.
Where it comes from
\[\vec{\hat p}=-i\hbar\vec\nabla,\qquad \vec{\hat L}=(-i\hbar)\vec r\times\vec\nabla\]
\[\vec\nabla=\hat r{\partial\over\partial r}+{\hat\theta\over r}{\partial\over\partial\theta}+{\hat\phi\over r\sin\theta}{\partial\over\partial\phi}\]
\[\vec r\times\vec\nabla=\hat\phi{\partial\over\partial\theta}-{\hat\theta\over\sin\theta}{\partial\over\partial\phi}\]
\[\vec{\hat L}=(-i\hbar)\left(\hat\phi{\partial\over\partial\theta}-{\hat\theta\over\sin\theta}{\partial\over\partial\phi}\right)\]
Cartesian unit vectors
The spherical-coordinate unit vectors are then written in the Cartesian directions.
\[\hat x=\hat r\sin\theta\cos\phi+\hat\theta\cos\theta\cos\phi-\hat\phi\sin\phi\]
\[\hat y=\hat r\sin\theta\sin\phi+\hat\theta\cos\theta\sin\phi+\hat\phi\cos\phi\]
\[\hat z=\hat r\cos\theta-\hat\theta\sin\theta\]
The hat denotes unit vectors in this card; the same symbol also denotes operators when attached to observables.
Component operators
\[\hat L_z=\hat z\cdot\vec{\hat L}=(-i\hbar)(\hat r\cos\theta-\hat\theta\sin\theta)\cdot\left(\hat\phi{\partial\over\partial\theta}-{\hat\theta\over\sin\theta}{\partial\over\partial\phi}\right)=-i\hbar{\partial\over\partial\phi}\]
\[\hat L_x=\hat x\cdot\vec{\hat L}=i\hbar\left(\cot\theta\cos\phi{\partial\over\partial\phi}+\sin\phi{\partial\over\partial\theta}\right)\]
\[\hat L_y=\hat y\cdot\vec{\hat L}=i\hbar\left(\cot\theta\sin\phi{\partial\over\partial\phi}-\cos\phi{\partial\over\partial\theta}\right)\]
Operators \(\hat L_\pm\) and \(\hat L^2\)
\[\hat L_\pm=\hbar e^{\pm i\phi}\left(\pm{\partial\over\partial\theta}+i\cot\theta{\partial\over\partial\phi}\right)\]
\[\hat L^2=-\hbar^2\left[{\partial^2\over\partial\theta^2}+\cot\theta{\partial\over\partial\theta}+{1\over\sin^2\theta}{\partial^2\over\partial\phi^2}\right]\]
\[-{\hat L^2\over\hbar^2}={1\over\sin\theta}{\partial\over\partial\theta}\left(\sin\theta{\partial\over\partial\theta}\right)+{1\over\sin^2\theta}{\partial^2\over\partial\phi^2}\]
Spherical harmonics as eigenfunctions
Recovering the angular equation from Chapter 5 identifies the spherical harmonics as eigenfunctions of the angular-momentum operators.
\[Y_l^m(\theta,\phi)=A_l^m\Theta(\theta)\Phi(\phi)\]
\[\hat L^2Y_l^m(\theta,\phi)=l(l+1)\hbar^2Y_l^m(\theta,\phi)\]
\[\hat L_zY_l^m(\theta,\phi)=m\hbar Y_l^m(\theta,\phi)\]
Complete example: \(Y_1^{-1}\)
For \((l,m)=(1,-1)\), start with the \(\hat L_z\) eigenvalue equation and separate \(Y_1^{-1}(\theta,\phi)=\Theta(\theta)\Phi(\phi)\).
\[\hat L_zY_1^{-1}(\theta,\phi)=-\hbar Y_1^{-1}(\theta,\phi)\]
\[-i\hbar\,\Theta(\theta){d\Phi(\phi)\over d\phi}=-\hbar\Theta(\theta)\Phi(\phi)\quad\Rightarrow\quad {d\Phi\over d\phi}=-i\Phi\]
\[\Phi(\phi)=e^{-i\phi}\]
To obtain the polar factor, use the endpoint condition \(\hat L_-Y_1^{-1}=0\).
\[\hat L_-Y_1^{-1}=0\quad\Rightarrow\quad -{d\Theta\over d\theta}+\cot\theta\,\Theta=0\]
\[\Theta(\theta)=\sin\theta\]
Normalization with \(\int_0^\pi\int_0^{2\pi}|Y_l^m|^2\sin\theta\,d\phi\,d\theta=1\) gives
\[Y_1^{-1}(\theta,\phi)=\sqrt{3\over8\pi}\,\sin\theta\,e^{-i\phi}\]
Exercise-ready boundary
This page supports guided exercises on: position-space angular-momentum operators and spherical harmonics as angular-momentum eigenfunctions.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed for a compact calculation or explanation.
- Long-form material: complete proofs, long demonstrations, extended examples and the full exercise set remain in the textbook.
- Boundary: do not introduce results, notation or physical claims outside the sequence presented here and in Chapter 6.
Practice anchors
Use these anchors only within the material introduced on this page.
- Focus: position-space angular-momentum operators and spherical harmonics as angular-momentum eigenfunctions.
- Conceptual check: explain how the angular equation from central potentials reappears as the \(\hat L^2\) eigenvalue equation.
- Key equation 1:
\[\vec{\hat L}=(-i\hbar)\vec r\times\vec\nabla\]
- Key equation 2:
\[\hat L^2Y_l^m=l(l+1)\hbar^2Y_l^m,\qquad \hat L_zY_l^m=m\hbar Y_l^m\]
- Typical task: apply \(\hat L_z\), \(\hat L^2\), or \(\hat L_\pm\) to an angular function.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 6 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, proofs, examples and exercises.