Spherical harmonics

Harmoniques sphériques


\(\ell\)
      \(m\)
      Plot type       Number of segments (meridian)       Number of segments (parallel)

L'applet ci-dessus permet de représenter l'harmonique sphérique \(Y_{\ell,m}(\theta,\varphi)\), fonction des coordonnées sphériques \(\theta\) et \(\varphi\), pour des valeurs de \(\ell\) et \(m\) spécifiées à l'aide des contrôles situées à gauche de la barre d'outil. Il est possible de changer l'angle de vue et le facteur de zoom à l'aide de la souris. Trois type de représentation sont proposés, comme montré dans l'exemple ci-dessous pour \(Y_{5,2}(\theta,\varphi)\).
                       
Finally, you can adjust the number of segments used to compute the surface along meridians and parallels. A greater number of segments provides a higher-quality surface but with a slower update rate.

Background

Spherical harmonics \(Y_{\ell,m}(\theta,\varphi)\) are the common eigenvectors of the two commuting observables \(\hat L^2\) and \(\hat L_z\), where \(\hat {\vec L}\) is the angular momentum operator. We thus have \[ \hat L^2 Y_{\ell,m}(\theta,\varphi) = \ell(\ell+1)\hbar^2 Y_{\ell,m}(\theta,\varphi)\qquad\qquad\mbox{and}\qquad\qquad\hat L_z Y_{\ell,m}(\theta,\varphi) = m \hbar Y_{\ell,m}(\theta,\varphi), \] where \(\ell\in \mathbb{N}\) and \(m\) is an integer such that \(|m|\le\ell\). In spherical coordinates, these two operators read \[ \hat L^2 = -\hbar^2\left( \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \sin\theta\frac{\partial}{\partial \theta} + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \right) \] and \[ \hat L_z = \frac{\hbar}{i} \frac{\partial}{\partial \varphi}, \] so that common eigenfunctions read \[ Y_{\ell,m}(\theta,\varphi) = F_{\ell,m}(\theta) \exp(im\varphi). \] It can be shown that the real function \( F_{\ell,m}(\theta)\) has exactly \(\ell-|m|\) nodes in the interval \(]0,\pi[\), as can be checked using the above applet. Also evident in the representation is the wrapping of \(|m|\) period along a parallel when \(\varphi\) varies from \(0\) to \(2\pi\), due to the \(m\varphi\) phase factor.

Gallery

The first spherical harmonics are shown below using the three available representations.

The above applet provides a 3D representation of spherical harmonics \(Y_{\ell,m}(\theta,\varphi)\) as a function of spherical coordinates \(\theta\) and \(\varphi\), for specified values of \(\ell\) and \(m\). The mouse can be used to change the point of view and the zooming factor. Three plot types are available, as shown in the example below for \(Y_{5,2}(\theta,\varphi)\).
                       
Finally, you can adjust the number of segments used to compute the surface along meridians and parallels. A greater number of segments provides a higher-quality surface but with a slower update rate.

Background

Spherical harmonics \(Y_{\ell,m}(\theta,\varphi)\) are the common eigenvectors of the two commuting observables \(\hat L^2\) and \(\hat L_z\), where \(\hat {\vec L}\) is the angular momentum operator. We thus have \[ \hat L^2 Y_{\ell,m}(\theta,\varphi) = \ell(\ell+1)\hbar^2 Y_{\ell,m}(\theta,\varphi)\qquad\qquad\mbox{and}\qquad\qquad\hat L_z Y_{\ell,m}(\theta,\varphi) = m \hbar Y_{\ell,m}(\theta,\varphi), \] where \(\ell\in \mathbb{N}\) and \(m\) is an integer such that \(|m|\le\ell\). In spherical coordinates, these two operators read \[ \hat L^2 = -\hbar^2\left( \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \sin\theta\frac{\partial}{\partial \theta} + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \right) \] and \[ \hat L_z = \frac{\hbar}{i} \frac{\partial}{\partial \varphi}, \] so that common eigenfunctions read \[ Y_{\ell,m}(\theta,\varphi) = F_{\ell,m}(\theta) \exp(im\varphi). \] It can be shown that the real function \( F_{\ell,m}(\theta)\) has exactly \(\ell-|m|\) nodes in the interval \(]0,\pi[\), as can be checked using the above applet. Also evident in the representation is the wrapping of \(|m|\) period along a parallel when \(\varphi\) varies from \(0\) to \(2\pi\), due to the \(m\varphi\) phase factor.

Gallery

The first spherical harmonics are shown below using the three available representations.

\(\ell=0\)
\(\ell=1\)
\(\ell=2\)
\(\ell=3\)
\(\ell=4\)
\(\ell=5\)
\(\ell=6\)
\(m=-6\) \(m=-5\) \(m=-4\) \(m=-3\) \(m=-2\) \(m=-1\) \(m=0\) \(m=1\) \(m=2\) \(m=3\) \(m=4\) \(m=5\) \(m=6\)
\(\ell=0\)
\(\ell=1\)
\(\ell=2\)
\(\ell=3\)
\(\ell=4\)
\(\ell=5\)
\(\ell=6\)
\(m=-6\) \(m=-5\) \(m=-4\) \(m=-3\) \(m=-2\) \(m=-1\) \(m=0\) \(m=1\) \(m=2\) \(m=3\) \(m=4\) \(m=5\) \(m=6\)
\(\ell=0\)
\(\ell=1\)
\(\ell=2\)
\(\ell=3\)
\(\ell=4\)
\(\ell=5\)
\(\ell=6\)
\(m=-6\) \(m=-5\) \(m=-4\) \(m=-3\) \(m=-2\) \(m=-1\) \(m=0\) \(m=1\) \(m=2\) \(m=3\) \(m=4\) \(m=5\) \(m=6\)

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