Quanty - documentation:standard_operators:orbital_angular_momuntum
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2026-04-14T23:21:18+00:00Quanty
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https://www.quanty.org/_media/wiki/dokuwiki.svgtext/html2025-11-20T02:29:32+00:00Anonymous (anonymous@undisclosed.example.com)Lmin
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/lmin?rev=1763605772&do=diff
Lmin
The $L^-$ operator is defined as:
\begin{equation}
L^- = \sum_{m=-l}^{m=l}\sum_{\sigma} \sqrt{l-m+1}\sqrt{l+m} \, a^{\dagger}_{m-1,\sigma}a^{\phantom{\dagger}}_{m,\sigma}.
\end{equation}
The equivalent operator in Quanty is created by:
OppLmin = NewOperator("Lmin", NF, IndexUp, IndexDn)
Table of contents
orbital_angular_momuntum indextext/html2025-11-20T02:29:33+00:00Anonymous (anonymous@undisclosed.example.com)Lplus
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/lplus?rev=1763605773&do=diff
Lplus
The $L^+$ operator is defined as:
\begin{equation}
L^+ = \sum_{m=-l}^{m=l}\sum_{\sigma} \sqrt{l+m+1}\sqrt{l-m} \, a^{\dagger}_{m+1,\sigma}a^{\phantom{\dagger}}_{m,\sigma}.
\end{equation}
The equivalent operator in Quanty is created by:
OppLplus = NewOperator("Lplus", NF, IndexUp, IndexDn)
Table of contents
orbital_angular_momuntum indextext/html2025-11-20T02:29:33+00:00Anonymous (anonymous@undisclosed.example.com)Lsqr
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/lsqr?rev=1763605773&do=diff
Lsqr
The $L^2$ operator is defined as:
\begin{eqnarray}
L^2 = \sum_{m=-l}^{m=l}\sum_{\sigma} && l(l+1) a^{\dagger}_{m,\sigma}a^{\phantom{\dagger}}_{m,\sigma}\\
\nonumber + \sum_{m_1,m_2=-l}^{m_1,m_2=l}\sum_{\sigma_1,\sigma_2}&& \sqrt{l+m_1+1}\sqrt{l-m_1}\\
\nonumber &&\times\,\sqrt{l+m_2+1}\sqrt{l-m_2}\\
\nonumber &&\times \, a^{\dagger}_{m_1+1,\sigma_1}a^{\dagger}_{m_2,\sigma_2}a^{\phantom{\dagger}}_{m_1,\sigma_1}a^{\phantom{\dagger}}_{m_2+1,\sigma_2}.
\end{eqnarray}
The equivalent operator in …text/html2025-11-20T02:29:33+00:00Anonymous (anonymous@undisclosed.example.com)Lx
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/lx?rev=1763605773&do=diff
Lx
The $L_x$ operator is defined as:
\begin{eqnarray}
L_x = \sum_{m=-l}^{m=l}\sum_{\sigma} && \frac{1}{2}\sqrt{l+m+1}\sqrt{l-m}\\
\nonumber &&\times\left(a^{\dagger}_{m+1,\sigma}a^{\phantom{\dagger}}_{m,\sigma} + a^{\dagger}_{m,\sigma}a^{\phantom{\dagger}}_{m+1,\sigma}\right).
\end{eqnarray}
The equivalent operator in Quanty is created by:
OppLx = NewOperator("Lx", NF, IndexUp, IndexDn)
Table of contents
orbital_angular_momuntum indextext/html2025-11-20T02:29:33+00:00Anonymous (anonymous@undisclosed.example.com)Ly
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/ly?rev=1763605773&do=diff
Ly
The $L_y$ operator is defined as:
\begin{eqnarray}
L_y = \sum_{m=-l}^{m=l}\sum_{\sigma} && \frac{\imath}{2}\sqrt{l+m+1}\sqrt{l-m}\\
\nonumber &&\times\left(-a^{\dagger}_{m+1,\sigma}a^{\phantom{\dagger}}_{m,\sigma} + a^{\dagger}_{m,\sigma}a^{\phantom{\dagger}}_{m+1,\sigma}\right).
\end{eqnarray}
The equivalent operator in Quanty is created by:
OppLy = NewOperator("Ly", NF, IndexUp, IndexDn)
Table of contents
orbital_angular_momuntum indextext/html2025-11-20T02:29:33+00:00Anonymous (anonymous@undisclosed.example.com)Lz
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/lz?rev=1763605773&do=diff
Lz
The $L_z$ operator is defined as:
\begin{eqnarray}
L_z = \sum_{m=-l}^{m=l}\sum_{\sigma} m a^{\dagger}_{m,\sigma}a^{\phantom{\dagger}}_{m,\sigma}.
\end{eqnarray}
The equivalent operator in Quanty is created by:
OppLz = NewOperator("Lz", NF, IndexUp, IndexDn)
Table of contents
orbital_angular_momuntum indextext/html2025-11-20T02:29:34+00:00Anonymous (anonymous@undisclosed.example.com)Orbital angular momentum operators (L)
https://www.quanty.org/documentation/standard_operators/orbital_angular_momuntum/start?rev=1763605774&do=diff
Orbital angular momentum operators (L)
The angular momentum operators need as a basis set two lists defining the spin up and spin down orbitals. These lists must have length $2l+1$ whereby $l$ is the angular momentum of the shell under consideration. The orbital quantum numbers $m_l=-l$$m_l=l$$p$$0$$l=1$$m_l=-1$$\sigma=-1/2$orbital_angular_momuntum index