Zeeman interaction¶
Zeeman interaction involves two entities. Therefore, it can be expressed as a \(\mathcal{H}_2\) term.
The summary of the mapping for each code is given in the table below.
Code |
McPhase (v1) |
McPhase (v2) |
|||
|---|---|---|---|---|---|
\(\mathcal{H}\) |
\[\mathcal{H}
=
\mu_B
\mathbf{H}
\sum_{i}
g_i
\mathbf{S}_i\]
|
\[\hat{\mathcal{H}}_{Z-J}
=
-
\sum_{n, \alpha=1,2,3}
g_n
\mu_B
\hat{J}^{n}_{\alpha}
H_{\alpha}\]
|
\[\hat{\mathcal{H}}_{Z-LS}
=
-
\sum_{n, \alpha=1,2,3}
\mu_B
\Bigl(
2 \hat{S}^n_{\alpha} + \hat{L}^n_{\alpha}
\Bigr)
H_{\alpha}\]
|
\[\mathcal{H}_{\rm Zeeman}
=
-\sum_i \mu_i \vec{B}\cdot \vec{n}_i\]
|
\[\mathcal{H}
=
\mu_B
\mathbf{B}
\sum_{i}
g_i
\mathbf{S}_i\]
|
Renaming of indices |
\(i \rightarrow (\mu_2, \alpha_2)\), \(i_1, i_2 = x, y, z\) |
\(n \rightarrow (\mu_2, \alpha_2)\), \(\alpha \rightarrow i_1\) and \(\alpha \rightarrow i_2\) |
\(n \rightarrow (\mu_2, \alpha_2)\), \(\alpha \rightarrow i_1\) and \(\alpha \rightarrow i_2\) |
\(i \rightarrow (\mu_2, \alpha_2)\), \(i_1, i_2 = x, y, z\) |
\(i \rightarrow (\mu_2, \alpha_2)\), \(i_1, i_2 = x, y, z\) |
\(C_2\) |
1 |
-1 |
-1 |
-1 |
1 |
\(V_{\mu_1, \mu_2; \alpha_1, \alpha_2}^{i_1, i_2}\) |
\(\mu_B g_{\mu_2, \alpha_2} \delta_{i_1, i_2} \delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\) |
\(g_{\mu_2, \alpha_2} \mu_B \delta_{i_1, i_2} \delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\) |
\(\mu_B \delta_{i_1, i_2} \delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\) |
\(\mu_{\mu_2, \alpha_2} \delta_{i_1, i_2} \delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\) |
\(\mu_B g_{\mu_2, \alpha_2} \delta_{i_1, i_2} \delta_{\mu_1, \mu_2} \delta_{\alpha_1, \alpha_2}\) |
\(X_{\mu_1, \alpha_1}^{i_1}\) |
\(H^{i_1}\) |
\(H^{i_1}\) |
\(H^{i_1}\) |
\(B^{i_1}\) |
\(B^{i_1}\) |
\(X_{\mu_2; \alpha_2}^{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |
\(\hat{J}^{\mu_2, \alpha_2}_{i_2}\) |
\(2 \hat{S}^{\mu_2, \alpha_2}_{i_2} + \hat{L}^{\mu_2, \alpha_2}_{i_2}\) |
\(n_{\mu_2, \alpha_2}^{i_2}\) |
\(S_{\mu_2, \alpha_2}^{i_2}\) |