Past exam of the advanced quantum physics course of the University of Cambridge / 2014 / Paper 1 / 3

For spin one-half, $s_{\pm }=s_x\pm i{s}_y=(\hslash /2)({\sigma }_x\pm i{\sigma }_y)$, henceThey raise or lower the spin projection by one unit of $\hslash$ and annihilate the corresponding endpoint. The exchange identity ${\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2={\sigma }_{1z}{\sigma }_{2z}+2({\sigma }_{1+}{\sigma }_{2-}+{\sigma }_{1-}{\sigma }_{2+})$ preserves total spin projection.

Write $\Delta =\Delta E$, $u={\mu }_{0}B(g_1+g_2)$ and $w={\mu }_{0}B(g_1-g_2)$. In the listed order $(\uparrow \uparrow ,\downarrow \downarrow ,\uparrow \downarrow ,\downarrow \uparrow )$, direct action givesThe printed matrix instead uses $(\uparrow \uparrow ,\uparrow \downarrow ,\downarrow \uparrow ,\downarrow \downarrow )$. Permuting the listed indices to $(1,3,4,2)$ recovers its central block and diagonal entries. This is a basis-order mismatch in the source, not a different Hamiltonian.

The two-spin exchange spectrum with unequal signed magnetic moments isUse the products $g_j{\mu }_0$ as the signed magnetic coefficients in the stated Hamiltonian. Opposite physical charges must be represented in those signs; they do not by themselves imply equal moment magnitudes. Here ${\mu }_0$ denotes the source's magneton, not vacuum permeability.

At zero field, the spin-one-half triplet state has energy $\Delta /4$ and degeneracy three; the spin-one-half singlet state has energy $-3\Delta /4$ and degeneracy one:For $\Delta >0$ the singlet is lower; for $\Delta <0$ the triplet is lower. If $\Delta =0$, all four states coincide.

For $|w|\gg |\Delta |$, the mixed states approach opposite-spin product states andThe omitted mixed-branch correction is of order ${\Delta }^2/|w|$. The four levels are generically nondegenerate, apart from special moment choices or crossings.

For equal-and-opposite signed moments, $g_2=-g_1$, $u=0$ and the parallel-spin level remains exactly twofold degenerate at $\Delta /4$; the other two approach $-\Delta /4\pm |g_1{\mu }_{0}B|$. Equal moment magnitudes are an extra assumption, not implied by opposite charges alone. If $g_1=g_2$, the differential Zeeman term vanishes and the singlet/triplet-zero block stays unmixed by the field. This is the strong-field limit of a two-spin exchange spectrum.

The square-root branches avoid one another when $\Delta \ne 0$; crossings with a parallel-spin branch are allowed because total spin projections differ. At zero field the positive-$\Delta$ sketch has one lower singlet and a triplet above it. The source supplies no numerical moment ratio, so the two panels show explicit illustrative signed choices, including the exactly compensated case.