Two-dimensional isotropic harmonic oscillator

For unit mass and angular frequency,The level with total occupation $N=n_x+n_y$ has energy $N+1$ and degeneracy $N+1$ in units where $\hslash =1$.

For Cartesian creation and annihilation operators $a_x,a_y$, defineTheir Hermitian adjoints define $a_{\pm }$. The canonical commutation relations imply $[a_{\pm },a_{\pm }^{†}]=1$, with mixed commutators zero. Direct expansion givesEach circular creation adds one energy quantum and angular momentum $+\hslash$ or $-\hslash$. At total occupation $N$, the allowed angular momentum values are $N\hslash ,(N-2)\hslash ,\dots ,-N\hslash$, each once in this energy eigenspace.

Completing the square in $H^{\prime }=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)-e\mathcal{E}x$ shifts the oscillator centre to $(a,0)$ and lowers every energy by $a^2/2$. Translated eigenfunctions keep their normalization. The angular momentum about the new centre is $L^{\prime }=(x-a)p_y-y{p}_x$, which commutes with $H^{\prime }$, whereas the angular momentum about the old centre has $[L,H^{\prime }]=-i\hslash ay$. For $a\ne 0$, a common normalizable eigenfunction of $L,H^{\prime }$ would have $y\psi =0$ and hence vanish in $L^2({\mathbb{R}}^2)$. This stronger argument excludes individual common eigenstates; noncommutativity alone generally excludes only a complete common eigenbasis.

For bosonic modes satisfying $[a_i,a_j^{†}]={\delta }_{ij}$, the number-preserving bilinears $T_{ij}=a_i^{†}{a}_j$ obey

Contracting two-mode oscillator bilinears with the Pauli matrices,gives $[T^a,T^b]=i{\epsilon }_{abc}{T}^c$. The subspace with total occupation $N$ is the spin-$N/2$ irreducible representation and has dimension $N+1$.

The Schwinger boson representation satisfies $S_n^2=(N_n/2)(N_n/2+1)$, where $N_n$ is the local total boson number operator. Thus a prescribed local spin $s$ requires the displayed operator constraint at every site. A condition on $⟨N_n⟩$ alone is insufficient: for $⟨N_n⟩=2s$, $⟨S_n^2⟩=s(s+1)+\mathrm{Var}(N_n)/4$. Projecting onto the required eigenspace at each site removes the unwanted local irreducible spin representations.

For the two-dimensional isotropic harmonic oscillator, $H=\hslash \omega (N+1)$ and the Schwinger boson representation gives $J^2={\hslash }^{2}N(N+2)/4$. Substitution yields the displayed relation, including its quantum constant term. The fixed-total-occupation sector is a spin-$j=N/2$ representation, so energy is $\hslash \omega (2j+1)$ with degeneracy $2j+1$.

Effective exchange of internal states through virtual boson hopping into energetically costly occupation sectors. For two bosons initially one per site and charging energy $U{\Delta }^2/4$, the second-order operator is $-(t^2/U)(3+4{\mathbf{s}}_a\cdot {\mathbf{s}}_b)$ for dimensionless spins. The triplet is lowered and the singlet is unchanged in this example; signs for other energy functions require their actual denominators.