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Suppress the fixed individual quantum numbers $s_1=s_2=1/2$. In the coupled basis, ordered on both rows and columns as $|1,1⟩,|1,0⟩,|1,-1⟩,|0,0⟩$, the total-spin eigenvalues giveHere the first three entries form the triplet state sector and the last is the singlet state. For the product basis, useThe first term has diagonal entries ${\hslash }^2(1,-1,-1,1)/4$. Each nonzero exchange term flips $|\uparrow \downarrow ⟩$ to $|\downarrow \uparrow ⟩$ or back, with coefficient ${\hslash }^2/2$. With rows and columns ordered as $|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩$,The change of basis is made explicit by $|1,0⟩=(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$ and $|0,0⟩=(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$, with $|1,\pm 1⟩=|\uparrow \uparrow ⟩,|\downarrow \downarrow ⟩$. The resulting eigenvalues reproduce the diagonal coupled matrix.

The spin Hilbert space is a tensor product of two two-dimensional spaces and therefore has dimension four. Angular momentum addition decomposes it asThese are three orthogonal triplet states and one singlet state. For identical electrons the complete wavefunction must also satisfy fermionic exchange symmetry: a symmetric triplet spin factor accompanies an antisymmetric spatial factor, whereas the antisymmetric singlet spin factor accompanies a symmetric spatial factor. If both electrons occupy the same spatial orbital, only the singlet spin factor is allowed. The four-dimensional spin space does not assert four allowed spin states for that particular spatial occupation.

In the ordered product basis $(|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩)$, the two-spin XY Hamiltonian isThe transverse term exchanges opposite spins and leaves parallel spins unchanged. Diagonalizing its two-dimensional block gives the complete orthonormal eigenbasisThe antisymmetric state is the singlet state; the other three are triplet states. Accidental equality of energies does not invalidate this eigenbasis.

The reduced density matrix has two eigenvalues equal to $1/2$. Its Von Neumann entropy, which is the entanglement entropy of the bipartite pure state, isThis is one bit when base-two logarithms are used. It is the maximum entanglement entropy for a pair of two-dimensional systems, while the joint Von Neumann entropy of the singlet state remains zero.

Taking the partial trace over the left atom in the singlet state gives the right-hand reduced density matrixThus every measurement on the right stream alone has the same statistics as B or C. The joint pure state is compatible with locally mixed states; the correlations reside in the full density operator.

The normalized singlet state has column vector $(0,1,-1,0{)}^{\mathsf{T}}/\sqrt{2}$ in the specified product basis. Its outer product isIt obeys ${\rho }_D^2={\rho }_D$ and has eigenvalues $(1,0,0,0)$, hence purity of a density operator one. The joint two-atom system is therefore a pure state, even though the quantum entanglement prevents assigning a separate pure spin state to each atom.

The complete wavefunction of identical fermions changes sign on exchange of two particles, including their spatial and internal coordinates. The Pauli exclusion principle follows because an antisymmetrized product with two identical complete spin orbitals is zero. Antisymmetry concerns the total state; a spin-antisymmetric singlet state can have a symmetric spatial factor.

Suppression of charge transfer by spin symmetry and the Pauli exclusion principle. For a double quantum dot, a $(1,1)$ singlet state can enter the lowest $(0,2)$ singlet, whereas a $(1,1)$ triplet state is blocked if the required $(0,2)$ triplet lies above the accessible energy window. Spin mixing or relaxation can lift blockade.

Two equal spins $j$ have the unique total-spin-zero stateIt is invariant under joint rotations.

A singlet state is the one-dimensional rotation representation; this construction supplies the angular-momentum-zero case for two equal spins.

A bound-state configuration with one electron predominantly in $1s$ and the other in a different orbital $n\ell$. The two assignments of identical electrons to these orbitals must be combined with the appropriate exchange symmetry of two identical spin-one-half fermions. Symmetric spatial combinations accompany the singlet state; antisymmetric combinations accompany the triplet state.

A state with total spin angular momentum quantum number $S$ has $2S+1$ magnetic projections. A singlet state has $S=0$, a triplet state has $S=1$, and the $S=3$ chromium ground term is a septet.

In the product basis $|m_1,m_2⟩$ for two spin-one particles, total magnetic quantum number zero allows only $|1,-1⟩$, $|0,0⟩$ and $|-1,1⟩$. Applying the total ladder operators to $\alpha |1,-1⟩+\beta |0,0⟩+\gamma |-1,1⟩$ givesA singlet state is annihilated by $S_{+}$ and $S_{-}$, so $\alpha =\gamma =-\beta$. Wavefunction normalization gives the displayed state, up to an overall phase. Here the ket on the left of the title formula has total-spin labels, while its right-hand kets have individual magnetic labels. The state is symmetric under exchanging the particles, although no exchange constraint is needed for distinguishable particles.

Map states with unequal escape rates ${\Gamma }_f\gg {\Gamma }_s$ onto distinct charge outcomes after an interval between their characteristic escape times. Both removal energies can lie above the reservoir chemical potential, so their splitting need not be thermally resolved. A useful rate difference must have a physical origin: a demonstrated single-dot example uses a two-electron singlet state and triplet state with different occupied orbitals. A bare single-electron Zeeman pair in the same orbital does not automatically provide the required rate contrast.^[ref]

The spatial ground orbital of a three-dimensional quantum harmonic oscillator has $l=0$ and energy $3\hslash {\omega }_0/2$; its first excited shell has $l=1$ and energy $5\hslash {\omega }_0/2$. Two identical electrons both in the ground orbital must form the singlet state, giving one state at total energy $3\hslash {\omega }_0$.

At energy $4\hslash {\omega }_0$, one electron occupies each shell. There are two ground-shell spin orbitals and six first-excited-shell spin orbitals, hence twelve Slater determinants. Equivalently, $0\otimes 1$ gives $L=1$; symmetric spatial combinations with the singlet state give three states, while antisymmetric spatial combinations with the triplet states give nine. Distinct occupied orbitals allow both spatial exchange symmetries, so both spin sectors must be kept.

For free fermions on a ring, each one-particle orbital has integer angular momentum label $l$ and two spin states. Two particles occupy the $l=0$ orbital in a unique singlet state at zero kinetic energy. At the next distinct energy ${\hslash }^2/(2m{R}^2)$, orbitals $0,+1$ or $0,-1$ can be combined with one singlet state or three triplet state spin states, giving total quantum degeneracy eight. The lowest triplet state has six states and its antisymmetric spatial factor vanishes when the particles coincide.

The two-site quantum version of an XY model couples the transverse components of two spin one-half degrees of freedom. In the ordered basis $|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩$, its only off-diagonal entries are $H_{23}=H_{32}=-2J$. The parallel states have energies $-2h$ and $2h$; the symmetric and antisymmetric opposite-spin combinations have energies $-2J$ and $2J$ respectively. The latter is the singlet state. The formulas remain valid at accidental degeneracies.