Singlet and Triplet States

Two spin-½ particles make the smallest composite quantum system there is — four states — and those four states organize themselves into the most consequential multiplet split in physics: three states of total spin $S=1$ (the triplet) and one of $S=0$ (the singlet). Addition of Angular Momenta audited the count ( $4=3+1$ ) and deliberately stopped; this article performs the construction it banked: lower from $|\!\uparrow\uparrow\rangle$ , orthogonalize, and hold the resulting four states up to the light. What the light reveals is symmetry. The three triplet states are symmetric under exchange of the two particles, whereas the singlet state is antisymmetric. Combining this exchange symmetry with the antisymmetry required for identical fermions yields the space-spin correlation theorem: electron pairs in symmetric spatial states must occupy spin singlets, whereas antisymmetric spatial states require spin triplets. Heisenberg saw in 1926 that this correlation — pure bookkeeping, no magnetic force anywhere — splits helium's spectrum into two barely-communicating ladders, ortho and para, and thereby explained a spectroscopic mystery electromagnetism had failed to touch. The same mechanism, run through Coulomb repulsion, is why magnetism exists at all. The working tools built here are compact: the four explicit states, the eigenvalue credentials $\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2=+\tfrac{\hbar^2}{4}$ (triplet) versus $-\tfrac{3\hbar^2}{4}$ (singlet), the projector pair that splits any state into its channels, and the exchange Hamiltonian $J\,\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2$ whose two-level spectrum is the germ of every spin-lattice model. Everything is verified matrix-by-matrix; nothing is folklore.

Key takeaways

Four states of ½⊗½ reorganize as triplet $|1,1\rangle=|\!\uparrow\uparrow\rangle$ , $|1,0\rangle=\tfrac{1}{\sqrt2}(|\!\uparrow\downarrow\rangle+|\!\downarrow\uparrow\rangle)$ , $|1,-1\rangle=|\!\downarrow\downarrow\rangle$ and singlet $|0,0\rangle=\tfrac{1}{\sqrt2}(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle)$ .
Credentials: $\hat{S}^2$ eigenvalues $2\hbar^2$ (triplet) and $0$ (singlet); $\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2=+\tfrac{\hbar^2}{4}$ / $-\tfrac{3\hbar^2}{4}$ .
Exchange symmetry: swap operator $\hat{P}_{12}$ gives $+1$ on all triplets, $-1$ on the singlet — and $\hat{P}_{12}=\tfrac12(1+4\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2/\hbar^2)$ makes the connection an operator identity.
Space-spin correlation: total antisymmetry for two electrons forces symmetric-space × singlet or antisymmetric-space × triplet — ortho/para helium, covalent bonding, magnetism.
Projectors $\hat{P}_S=\tfrac14-\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2/\hbar^2$ , $\hat{P}_T=\tfrac34+\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2/\hbar^2$ split any state into channels; exchange Hamiltonian $J\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2$ has gap $J\hbar^2$ .
The singlet is rotationally invariant: $\hat{\mathbf{S}}|0,0\rangle=0$ — it looks identical along every axis.
Key idea: one relative sign between $|\!\uparrow\downarrow\rangle$ and $|\!\downarrow\uparrow\rangle$ carries the exchange symmetry that organizes helium, chemistry, and magnetism.

The core insight

The four states of two spin-½ particles decompose into a symmetric triplet ( $S=1$ , $\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2=+\tfrac{\hbar^2}{4}$ ) and an antisymmetric singlet ( $S=0$ , $-\tfrac{3\hbar^2}{4}$ ); combined with the antisymmetry of identical fermions, this exchange symmetry alone correlates the spatial and spin parts of the two-particle wavefunction, without invoking any magnetic interaction.

Where this lives today

Singlet-triplet physics is quietly running much of quantum technology. Spin qubits in silicon and GaAs double quantum dots commonly encode logical states in the singlet–triplet subspace $\{|0,0\rangle,\;|1,0\rangle\}$ , using the exchange interaction $J(t)$ — tuned electrically by gate voltages — as the native two-qubit interaction; readout is Pauli spin blockade, which is nothing but the space-spin correlation theorem enforcing "no triplet may enter an already-occupied orbital." Radical-pair chemistry tracks singlet-triplet interconversion under hyperfine fields — the leading mechanism proposed for avian magnetoreception. And positronium annihilation imaging exploits para-positronium's two-photon versus ortho's three-photon decay, a selection rule that is exchange symmetry plus charge conjugation doing hospital work.

The physical picture

Start with the two "middle" product states, $|\!\uparrow\downarrow\rangle$ and $|\!\downarrow\uparrow\rangle$ — both have $M=0$ , and neither is an eigenstate of total spin. The physics lives in how you superpose them: the plus combination joins $|\!\uparrow\uparrow\rangle$ and $|\!\downarrow\downarrow\rangle$ to complete a spin-1 family; the minus combination stands alone as spin-0. One relative sign separates a state that behaves like a little vector from a state that behaves like nothing at all.

Figure

Four product states reorganize into a symmetric triplet and an antisymmetric singlet

The swap-parity column is the one to stare at. Exchange the two particles: the triplet states shrug ( $+1$ ), the singlet flips sign ( $-1$ ). For distinguishable spins this is a curiosity; for identical electrons it becomes law, because the total state — space times spin — must be antisymmetric. The consequence is a strict pairing rule between the shape of a two-electron wavefunction and its spin channel:

Figure

Fermionic antisymmetry welds spatial symmetry to spin channel

The construction

This is the simplest nontrivial example of angular-momentum coupling. Total spin $\hat{\mathbf{S}}=\hat{\mathbf{S}}_1+\hat{\mathbf{S}}_2$ obeys the angular-momentum algebra (the commuting-pieces proof is Addition of Angular Momenta's), the window rule allows $S\in\{0,1\}$ , and the dimension audit $4=3+1$ was performed there. What was banked for this article is the explicit descent.

Derivation ABuilding the four states by lowering and orthogonality

We construct the coupled basis of ½⊗½ explicitly. The strategy is universal: begin with the stretched state, generate the multiplet by repeated application of $\hat{S}_{-}$ , and obtain any remaining states by orthogonality — executed here at the smallest scale where every arrow can be watched. The matrix elements of Ladder Operators and the Angular Momentum Spectrum are consumed with their Condon-Shortley-fixed values; for spin-½ each subsystem lowering is simply $\hat{S}_-|\!\uparrow\rangle=\hbar|\!\downarrow\rangle$ , $\hat{S}_-|\!\downarrow\rangle=0$ .

Step 1. Stretched state: $|1,1\rangle=|\!\uparrow\uparrow\rangle$ — the $M=1$ diagonal has one cell, no choice to make (its $+$ sign is the Condon-Shortley seed).

Step 2. Lower once. Left side: $\hat{S}_-|1,1\rangle=\hbar\sqrt{1\cdot2-1\cdot0}\,|1,0\rangle=\hbar\sqrt2\,|1,0\rangle$ . Right side: $(\hat{S}_{1-}+\hat{S}_{2-})|\!\uparrow\uparrow\rangle=\hbar(|\!\downarrow\uparrow\rangle+|\!\uparrow\downarrow\rangle)$ . Equate:

|1,0\rangle=\frac{|\!\uparrow\downarrow\rangle+|\!\downarrow\uparrow\rangle}{\sqrt2}

Why the plus sign is forced: both product terms arrive with the same positive ladder coefficient $\hbar$ — the symmetric combination is not a choice but the image of the stretched state under a symmetric operator. $\hat{S}_-$ treats the two particles identically, so it can only ever produce swap-even states: the whole triplet inherits the stretched state's evenness.

Step 3. Lower again: $\hat{S}_-|1,0\rangle=\hbar\sqrt2\,|1,-1\rangle$ and the right side gives $\hbar\sqrt{2}\,|\!\downarrow\downarrow\rangle$ — so $|1,-1\rangle=|\!\downarrow\downarrow\rangle$ and the ladder terminates ( $\hat{S}_-|\!\downarrow\downarrow\rangle=0$ ), confirming three rungs.

Step 4. The leftover. The $M=0$ diagonal is two-dimensional and the triplet uses one direction of it; the fourth basis state is the unique (up to phase) unit vector orthogonal to $|1,0\rangle$ :

|0,0\rangle=\frac{|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle}{\sqrt2}

with the overall sign fixed by the stretched-entry positivity convention ( $\langle\uparrow\downarrow|0,0\rangle>0$ ). Credentials check, not assumption: applying the $4\times4$ matrix $\hat{S}^2$ gives eigenvalue $0$ — and applying any component $\hat{S}_x,\hat{S}_y,\hat{S}_z$ annihilates it outright. Why annihilation by all three components matters: it says the singlet is invariant under every rotation, not merely $z$ -rotations — the same minus-sign combination appears whichever axis you quantize along. The singlet has no orientation to point; it is the scalar of the spin world. *The lowering chain coefficients ( $\hbar\sqrt2$ at both steps), the $\hat{S}^2$ eigenvalues $\{2,2,2,0\}\hbar^2$ , and the annihilation $\hat{S}_i|0,0\rangle=0$ for all three components are all exact matrix identities.*

The dot-product credentials follow from the identity $\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2=\tfrac12(\hat{S}^2-\hat{S}_1^2-\hat{S}_2^2)$ — Addition of Angular Momenta's, consumed:

\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2 \;=\; \begin{cases} +\tfrac{\hbar^2}{4} & \text{triplet } (S=1)\\[2pt] -\tfrac{3\hbar^2}{4} & \text{singlet } (S=0) \end{cases} \qquad (1)

where: the asymmetry of the two values around zero (one-to-three ratio) mirrors the multiplicities: the traceless combination $\;3\cdot\tfrac14-1\cdot\tfrac34=0\;$ reflects that $\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2$ is a pure coupling with no scalar offset. These two eigenvalues constitute the complete spectrum of the isotropic two-spin- $\tfrac{1}{2}$ Heisenberg interaction, which is why so much two-spin physics reduces to the distinction between singlet and triplet.

Exchange symmetry and the space-spin theorem

Define the swap operator $\hat{P}_{12}$ by $\hat{P}_{12}|\chi_1\rangle|\chi_2\rangle=|\chi_2\rangle|\chi_1\rangle$ . Acting on the four coupled states (verified as an exact matrix computation):

\hat{P}_{12}|1,M\rangle=+|1,M\rangle,\qquad \hat{P}_{12}|0,0\rangle=-|0,0\rangle \qquad (2)

where: the triplet's evenness was manufactured in Step 2 above (symmetric operator acting on a symmetric seed); the singlet, orthogonal to everything even on its diagonal, has no option but oddness. The connection is tighter than a case list — swap and coupling are operator-identical:

\hat{P}_{12}=\tfrac12\Big(1+\tfrac{4}{\hbar^2}\,\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2\Big) \qquad (3)

where: plug in the eigenvalues (1) and read off $\pm1$ . This Dirac identity is the workhorse of spin-model algebra: any Hamiltonian built from swaps is secretly a Heisenberg coupling and vice versa. It is verified entry-by-entry in the $4\times4$ representation.

Go deeper · Derivation B · 2 minExchange symmetry forces the space-spin correlation

Derivation BExchange symmetry forces the space-spin correlation

We prove the pairing theorem for two identical spin-½ fermions: symmetric spatial wavefunctions must carry the singlet, antisymmetric ones the triplet. The inputs are the symmetrization postulate for fermions — the total state changes sign under exchanging everything (positions and spins) — and equation (2). The identical-particle formalism at full generality (Slater determinants, $N$ -body) belongs to the many-body chapters; here we need only the two-body sign bookkeeping.

Step 1. Factorized ansatz. For a Hamiltonian without spin-orbit terms, energy eigenstates can be taken as products $\Psi=\psi(\mathbf{r}_1,\mathbf{r}_2)\otimes|\chi\rangle$ with $|\chi\rangle$ one of the four spin states. Total exchange acts as (spatial swap) × (spin swap).

Step 2. Impose antisymmetry: $\Psi$ must acquire $-1$ under total exchange. With spatial parity $\epsilon_{\text{space}}=\pm1$ and spin parity $\epsilon_{\text{spin}}=\pm1$ from (2), the requirement is $\epsilon_{\text{space}}\cdot\epsilon_{\text{spin}}=-1$ : the two parities must be opposite.

Step 3. Read off the pairing: symmetric space ( $+$ ) ⇒ singlet ( $-$ ); antisymmetric space ( $-$ ) ⇒ triplet ( $+$ ). These are the only two possibilities for identical spin- $\tfrac{1}{2}$ fermions, since the total wavefunction must be antisymmetric. Why energy cares: an antisymmetric spatial function has a node at $\mathbf{r}_1=\mathbf{r}_2$ — triplet electrons are never found at the same point, so they feel less Coulomb repulsion than singlet electrons in the corresponding symmetric state. The energy split between the channels is an electrostatic integral (the exchange integral), yet it selects a spin channel: electrostatics impersonating a huge spin-spin force. That impersonation is the origin of magnetic ordering scales — magnets are held together by the Coulomb interaction plus a minus sign.

Step 4. Name the consequence in helium: two 1s electrons must pair to a singlet (both in the same orbital makes the spatial part automatically symmetric) — the para ground state; excited configurations split into ortho (triplet, lower — the node cuts repulsion) and para (singlet, higher) ladders whose weak intercombination makes helium behave like two gases. The quantitative variational theory of those levels is the many-body chapters' property; the mechanism — this derivation — is complete already. The swap eigenvalues (2) and the Dirac identity (3) are exact in the $4\times4$ representation; the projector algebra of the channel splitters below is likewise exact.

For calculations, package the channels as projectors — both verified idempotent, orthogonal, and complete, with traces 1 and 3:

\hat{P}_S=\tfrac14-\frac{\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2}{\hbar^2},\qquad \hat{P}_T=\tfrac34+\frac{\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2}{\hbar^2} \qquad (4)

where: sandwich any two-spin state and $\langle\hat{P}_S\rangle$ , $\langle\hat{P}_T\rangle$ are their channel probabilities; and any isotropic two-spin Hamiltonian is a linear combination $E_S\hat{P}_S+E_T\hat{P}_T$ . The canonical instance is the exchange Hamiltonian $\hat{H}=J\,\hat{\mathbf{S}}_1\!\cdot\!\hat{\mathbf{S}}_2/\hbar^2$ (defining $J$ as an energy): spectrum $\{-\tfrac{3J}{4},+\tfrac{J}{4}\}$ , gap exactly $J$ , singlet ground state for $J>0$ (antiferromagnetic sign convention). This is precisely the two-spin building block from which the Heisenberg chain is constructed.

Limiting cases

case	result	meaning
measure $\hat{S}_z^{\text{tot}}$ on any of the four	sharp $M$	both bases share $M$ ; only the $M=0$ pair distinguishes them
distinguishable particles	same four states, no pairing rule	exchange symmetry exists but is not enforced; the theorem needs identity
bosonic pair (spin-½ composite bosons)	pairing rule flips: symmetric total	symmetric space ⇒ triplet; the sign of the postulate drives everything
$J\to0$ exchange coupling	fourfold degeneracy restored	singlet-triplet split is entirely interaction, never kinematics
add a uniform field $b\hat{S}_z$	triplet fans into 3, singlet inert	the $M$ -blind singlet is the tell in level diagrams

Worked examples

Example 1 — measurement ladder on $|\!\uparrow\downarrow\rangle$ . A pair is prepared with particle 1 up and particle 2 down. What do total-spin measurements give?

Step 1. Expand (construction above): $|\!\uparrow\downarrow\rangle=\tfrac{1}{\sqrt2}|1,0\rangle+\tfrac{1}{\sqrt2}|0,0\rangle$ . So measuring $\hat{S}^2$ : outcomes $2\hbar^2$ and $0$ with probability $\tfrac12$ each — verified against the direct expectation $\langle\hat{S}^2\rangle=\hbar^2$ , which sits exactly midway.

Step 2. Suppose $\hat{S}^2$ returned $2\hbar^2$ (triplet). The state collapses to $|1,0\rangle$ (the Lüders projection of Measurement and Collapse, cited). A subsequent $\hat{S}_z^{\text{tot}}$ measurement gives $0$ with certainty — but each individual spin is now a fair coin, symmetrically correlated.

Step 3. Had it returned $0$ instead, the collapsed singlet answers every subsequent total-spin question — any axis — with zero. Answer: Thus $P(S=1)=P(S=0)=\tfrac{1}{2}$ . After projection onto the triplet, the total-spin projection $M=0$ is definite, while the individual spin projections remain maximally uncertain.

Example 2 — an exchange-coupled dimer at finite temperature. Two spins in a molecular dimer have antiferromagnetic exchange $J/k_B=5\,$ K. Find the singlet-triplet gap and the equilibrium singlet fraction at $T=4\,$ K.

Step 1. Gap: $\Delta=E_T-E_S=J=5k_B=6.90\times10^{-23}$ J $=0.431$ meV (computed: $0.4309$ meV) — deep microwave territory, typical of molecular magnets.

Step 2. Boltzmann census over $\{E_S=-\tfrac{3J}{4}\ (\times1),\ E_T=+\tfrac{J}{4}\ (\times3)\}$ : $f_S=\dfrac{e^{3J/4k_BT}}{e^{3J/4k_BT}+3e^{-J/4k_BT}}$ .

Step 3. At $T=4\,$ K, $J/k_BT=1.25$ : $f_S=0.538$ (computed). The singlet wins the energy but the triplet wins the entropy — three states versus one — so even below the gap temperature the channels stay comparably occupied; cooling to $T\ll J/k_B$ is what purifies the singlet. Answer: $\Delta=0.431$ meV; singlet fraction $\approx0.54$ at 4 K.

Origins · 1 minWerner Heisenberg and how this was discovered

Werner Heisenberg

1901–1976 · Copenhagen and Leipzig

By mid-1926 Heisenberg had already rebuilt mechanics; that June he used it on the first system with two identical electrons. The paper Mehrkörperproblem und Resonanz in der Quantenmechanik treated helium's electrons as a resonating pair and discovered that the states sort into symmetric and antisymmetric families that light barely connects — explaining, at a stroke, why spectroscopists had catalogued "two heliums" (ortho and para) for decades. The follow-up insight, that the electrostatic exchange integral acts like an enormous effective spin-spin coupling, became the Heisenberg model of magnetism — ferromagnetism explained without a single magnetic force. This band tells only the 1926 exchange story: the invention of matrix mechanics on Helgoland the year before, and the 1927 uncertainty paper, are told with their own articles (Ladder Operators and Standard Deviation and Quantum Uncertainty respectively).

History

1912

— Spectroscopists formalize the puzzle of "orthohelium" and "parahelium" — two non-communicating line systems in one gas

1925

— Pauli's exclusion principle; Uhlenbeck and Goudsmit's spin — the ingredients assemble

1926

— Heisenberg's resonance paper derives the ortho/para split from exchange symmetry; Dirac independently formalizes the symmetrization postulate

1928

— Heisenberg's exchange model of ferromagnetism: Coulomb plus antisymmetry masquerading as spin coupling

1938

— Positronium-style singlet/triplet annihilation selection rules worked out (para decays to two photons, ortho to three)

Going further

Griffiths & Schroeter §4.4.3 — the construction at the gentlest register.
Sakurai & Napolitano §3.8 — coupling formalism; §7 for the identical-particle postulate this article's theorem consumes.
Cohen-Tannoudji, Diu & Laloë, Vol. 2, Ch. X — completeness on the two-spin problem.
Shankar Ch. 15; Townsend Ch. 5 — alternative routes.

Provenance note: All coefficients, eigenvalues ( $-\tfrac34\hbar^2$ , $+\tfrac14\hbar^2$ ), and ladder normalizations are standard textbook results. The singlet/triplet decomposition follows from $\hat{S}_-$ acting on $|\!\uparrow\uparrow\rangle$ , and the $\mathbf{S}_1\cdot\mathbf{S}_2$ eigenvalues are confirmed from $\tfrac12(S^2-S_1^2-S_2^2)$ . The 21 cm hyperfine frequency 1420.405751 MHz is an anchor value from standard atomic data, carried by the hydrogen treatment in Coupled and Uncoupled Bases.

Is the singlet the same as a pair of opposite spins?

No — and the difference is the relative sign. $|\!\uparrow\downarrow\rangle$ has opposite spins but is half singlet, half triplet; measuring $\hat{S}^2$ on it is a coin flip. The singlet is the specific antisymmetric superposition, with anticorrelation along every axis, not just $z$ . "Opposite along one axis" is a product-state property; "opposite along all axes" is the singlet's — and no product state can do it.

Why is the symmetric M=0 state grouped with the stretched states rather than the singlet?

Because rotations say so: applying total $\hat{S}_\pm$ walks $|1,0\rangle$ up to $|\!\uparrow\uparrow\rangle$ and down to $|\!\downarrow\downarrow\rangle$ , while the singlet is annihilated. The three triplet states are one object viewed at three orientations; the grouping is by $\hat{S}^2$ eigenvalue, which rotations cannot change, not by how the state "looks" in the $z$ -basis.

Does the space-spin correlation require a spin-dependent force?

No — that is its entire charm. The Hamiltonian in the theorem's derivation is spin-blind; the correlation is imposed by the symmetrization postulate, a statement about which states exist for identical fermions. The energy difference between channels then comes from ordinary electrostatics evaluated on the differently-shaped spatial states. Spin-dependent forces (spin-orbit, dipole-dipole) exist too, but they are small corrections layered on top.

Ortho-helium lies below para-helium in excited states — but the ground state is para. Why no ortho ground state?

The would-be ortho ground state needs an antisymmetric spatial function of two 1s orbitals — and the antisymmetric combination of two identical orbitals vanishes identically. With both electrons in 1s the spatial part is forcibly symmetric, so the spin part must be the singlet. That is the exclusion principle in its original spectroscopic clothing: no triplet 1s².

Is the singlet entangled, and does that matter here?

Maximally, in the spin factorization — it is the canonical Bell state. In this article the entanglement shows up operationally as all-axis anticorrelation (Example 1's collapsed statistics). What that correlation can and cannot be explained by — hidden variables, Bell inequalities — is the entanglement chapters' territory; here the singlet is simply the $S=0$ member of a coupling decomposition.

What plays the role of singlet/triplet for two spin-1 particles?

The same machinery with a bigger window: $1\otimes1=2\oplus1\oplus0$ — quintet, triplet, singlet of dimensions 5, 3, 1. Exchange parity alternates down the ladder ( $+,-,+$ ): the swap eigenvalue is $(-1)^{2j-S}$ in general, which for two spin-½ reduces to the $+$ / $-$ split used here. The counting theorem behind all of this is Addition of Angular Momenta's; the tables are Clebsch-Gordan Coefficients'.

How do experiments tell singlet from triplet without measuring every spin?

Three standard handles. Energy: an exchange or hyperfine splitting makes the channels spectroscopically distinct (Example 2). Multiplicity: apply a field — triplets fan into three Zeeman components, singlets stay put. Selection rules: transitions and decays conserve exchange symmetry to high accuracy (helium's weak intercombination lines, positronium's two- versus three-photon decays). Each is the abstract $\pm1$ made visible.