Tensor product of quantum systems

Independent quantum systems combine by tensor product, with local operators acting as $A\otimes I$ or $I\otimes B$.

The Jaynes-Cummings model couples a two-level atom to one quantized oscillator mode throughThe first term excites the atom while annihilating one field quantum; the second performs the Hermitian-conjugate process.

For distinguishable particles with one-particle Hilbert spaces ${\mathcal{H}}_i$, the multiparticle state space is ${⨂}_i{\mathcal{H}}_i$. A product state is a tensor product of one-particle states, while a general state is a linear combination of product states.

For two particles, the exchange operator sends $|\alpha ⟩\otimes |\beta ⟩$ to $|\beta ⟩\otimes |\alpha ⟩$. It has eigenvalues $+1$ on symmetric states and $-1$ on antisymmetric states.

Physical states of identical particles have definite symmetry under every particle exchange operator: symmetric for bosons and antisymmetric for fermions.

Bosons are identical particles whose total multiparticle state is symmetric under exchange. In three-dimensional relativistic quantum theory they have integer spin.

Bose statistics is the symmetric exchange statistics of identical bosons. Commuting creation operators build the symmetric bosonic Fock space; a normalized one-particle mode has occupation states $(a^{†}{)}^n|0⟩/\sqrt{n!}$ for every nonnegative integer $n$. At thermal equilibrium these statistics lead to the Bose-Einstein distribution. Exchange symmetry itself does not require thermal equilibrium.

Commuting creation operators make multiparticle states invariant under exchange of their labels. With $[a,a^{†}]=1$, the normalized number states $(a^{†}{)}^r|0⟩/\sqrt{r!}$ exist for all $r\ge 0$, giving unrestricted bosonic occupation numbers. The single-mode thermal sum is $Z={\sum }_{r\ge 0}{e}^{-\beta Er}=(1-e^{-\beta E}{)}^{-1}$ and the mean occupation is $(e^{\beta E}-1{)}^{-1}$. Thus the exchange symmetry and the Bose-Einstein distribution are two expressions of the same bosonic algebra.

Fermions are identical particles whose total multiparticle state is antisymmetric under exchange. In three-dimensional relativistic quantum theory they have half-integer spin.

For two identical spin-one-half fermions with no additional antisymmetric internal factor, the antisymmetric spin-one-half singlet state combines with an even-$\ell$ symmetric orbital state, while the symmetric spin-one-half triplet state combines with an odd-$\ell$ antisymmetric orbital state.

No two identical fermions may occupy the same one-particle quantum state. At zero temperature, noninteracting fermions fill the available one-particle states in increasing order of energy.

The spin-statistics theorem associates integer-spin particles with bosonic exchange symmetry and half-integer-spin particles with fermionic exchange symmetry in relativistic quantum field theory under the standard locality and positivity assumptions.

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

If $E_n=E_0+n\Delta$, the two-particle level $2E_0+N\Delta$ has degeneracyEach unequal orbital pair contributes six symmetric-spin states and three antisymmetric-spin states; an equal orbital pair exists only for even $N$ and contributes six states.

Two decoupled Fermi oscillators have product occupation states and additive energies $E_1{n}_1+E_2{n}_2$.

In a product basis, an operator local to one subsystem is represented by its matrix Kronecker the identity on every other subsystem.

Commuting Hamiltonian terms on separate factors are simultaneously diagonalized by tensor products of their eigenvectors and have additive eigenvalues.