In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism. In this model, for a single spin, four base operators exist: I x {\displaystyle I_{x}} , I y {\displaystyle I_{y}} , I z {\displaystyle I_{z}} and E / 2 {\displaystyle E/2} which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for coupled systems. Using this approach, the evolution of the magnetization under free precession is represented by I z {\displaystyle I_{z}} and corresponds to a rotation about the z-axis with a phase angle proportional to the chemical shift of the spin in question:
I x → ω τ I z cos ( ω τ ) I x − sin ( ω τ ) I y {\displaystyle I_{x}{\xrightarrow {\omega \tau I_{z}}}\cos(\omega \tau )I_{x}-\sin(\omega \tau )I_{y}}
Pulses about the x and y axis can be represented by I x {\displaystyle I_{x}} and I y {\displaystyle I_{y}} respectively; these allow to interconvert the magnetization between planes and ultimately to observe it at the end of a sequence. Since every spin will evolve differently depending on its shift, with this formalism it is possible to calculate exactly where the magnetization will end up and hence devise pulse sequences to measure the desired signal while excluding others. The product operator formalism is particularly useful in describing experiments in two-dimensions like COSY, HSQC and HMBC.
Motivation for sets of spin-1/2 particles Throughout this section, the reduced Planck constant ℏ = 1 {\displaystyle \hbar =1} for convenience. The product operator formalism is usually applied to sets of spin-1/2 particles, since the fact that the individual operators satisfy L x 2 = L y 2 = L z 2 ∝ 1 {\displaystyle L_{x}^{2}=L_{y}^{2}=L_{z}^{2}\propto \mathbf {1} } , where 1 {\displaystyle \mathbf {1} } is the identity operator, makes the commutation relations of product operators particularly simple. In principle the formalism could be extended to higher spins, but in practice the general irreducible spherical tensor treatment is more often used. As such, we consider only the spin-1/2 case below. The main idea of the formalism is to make it easier to follow the system density operator ρ {\displaystyle \rho } , which evolves under a Hamiltonian H {\displaystyle H} according to the Liouville-von Neumann equation as
∂ ρ ∂ t = − i [ H , ρ ] . {\displaystyle {\frac {\partial \rho }{\partial t}}=-\mathrm {i} [H,\rho ].}
For a time-independent Hamiltonian, the density operator inherits its solutions from the Schrödinger time-evolution operator U ( t ) = exp ( − i H t ) {\displaystyle U(t)=\exp(-\mathrm {i} Ht)} as
ρ ( t ) = U ( t ) ρ ( 0 ) U − 1 ( t ) = exp ( − i H t ) ρ ( 0 ) exp ( + i H t ) {\displaystyle \rho (t)=U(t)\,\rho (0)\,U^{-1}(t)=\exp(-\mathrm {i} Ht)\,\rho (0)\,\exp(+\mathrm {i} Ht)}
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