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.
which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator.
and corresponds to a rotation about the z-axis with a phase angle proportional to the chemical shift of the spin in question:
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.
In principle the formalism could be extended to higher spins, but in practice the general irreducible spherical tensor treatment is more often used.
The main idea of the formalism is to make it easier to follow the system density operator
The factors of two in the 'true' two-spin operators are to allow for convenient commutation relations in this specific spin-1/2 case - see below.
that obey the cyclic commutation relations In fact only the first two relations are necessary for the following derivation, but since we are usually working with operators associated with Cartesian directions, such as the individual angular momentum operators, the third commutator follows by a symmetry argument.
, and noting that both sides satisfy the same differential equation in that parameter, with the same initial condition at
, we have where the final equality follows from recognising the Taylor series for sine and cosine.
will be given by The interpretation of this is that although nuclear spin angular momentum itself is not connected to rotations in three-dimensional space in the same way that angular momentum is, the evolution of the density operator can be viewed as rotations in an abstract space, in which the operators
Note that a more complicated calculation has now been reduced to a simpler procedure that requires no knowledge of the underlying quantum mechanics, especially since the subspaces of cyclic operators can be tabulated in advance.
is which means that in an ensemble of many such spins with slightly different chemical shifts, there is a dephasing of the magnetisation in the
-axis, the evolution of an individual spin in the ensemble is Hence this sequence refocuses the transverse magnetisation produced by the first pulse, independent of the value of the chemical shift.
As an indication of the utility of the formalism, suppose instead that we tried to reach the same result using states only and therefore the Schrödinger time evolution operators.
Since the arguments of the exponentials in the original form of the propagator do not commute, this amounts to solving a specific example of the Baker–Campbell–Hausdorff (BCH) problem.
, as well as the mathematical similarity of the spin operators with the physical rotation generators, which allow us to write Hence
For larger sequences of pulses this state treatment quickly becomes even more unwieldy, unless more advanced methods such as exact effective Hamiltonian theory (which gives closed-form expressions for the entangled propagators via the Cayley–Hamilton theorem and eigendecompositions) are used.
After the second pulse, we can decompose the remaining terms into a sum of two spin populations differing only in the sign of the
In reality there will be further transverse components originating from the tipping of the longitudinal magnetisation that remained after the first pulse.
To instead arrive at this result using the state formalism, we would have had to non-trivially evaluate the rotation propagator as and then evaluate a transition probability by considering the result of applying this to a state representing polarisation in the transverse plane.
DEPT (Distortionless Enhancement by Polarisation Transfer) is a pulse sequence used to distinguish between the multiplicity of hydrogen bonded to carbon, that is it can separate C, CH, CH2 and CH3 groups.
) such that it satisfies because then the first term in the evolved density operator in Equation 2 vanishes under the pure coupling evolution between the pulses.
where 'others' denotes various terms that can safely be ignored because they will not evolve into observable transverse polarisation on the target spin
The key observation is that since we can again ignore the refocussed chemical shift, the only relevant dynamics occur in the interval with no hydrogen decoupling, where we can consider solely the
separate neighbouring hydrogen atoms commute, so the overall effect is to multiply by a factor