# Physics¶

## Physical Background¶

SpinDrops is primarily interested in the state evolution of coupled two-level (ie spin $$\frac{1}{2}$$) quantum systems, typical in solution NMR (as well as in other contextx). The simulation is a solution of the Liouville–von Neumann equation

$i \hbar \frac{\partial}{\partial t}\rho(t) = [H, \rho(t) ]$

(with $$\hbar$$ set to 1 for NMR) over the duration of the pulse sequence, with $$\rho(0)$$ the defined initial state. The pulse sequence is defined as a series of time-constant Hamiltonians incorporating rf-pulses, offset effects, or simply user-defined operators.

## Limitations¶

Because version 2.0+ of SpinDrops calculates the closed-system propagation of the density operator ($$\rho$$) under the Liouville–von Neumann equation (with $$\hbar$$ set to 1 for NMR), the following effects can not yet be taken into account:

• T1 relaxation
• T2 relaxation
• NOE

Incorporation of these effects in the simulation is planned for a future version of SpinDrops, which will calculate the time propagation taking into account relaxation ($$R$$) and chemical-exchange ($$X$$) terms:

$\frac{\partial}{\partial t}\rho(t) = (-iL - R - X) \rho(t)$

### Version 1.x¶

In version 1.2 of SpinDrops, simulations were based on the standard Cartesian Product Operator formalism. This elegant formalism provides simple analytical expressions to calculate the dynamics of coupled spins. However, it also has some limitations:

• Ideal pulses were assumed, i.e. the effects of frequency offsets and couplings could not be taken into account during the pulses.
• During delays, the weak coupling limit was assumed. For heteronuclear spins, this is always an excellent approximation. However, the simulations of homonuclear spin systems is not exact if strong coupling effects play a role (i.e. if the offset-difference of two spins is on the same order of magnitude as the J coupling between them). This was not a problem as long as you were aware of this limitation. In fact, it allows you to see and study the effects of simultaneous offset and weak coupling evolution on a comparable time-scale, which would not be possible otherwise. Although strong coupling effects were neglected during delays, they were fully taken into account in the simulation of spin dynamics of two coupled spins under isotropic mixing conditions (in TOCSY and TACSY experiments).