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Physics
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Physical Background
-------------------
SpinDrops is primarily interested in the state evolution of coupled
two-level (ie spin :math:`\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
.. math::
i \hbar \frac{\partial}{\partial t}\rho(t) = [H, \rho(t) ]
(with :math:`\hbar` set to 1 for NMR) over the duration of the pulse
sequence, with :math:`\rho(0)` the defined :ref:`initial state
`. The :ref:`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 :term:`density operator` (:math:`\rho`) under the
`Liouville–von Neumann equation
`_
(with :math:`\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 (:math:`R`) and chemical-exchange
(:math:`X`) terms:
.. math::
\frac{\partial}{\partial t}\rho(t) = (-iL - R - X) \rho(t)
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.. image:: _images/blurb.png
:width: 40%
:align: right
Version 1.x
+++++++++++
In **version 1.2** of SpinDrops, simulations were based on the
standard :term:`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
:term:`TOCSY` and TACSY experiments).