Today’s Lecture 5) Wed, Oct 8: Product operators I (tools to simplify the quantum mechanics) a. RF pulses b. Chemical shift Download Mathematica Player and Bloch Equation demo http://demonstrations.wolfram.com/MagneticResonanceAndBlochEquations/
5) Wed, Oct 8: Product operators I (tools to simplify the quantum mechanics)
a. RF pulses
b. Chemical shift
Download Mathematica Player and Bloch Equation demo
NMR is all rotations.
Notice that the coordinate system satisfies the “right hand rule”. If you point your right thumb along the z-axis, you fingers will close from x to y.
These are 3D rotation matrices. When they act on a vector, they rotate the vector around the axis that defines the vector (Rx around x; Ry around y; Rz around z).
Here is an example of a rotation of Mz around the x-axis by an angle f:
Notice that if f=0, the vector stays the same. If f=p/2, the vector is rotated to the –y axis.
Every rotation in 3D space leads to 2 terms. The first term is pointing in the same direction and is multiplied by Cos(f) where f is the rotation angle. The second term is along the axis that the vector is rotated into and is multiplied by Sin(f).
For example, in the rotation above, a rotation of f around the x-axis (because it is the Rx rotation matrix) of Mz produces Mz*Cos(f)-My*Sin(f).
For example, the rotation we just saw:
Just like a pulse that produces a torque, operators rotate around a given axis but do not act on things along the axis of rotation. Corresponding to B1 fields along the x and y axes, we have 2 operators for pulses:
Pulse along the x-axis by f degrees
Operators have hats.
Pulse along the y-axis by f degrees
Chemical shift! The chemical shift operator works exactly like a pulse operator, but it only acts along the z-axis.
The only difference between the convention for a pulse and chemical shift is that we put in the frequency (W) times time for the shift and only the rotation angle for the pulse.
6) Fri, Oct 10: Product operators II
a. Scalar (J) coupling
b. Multiple pulse experiments