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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

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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


Vector representations of Mx, My, and Mz



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.









Rotation Matrices

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).


What is a rotation?

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.


General Rule of Rotations

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).


Product Operators are a mathematical shorthand to do rotations.

For example, the rotation we just saw:



Product Operator Rules

  • The thing above the arrow is the operator. The one shown above is a f pulse along the positive x-axis. The nucleus that we are worrying about is given by a capital letter like “I”. The orientation of the magnetization is given by the subscript. If we just have one nuclear spin to worry about (e.g. water), then there are 3 spatial orientations for that nucleus:
    • Iz (pointing along the z-axis)
    • Ix (pointing along x)
    • Iy (pointing along y)

Product Operator Rules

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


What is a rotation around the z-axis?

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.


1D NMR experiment





(+ relaxation)





Product Operator I Summary

  • Operators act on magnetization vectors and cause rotations.
  • Rotations always have the starting term multiplied by cosine and the resulting term multiplied by sine of the angle.
  • We have discussed two kinds of operators, pulse and chemical shift. These both do “normal” rotations in 3D and leave the magnetization unaltered.
  • Operators can act sequentially (e.g. pulse followed by evolution). This becomes critical in analyzing 2D or multiple pulse experiments.

Next Lecture

6) Fri, Oct 10: Product operators II

a. Scalar (J) coupling

b. Multiple pulse experiments