Today’s Lecture
1 / 14

Lecture 4 Notes powerpoint: Product Operators: I - PowerPoint PPT Presentation

  • Updated On :

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

Related searches for Lecture 4 Notes powerpoint: Product Operators: I

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

PowerPoint Slideshow about 'Lecture 4 Notes powerpoint: Product Operators: I' - johana

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Slide1 l.jpg

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

Slide3 l.jpg

Vector representations of Mx, My, and Mz time!



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.








Slide4 l.jpg

Rotation Matrices time!

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).

Slide5 l.jpg

What is a rotation? time!

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.

Slide6 l.jpg

General Rule of Rotations time!

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).

Slide7 l.jpg

Product Operators are a mathematical shorthand to do rotations.

For example, the rotation we just saw:


Slide8 l.jpg

Product Operator Rules rotations.

  • 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)

Slide9 l.jpg

Product Operator Rules rotations.

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

Slide10 l.jpg

Examples of pulses rotations.

Slide11 l.jpg

What is a rotation around the z-axis? rotations.

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.

Slide12 l.jpg

1D NMR experiment rotations.





(+ relaxation)




Slide13 l.jpg

Product Operator I Summary rotations.

  • 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.

Slide14 l.jpg

Next Lecture rotations.

6) Fri, Oct 10: Product operators II

a. Scalar (J) coupling

b. Multiple pulse experiments