1. 3.2.2 Magnetic Field in general direction Larmor frequencies: Hamiltonian: In matrix representation 

2. Rewriting the Hamiltonian:    Characteristic equation:

3. Eigenstates: Spin Flip: or Initial state Time evolved state z B

4.  z BProbability of a spin flip

5. Sinceand Rabi Formula t z B

6. t     (a) (b) t     (c) z B z B B z

7. The most general initial state: In matrix formalism 0 T       T

8. Probability for measuring the spin projection along the z-axis: Probability for measuring the spin projection along the x-axis:

9. 3.4 Magnetic Resonance Magnetic Resonance Image showing a vertical cross section through a human head. Modern 3 tesla clinical MRI scanner. Magnetic Resonance Imaging (MRI) http://en.wikipedia.org/wiki/Magnetic_resonance_imaging

10.  Uniform magnetic field: Additional rotating magnetic field: Total magnetic field: Time dependent Hamiltonian: x y z

11. Matrix representation of the Hamiltonian: Schrödinger Equation :

12. State vector as viewed from the rotation frame: Schrödinger Equation becomes:

13. where In the rotating frame the Hamiltonian is time independent!!

14. Rabi Formula When Rabi Formula with   

15. Rabi flopping equation where Generalized Rabi frequency  When   (resonance condition),  Rabi frequency.

16. Amplitude of the spin flip probability: Lorentzian curve   Magnetic resonance curve   FWHM emissionabsorptionemission absorption t   Time dependent spin flip probability: