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

2. Characteristic equation:w0 w1
Rewriting the Hamiltonian:

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

4. Probability of a spin flipq z B

5. t z B
Since
and
Rabi Formula

6. z B
(a)
(b) t
0.25
0.5
0.75 1 B z
(c) t
0.25
0.5
0.75 1 z B

7. The most general initial state:
In matrix formalism T
q  q
f  f+w0 T

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

9. Magnetic Resonance Imaging (MRI)
Modern 3 tesla clinical MRI scanner.
Magnetic Resonance Imaging (MRI)
Magnetic Resonance Image showing a vertical cross section through a human head.

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

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 w0  -Dw

15. Rabi flopping equation
where
Generalized Rabi frequency
When w = w0 (resonance condition), : Rabi frequency. E

16. 2w1 w0 w t Amplitude of the spin flip probability: Lorentzian curve
Magnetic resonance curve
2w1
FWHM t
0.5 1
Time dependent spin flip probability:
emission
absorption
emission
absorption
emission