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3.2.2 Magnetic Field in general direction

q. Larmor frequencies:. Hamiltonian:. In matrix representation. 3.2.2 Magnetic Field in general direction. Characteristic equation:. w 0. q. Rewriting the Hamiltonian:. w 1. Eigenstates:. or. Spin Flip:. Initial state. z. B. Time evolved state. Probability of a spin flip. q. z.

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3.2.2 Magnetic Field in general direction

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

  2. Characteristic equation: w0 q Rewriting the Hamiltonian: w1

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

  4. Probability of a spin flip q z B

  5. t z B Since and Rabi Formula

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

  7. In matrix formalism 0 T q  q f f+w0T The most general initial state:

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

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

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

  11. Schrödinger Equation : Matrix representation of the Hamiltonian:

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

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

  14. When Rabi Formula Rabi Formula with w0 -Dw

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

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

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