Representations of the Precession

• Assume static magnetic field along $\hat{z}$ and write out components of vector equation in cartesian space. (see HW)

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  $$\frac{d\mu_{x}}{dt} = \gamma \mu_{y} B_{0} = \omega_{0} \mu_{y}$$

  $$\frac{d\mu_{y}}{dt} = -\gamma \mu_{x}B_{0} = - \omega_{0} \mu_{x}$$

  $$\frac{d\mu_{z}}{dt} = 0$$ (2.13)

  
  

• sets of differential equations where derivative of a = b and derivative of b=a have sine and cosines as solutions. See solutions in the book.

  
  
  
  
  
  
  
  

• can combine the first two lines of (2.13) to get an interesting complex representation
  

  $$\frac{d\mu_{x}}{dt} + i \frac{d\mu_{y}}{dt} = -i \left( i \omega_{0} \mu_{y} + \omega_{0} \mu_{x} \right)$$

  $$\frac{d\mu_{+}}{dt} = -i\omega_{0} \mu_{+}$$ (2.14)

  
  
  where have defined
  

  $$\mu_{+} (t) = \mu_{x}(t) + i\mu_{y}(t) \\$$ (2.15)

  
  

• solution for derivate of something is equal to itself, i.e., (2.14) is an exponential
  

  $$\mu_{+}(t) = \mu_{+}(0)e^{-i\omega_{0}t}$$ (2.16)

  
  

  
  
  
  
  
  
  
  

• Please Pay Attention: Observing phase of the spins is how we make images, and explains the bulk of the physical phenomenon we make use of in MRI. Phase is going to come up repeatedly. Won't always be talking about the differential equations, but will concentrate on phase as a function of field strength, and initial phase under special circumstances
  

  $$\phi_{0}(t) = - \omega_{0} t + \phi_0(0)$$ (2.17)

  
  

(IN CLASS QUESTION)