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The electron magnetic dipole, due to spin, interaction with the magnetic flux
density is described by the potential:

r r
V = µ σ ⋅B (8.59)
B
and the dynamics of the electron spin state in the magnetic flux density is
described by Schrodinger’s equation:

d r r
jh ψ = µ σ ⋅ B ψ (8.60)
B
dt
where, as previously mentioned, the Dirac ket-vector is two-dimensional.

Mathematical Problem: To put the problem in purely mathematical form, we
are asked to find the time development of the two-dimensional vector ψ if
this vector obeys the system of equations:


d at() =− Ω r  at()
( ⋅σ

dt bt()  j 2 ê)   bt()  (8.61)

Ω µ B
where = B 0 , and is called the Larmor frequency, and the magnetic flux
2 h
r
density is given by B = B ê. The solution of Eq. (8.61) can be immediately
0
written because the magnetic flux density is constant. The solution at an arbi-
trary time is related to the state at the origin of time through:

 at()  Ω r   0
a()
  = exp  j − ( ⋅σ ê t)    (8.62)
 bt()  2  0
b()
which from Eq. (8.55) can be simplified to read:

 at()   Ω  r  Ω    0
a()
  =  cos tI  − j( ⋅σ ê)sin t   (8.63)

 bt()   2   2   0
b()

If we choose the magnetic flux density to point in the z-direction, then the
solution takes the very simple form:
 j − Ω 
 at() e 2 t a() 0
  =  Ω  (8.64)
 bt()  j t 
 e 2 b() 
0

© 2001 by CRC Press LLC