2 Magnetic Field in general direction Larmor frequencies: c m eB c m eB e e 1 1 0 0 , x z z x e e S S B S B S c m e B S c m e B H 1 0 0 1 ) ( Hamiltonian: 0 1 1 0 1 0 2 0 1 1 0 2 1 0 0 1 2 H In matrix representation x B z B B ˆ ˆ 1 0 B 0 B 1 B 0 1 tan B B
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. - PowerPoint PPT Presentation
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3.2.2 Magnetic Field in general direction
Larmor frequencies:cm
eB
cm
eB
ee
11
00 ,
xzzxee
SSBSBScm
eBS
cm
eBH 1001 )(
Hamiltonian:
01
1010
201
10
210
01
2
H
In matrix representation
xBzBB ˆˆ 10 B
0B
1B
0
1tanB
B
21
20
2
1
2
02
2
100
01
10
20
22
0222
0
22
22
cossin,cossin
sincos
2
22
21
20
21
20
21
20
0
21
20
1
21
20
1
21
20
0
21
20
01
10
zxnn SSnSSS
H
Rewriting the Hamiltonian:0
1
0
1tan
B
B
21
20
1
21
20
0
sin
cos
Characteristic equation: 0 IH
2cos
2sin
2sin
2cos
n
nEigenstates:21
202
E
21
202
E
Spin Flip: or
Initial state
nn
nnnn
nnnn
nnnn
2sin
2cos
)0(
n
tEi
n
tEi
eet
2sin
2cos)(
Time evolved state
zB
0B B
1B
x
z
zB
)(t
2cos
2sin
2sin
2cos
n
n
2)()( tP
Probability of a spin flip
)0(
tE
tEE
tEE
e
ee
ee
ee
tEE
i
tEit
Ei
n
tEi
n
tEi
n
tEi
n
tEi
2sinsin
2
)(sinsin
)(cos1sin
2
11
2cos
2sin
2cos
2sin
2sin
2cos
2sin
2cos
2sin
2cos
2222
2
2)(22
2
2
2
21
20
1sin
21
202
ESince and
tP2
sin)(21
202
21
20
21
Rabi Formula
t
2sin
)( P
E2
tE
P2
sinsin)( 22 z
B
t
)(P
1/2 1/4 1/6
(a) 0)(0ˆ 10 PzBB
(b)
tPxBB
2sin)(0ˆ 12
01
t
)(P
0/2 0/4 0/6
(c)
tPBB
2sin)( 02
20
21
0101
tP2
sin)(21
202
21
20
21
zB
z B
B
z
The most general initial state:
2
sin2
cos)0( i
ne
In matrix formalism
2sin
2cos
2sin
2cos
2sin
2cos
)(
2sin
2cos
)0(
0
0
0
0
2
2
2
Ti
Ti
iT
i
Ti
iTEi
TEi
i
ee
ee
e
ee
eT
e
0 T T
00 Bcm
eTT
e
vLTeeeT iTEiT
Ei
/,2
sin2
cos)(
2
cos2
cos
2sin
2cos
01)()( 2
2
2
2
)(
22 0
0
0
T
i
Ti
Ti
ee
eTP
Probability for measuring the spin projection along the z-axis:
)cos(sin12
1)cos(sin1
2
1
2sin
2sin
2cos
2cos
2
1
2sin
2cos
2
1
2sin
2cos
112
1)()(
00
2)()(2
2)(
2
)(
22
00
0
0
0
Bcm
eTT
ee
ee
eTP
e
TiTi
Ti
Ti
Ti
xx
Probability for measuring the spin projection along the x-axis:
3.4 Magnetic Resonance
Magnetic Resonance Image showing a vertical cross section through a human head.