3.2.2 magnetic field in general direction larmor frequencies: hamiltonian: in matrix representation
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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.
Modern 3 tesla clinical MRI scanner.
Magnetic Resonance Imaging (MRI)
http://en.wikipedia.org/wiki/Magnetic_resonance_imaging
20
E
20
E 0 EEE
cm
eBSH
ez
0000 ,
2
2
00
00
H
H
zBB
00 Uniform magnetic field:
ytBxtBB ˆsinˆcos 111
Additional rotating magnetic field:
Total magnetic field: ytxtBzBBBB ˆsinˆcosˆ 1010
Time dependent Hamiltonian:
cm
eB
cm
eB
StStStHHBtH
ee
yxz
11
00
1010
,
,sincos)()(
0B
)(1 tB
x
y
z
Matrix representation of the Hamiltonian:
yxz StStStH sincos)( 10
,)()()( ttHtdt
di Schrödinger Equation :
)(
)()()()(
tc
tctctct
)(2
)(2
)(
)(2
)(2
)(
01
10
tctcedt
tdci
tcetcdt
tdci
ti
ti
0sin
)sin0
20cos
cos0
20
0
2 1
1
1
1
0
0
ti
ti
t
t
01
10
2
ti
ti
e
e
)(
)(
2)(
)(
01
10
tc
tc
e
e
dt
tdci
dt
tdci
ti
ti
State vector as viewed from the rotation frame:
2
2
2
2
2
2
)()(
)()(,
)(
)()()(
)(
)()(
)(
0
0)()(~
ti
ti
ti
ti
ti
ti
etct
etctt
ttt
etc
etctc
tc
e
etRt
2
222
)(
)()()()( t
i
tit
it
i
et
etetett
Schrödinger Equation becomes:
titititi
titititi
etetetiiedt
tdi
etetetiiedt
tdi
202122
212022
)(2
)(2
)(2
)(
)(2
)(2
)(2
)(
)(
)()(
tc
tct
2
2
0
0)( ti
ti
e
etR
)(
)(
22
22)(
)(
1
1
t
t
dt
tdi
dt
tdi
)(2
)(2
)(
)(2
)(2
)(
1
1
ttdt
tdi
ttdt
tdi
0 where
In the rotating frame the Hamiltonian is time independent!!
)(
)(
2)(
)(
1
1
t
t
t
t
dt
di
1
1
2
~,)(~~
)(~ HtHtdt
di
22
2
2
22
)(~)()(
)()()(
ttte
tctP
ti
xBzBB ˆˆ 10
cm
eB
cm
eB
ee
11
00 ,
01
10
2
H
tP2
sin)(21
202
21
20
21
Rabi Formula
When
Rabi Formula with
t
tP
2sin
2sin)(
21
202
21
20
21
21
22
21
2
21
1
1
2
~ H
21
20
tP
2sin)( 2
2
21
Rabi flopping equation
where
Generalized Rabi frequency
20
E
20
E
0 EEE
tP
2sin)( 12
When (resonance condition), Rabi frequency.cm
eB
e
11
Amplitude of the spin flip probability: Lorentzian curve 21
20
21
PMagnetic resonance curve
FWHM
emission absorption emission
emission
absorption
t
)(P
1
Time dependent spin flip probability:
1
2
1
3
tP
2sin)( 12