Product Operators: Connection to
Density Matrix Properties
• For small deviations it is convenient to work with a deviation density matrix, |

| = |

| - 1/Z| E |
(E is identity matrix)
• |

| is also just a collection of numbers; product operators are specific subsets of these numbers.
• Some subsets are associated with observables; ie.

12
,

21
,

34
,

43
, dictate Mx and My for A spin of an
AX system
• Some sets transform cleanly to other sets under rf pulses; ie.

11
,

22
->

12
,

21 under a 90°x pulse
(actually linear combinations of these)

Expressing
 in a basis set of matrices
• Rationale: collect elements with common evolution and transformation properties in each basis matrix
•

(t) =
 s b s
(t) B
S
. But with certain requirements: Hermitian;
 ij =
 ij*;
 ii real , # basis matrices = # elements, orthogonal
• Example for a single spin ½:
I
½
Y
E =
1
2


1
0
0
1


I
Z
=
1
2

 i
0

0 i


I
X
=
1
2


1
0
1
2


0
1
0

1


1
0


• Look just like matrix representation of corresponding operators – hence name product operators – but actually
 elements

Representation of some simple density matrices
• What about
 eq
?
 eq
 1
2



0

0



 
I
Z
• What about
 after a

/2 pulse on +X axis?

(

2
)
 1
2



0 i

0
 i


 
I
Y
• Note the simple conversion of one product operator (part of a density matrix) to another operator (part of a density matrix) under an rf pulse

Transformation for an arbitrary pulse on X
• exp(-i

1 t I
X
) I
Z exp(i
• Note: H’ = -

B
1
I
X t

1 t I
X
) = I
Z cos(

1 t) I
Y sin(

1 t)
• Hence, above

1 t for positive B
1 would be negative. Convention is not to specify sign by B
1 direction on axis but to do so by sign of the rotation that occurs. This makes product operator transformations look opposite of our Bloch equation description.
• A positive 90 degree rotation about X converts I
Z
– I
Y converts M
Z to M
Y
.
to
; a 90 degree rotation with field in +X direction

Product Operators for Two Spin Case
• q = 0 ½ E
• q = 1 I
1X
, I
1Y
, I
1Z
, I
2X
, I
2Y
, I
2Z
• q = 2 2 I
1X
I
2X
, 2 I
1X
I
2Y
, 2 I
1X
I
2Z
,
2 I
1Y
I
2X
, 2 I
1Y
I
2Y
, 2 I
1Y
I
2Z
,
2 I
1Z
I
2X
, 2 I
1Z
I
2Y
, 2 I
1Z
I
2Z
,
• Note: 16 operators (pieces of

)
16 elements in 2 spin (4X4) density matrix

Two-Spin Cartesian Product Operators
Howarth et al., JMR 68, 433-452 (1986)
½ E
½
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
I1z
½
1 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 -1
I2z
½
1 0 0 0
0 -1 0 0
0 0 1 0
0 0 0 -1
2 I1zI2z
½
1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 1
½
I1x
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
I1y
½
0 0 -i 0
0 0 0 -i i 0 0 0
0 i 0 0
2 I1xI2z
½
0 0 1 0
0 0 0 -1
1 0 0 0
0 -1 0 0
2 I1yI2z
½
0 0 -i 0
0 0 0 i i 0 0 0
0 -i 0 0

Two-Spin Cartesian Product Operators
(continued)
Note: for heteronuclear often use I and S instead of I1 and I2
I2x
½
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
I2y
½
0 -i 0 0 i 0 0 0
0 0 0 -i
0 0 i 0
2 I1zI2x
½
0 1 0 0
1 0 0 0
0 0 0 -1
0 0 -1 0
2 I1zI2y
½
0 -i 0 0 i 0 0 0
0 0 0 i
0 0 -i 0
½
2 I1xI2x
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
2I1yI2y
½
0 0 0 -1
0 0 1 0
0 1 0 0
-1 0 0 0
2 I1xI2y
½
0 0 0 -i
0 0 i 0
0 -i 0 0 i 0 0 0
2 I1yI2x
½
0 0 0 -i
0 0 -i 0
0 i 0 0 i 0 0 0

Physical interpretation of Product Operators
I1z + 12z is proportional to equilibrium population
(basis set order:

,

,

,

) :

11
=

 has excess
1 0 0 0 1 0 0 0
 
½ 0
0 0 -1 0 0 0 1 0
0 0 0 -1 0 0 0 -1
=

1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 -1
I1x is obvious; if
 is proportional to Ix , x magnetization exists.
Mx = Tr {|

|

(h/2

)

|Ix|} =

(h/2

)

/4 Tr{
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
} =

(h/2

)

/4 Tr
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
=

(h/2

)


What about 2 I1xI2z?
Note: we can generate 2 I1xI2z from I1x and I2z by multiplication
0 0 1 0
½
0 0 0 1
1 0 0 0
0 1 0 0
1 0 0 0 0 0 1 0
0 -1 0 0
0 0 1 0
0 0 0 -1
• ½ =
1 0 0 0
0 0 0 -1 0 -1 0 0
Result is much like I1x , but with reversal of some signs.
We can associate elements with particular lines;

=


This is transition of the first spin with second

.

24
13
=

 is a transition of the first with the second spin

: first doublet
| I1x | = | 2 I1xI2z | =

I
1X
I
2X
?
2
 1
2
 1
2


0

 0


1
0
0
0
0
1
1
0
0
0
0

1


0
0








0
1
0
0
1
0
0
0
0
0
0
1
0

0


1
0


 1
2

0

 0


0
1
0
0
1
0
0
1
0
0
1
0



0
0


2 I
1 Y
I
2 Y







0
0
0
1
0
0
1
0
0
1
0
0

0
0
0
1





Two Q, Zero Q
• Linear combination of 2 is pure Zero Q or Two Q
2 I
1X
I
2X
+ 2 I
1X
I
2X
= Two Q
• 2 I
1X
I
2y
, 2 I
1Y
I
2X
, are imaginary components

Transformation properties come from rotation operators
• X pulse by angle 
1 t =

I
Z
I
Y
I
X
-( I
X
)

I
Z
-( I
X
)

I
Y
-( I
X
)

I
X cos(

1 t) I
Y cos(

1 t) + I
Z sin(

1 t) sin(

1 t)
• Free precession:
I
I
X
X
-(

I
Z
)

I
-(2

J I
1Z
I
2Z
X
) cos(
 t) + I
Y

I
X sin(
 cos(

Jt) + I
1Y
I t)
2Z sin(

Jt)

Transformations Caused by Various Evolution Operators
Product Oper. I1x + I2x I1y +I2y I1z +I2z 2 I1zI2z
½ E
I1z
½ E
-I1y
½ E
I1x
½ E
I1z
½ E
I1z
I2z
2 I1zI2z
I1x
I1y
I2x
I2y
2 I1xI2z
2 I1yI2z
2 I1zI2x
2 I1zI2y
2 I1xI2x
2 I1yI2x
-I2y
(2 I1yI2y)
I1x
I1z
I2x
I2z
-2 I1xI2y
(-2 I1zI2y)
-2 I1yI2x
(-2 I1yI2z)
2 I1xI2x
2 I1zI2x
I2x
(2 I1xI2x)
-I1z
I1y
-I2z
I2y
(-2 I1zI2x)
2 I1yI2x
(-2 I1xI2z)
2 I1xI2y
(2 I1zI2z)
-2 I1yI2z
I2z
2 I1zI2z
I1y
-I1x
I2y
-I2x
2 I1yI2z
-2 I1xI2z
2 I1zI2y
-2 I1zI2x
---
---
2 I1xI2y 2 I1xI2z -2 I1zI2y --2 I1xI2y
2 I1yI2y (2 I1zI2z) 2 I1yI2y --2 I1yI2y
Evolution is to 2, 4( ), or more ---. Coefficient is sin of

,

Jt, or
 t.
I2z
2 I1zI2x
2 I1yI2z
-2 I1xI2z
2I1zI2x
-2 I1zI2x
I1y
-I1x
I2y
-I2x
2 I1xI2x
2 I1yI2x

Application of Product Operators:
2D, 2Q Spectrum
• Can we excite a 2Q coherence? Can we detect it?
• Consider: 90
X

180
Y

90
X with

= 1/(4J)
(removes chem shift)
• I
1Z
+ I
2Z
-( I
X
+I
X
, 90)

I
1Y
– I
2Y
-(J I
1Z
I
2Z
,

)

2 I
1X
I
2Z
+ 2 I
1Z
I
2X
-( I
X
+I
X
, 90)

-2 I
1X
I
2Y
- 2 I
1Y
I
2X
ZQ and 2Q evolution -- also gives -2 I
1X
I
2X etc
• Consider detection:
• 2 I
1X
I
2Y
90
X
-( I
X
+I
X
, 90)

-2 I
1X
I
2Z
+ others –> 1Q detect
• 2 I
1Y
I
2X
-( I
X
+I
X
, 90)

-2 I
1Z
I
2X
+ others –> 1Q detect

Example:

-Me-Galactose
OH
O
HO
4
3
OH
2
1
Me
OH