Matrix representation of Spin Operator

2

2

2

cos sin

cos sin

cos 2 sin

cos sin

cos 2 sin

2 2 cos

kX

Z

kl kZ lZ

Z

kl kZ lZ

kl kZ lZ

IkZ kZ kY

tIkX kX k kY k

J I IkX kX kY lZ

tIkY kY k kX k

J I IkY kY kX lZ

J I IkX lZ kX lZ

I I I

I I t I t

I I Jt I I Jt

I I t I t

I I Jt I I Jt

I I I I

2

sin

2 2 cos sinkl kZ lZ

kY

J I IkY lZ kY lZ kX

Jt I Jt

I I I I Jt I Jt

J.IkzIlz

2IkyIlz

Ikx

x x

t1t2

Correlation Spectroscopy (COSY)

Considering two spin k and lˆ

2

ˆ

1 1

1

1

1

First under chemical shift hamiltonian during

considering evolution of only

cos sin

co

t period

under coupling hamiltonian during t period

s

X

Z

H

kZ lZ kY lY

kY

HkY kY k kX k

kY k kX

I I I I

I

I I

I

J

t I t

t I

ˆ

1 1 1 1

1 1 1

1 1 1

sin ( cos 2 sin )cos

(

puls

cos 2 sin )sin

( cos 2 sin )cos

( c s

e2

o

JHk kY kl kX lZ kl k

kx kl kY lZ kl k

kZ kl kX lY kl k

kX kl

t I J t I I J t t

I J t I I J t t

I J t I I J t

x

t

I J

1 1 12 sin )sinkZ lY kl kt I I J t t

1 1 1

1 1

1

1 1 1

1

2

( cos 2 sin )cos

( cos 2 sin )sin

cos sin 2 s

in sin

kz kl kx ly kl k

kx kl kz ly kl k

kx

evolution under chemical shift hamiltonian

kl k kz ly kl

during t perio

k

d

Obser

I J t I I J t t

I J t I I J t t

I J t t I I J t t

vable term is

2 2 1

1

2 2 2

2

1

1

( cos sin )cos sin

cos 2 sin cos sin cos

cos

for the

evolution under J coupling hamiltonian dur

kx k ky k kl k

kx kl ky lz kl k k

first ter

ing t period for t

m

first termh

kl

ky kl

e

I t I t J t t

I J t I I J t t t J t

I J

2 2 2 1 1

2 2 1 1

2 sin sin sin cos

cos cos sin cos

kx lz kl k k kl

kx ky kl k k kl

t I I J t t t J t

I I J t t t J t

evolution under chemical shift hamiltonian during t perio second d for the 2

evolution under couplin

1 1

2

g hamiltonia

ter

2

m

1 1

2 sin sin

2 cos sin si

for the second term

n s

in

kZ lY kl k

kZ lY l lX l kl k

J

I I J t t

I I t I t J t t

2 2 2

1 1

2 2

secn during t ond term

1

period for the 2

cos 2 sin cos2 sin sin

cos 2 sin sin

lY kl lX kZ kl l

kZ kl k

lX kl lY kZ kl l

I J t I I J t tI J t t

I J t I I J t t

Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)

ϕ1= x, -x, x, –x

ϕ2= x, x, -x, –x

ϕR= x, -x, -x, x

1 (2 )ISJ

yyt2

ϕ1y

y

yϕ2

y

2

2

2

2

1

2

t 1

2

t

I

S

ϕR

A B C D E F

-y


I

S

90

At AXI

Z YI I

2 1 2co

At B

s 2 sin 2IS Z Z ISJ I S JY Y IS X Z IS X ZI I J I S J I S

y

y

2

2

A B


y

y

2

2

I

S A B

y

ϕ1

y

ϕ21

2

t 1

2

t

C D E

90 902 2

At

2

CY XI S

X Z Z Z Z YI S I S I S

-y

ϕ1= x, -x, x, –x

1( )1 1180

2 2 cos 2 sin

At Ds Z

X

t SZ Y Z Y S Z X SI

I S I S t I S t

90 ( )1 1 1 12 cos 2

At E

sin 2 cos 2 sinY XI SZ Y S Z X S X Z S X X SI S t I S t I S t I S t

2

At B

X ZI S

ϕ2= x, x, -x, –x


yy

ϕ1y

y

ϕ2

2

2

1

2

t 1

2

t

I

S A B C D E

-y

y

y

2

2

F

1 12 cos 2 si

At

n

E

X Z S X X SI S t I S t

21 1 1

1

2 cos 2 sin 2 cos sin cos

A

2 sin

t FIS Z ZI I S

X Z S X X S X Z IS Y IS S

X X S

I S t I S t I S J I J t

I S t

11 2 si

If 2

n

1

cosY S X X

I

S

S

I I St t

J


yy

ϕ1y

y

ϕ2

2

2

1

2

t 1

2

t

I

S A B C D E

-y

y

y

2

2

F

,

1

1

1

1 2 sincos

cos

;

2 s

F

in

is only observable

is multiple quantum term thus unobserva

X XY S

bl

Y

X

S

X S e

S

whil

I S tI t

I t

At

I S te

t2

ϕR

2( )1 2 2 1

2

cos cos sin cos

ZSt I

Y S Y I X I S

A

I t I

fter t term at the re

t I

ei

t

c ve

t

r

ϕ2= x, x, -x, –xϕ1= x, -x, x, –x ϕR= x, -x, -x, x


290 180 180 180 ( )2cosX Y Y Y I ZI I I I t I

Z Y Y Y Y Y II I I I I I t

yyt2

ϕ1y

y

yϕ2

y

2

2

2

2

1

2

t 1

2

t

I

S

ϕR

A B C D E F

-y

For protons NOT coupled to S spin

We need two step phase cycle to get rid of this magnetization ϕ1= x, -x ϕR= x, -x

Steps ϕ1 ϕR

Magnetization at point F

Protons coupled to S spin Protons NOT coupled to S spin

Step I x x

Step II -x -x

cos 2 sin1 1I t I S tY S X X S

( cos 2 sin )1 1I t I S tY S X X S

IYIY

Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)yy

t2

ϕ1y

y

yϕ2

y

2

2

2

2

1

2

t 1

2

t

I

S

ϕR

A B C D E F

-y

For complete removal of multiple quantum term we need four step phase cycle to get rid of this magnetization

ϕ1= x, -x, x, -x ϕ1= x, x, -x, -x ϕR= x, -x, -x, x

Steps ϕ1 ϕ2 ϕR

Magnetization at point F

Protons coupled to S spin Protons NOT coupled to S spin

Step I x x x

Step II -x x -x

Step III x -x -x

Step IV -x -x x





IYIYIYIY

Sensitive enhanced Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)

yyt2

ϕ1y

y

yϕ2

y

2

2

2

1

2

t 1

2

t

I

S

ϕR

A

-y

,

1

1

1

1 2 sincos

cos

;

2 s

A

in

is only observable

is multiple quantum term thus unobserva

X XY S

bl

Y

X

S

X S e

S

whil

I S tI t

I t

At

I S te

ϕ3= y, -y, y, –y

2

ϕ3

-y

y

y

2

y

2

SE- Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)

yyt2

ϕ1y

y

yϕ2

y

2

2

2

1

2

t 1

2

t

I

S

ϕR

-y

90 ( , )1 1 1 1cos 2 sin cos 2 in

B

sX XI SY S X X S X S X Z SI t I S t I t I S

At

t

DA B C

2

ϕ3

-y

y

y

2

y

2

ϕ3= y, -y, y, –y

21 1 1 11 2

cos 2 sin cos si

C

nIS Z Z

IS

J I SX S X Z S Z S Y SJ

I t I S t I

At

t I t

901 1 1 1cos sin cos sin

DYI

Z S Y S X S Y SI t I t I t I t

At

SE- Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)

yyt2

ϕ1y

y

yϕ2

y

2

2

2

1

2

t 1

2

t

I

S

ϕR

D

-y

A B C

2

ϕ3

-y

y

y

2

y

2

ϕ3= y, -y, y, –y

1 1 1

1 1 1

exp

cos sin

cos sin

X S n Y S

X S n Y S

For the first eriment

For the second

I t I t

I t

experi

I

n

t

me t

Matrix representation of Spin Operator

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Transcript of Matrix representation of Spin Operator