Product Operators.PDF

2 Product Operators

The product operator formalism is a complete and rigorous quantum

mechanical description of NMR experiments; the formalism is a version of

density matrix theory and is well suited to calculating the outcome of

modern multiple-pulse experiments.

One particularly appealing feature is the fact that the operators have a

clear physical meaning and that the effects of pulses and delays can be

thought of as geometrical rotations. To emphasise this connection the

discussion will start with a brief summary of the vector model.

spin down

spin up

9HFWRUPRGHORI105

Unequal populations of the two

energy levels give rise to a net

magnetization, represented as

a vector along the z-axis.

The vector model is a complete description of the behaviour of an ensemble

(a macroscopic sample) of non-interacting spin-half nuclei. Each spin has

two energy levels and at equilibrium the lower of these is more populated.

The result is a net magnetization of the sample along the direction of the

applied magnetic field (taken to be the z -direction). The vector model

focuses entirely on the behaviour of this magnetization, which can be

represented as a vector.

Radiofrequency pulses are represented as rotations about the x - or y -axes;

if the radiofrequency field strength is

1 (rad s –1 ) then a pulse applied for a

time t causes a rotation through an angle

, where

1 t . For example a

90° pulse about the x -axis has

1 t =

/2 and rotates magnetization from the

z -axis onto the – y -axis.

Free precession is represented as a rotation about the z -axis at frequency

is the offset (that is the difference between the Larmor

frequency and the transmitter frequency). Free precession for a time t causes

a rotation through an angle

(rad s –1 ), where

t .

Only x - and y -magnetization are directly observable in an NMR

experiment; it is the precession of the magnetization in the xy -plane which

gives rise to the free induction signal.

, where

A pulse about the x-axis rotates

the magnetization through an

angle

in the yz-plane. The

picture shows the view down

the x-axis.

2.1.1 Example – the conventional pulse-acquire experiment

Assume that the system starts at equilibrium; a pulse of flip angle a is

applied and then the free induction signal is recorded. Let the equilibrium

magnetization (aligned along the z -axis) have size M 0 . After the pulse the z -

and y -magnetization ( M z and M y , respectively) are

M z = cos

M 0 M y = - sin

M 0

Free precession, which is a rotation about the z -axis, has no effect on the z -

component. The y -component rotates in the xy -plane giving the following

transverse components after time t

2–1

M y ( t ) = -sin

cos

t M 0 M x ( t ) = sin

sin

t M 0

It is these transverse (that is, x and y ) components of the magnetization that

are detected in NMR experiments. It is seen that these are oscillating at

frequency W , and that their overall size depends on the sine of the flip angle

i.e. they are a maximum for a 90° pulse.

2.1.2 Example – the spin echo

()

delay

180

()

delay

acquire

y-component

x-component

After the delay, point b , the vector can be resolved into y - and x -components

as shown in c . The 180° pulse about the x -axis has no effect on the x -

component of the magnetization; in contrast the y -component is rotated by

180° in the yz -plane, ending up along the opposite axis. The individual

components after the 180° pulse are shown in d , and corresponding vector is

shown in e . The effect of the 180° pulse about the x -axis is to reflect the

vector in the xz -plane. During the second time

the vector precesses in the

same direction as it did during the first time

and through the same angle,

ending up along the y -axis.

At the end of the sequence the vector always ends up along the y -axis,

regardless of the time

and the offset; the sequence is said to "refocus the

offset (or shift)".

2SHUDWRUVIRURQHVSLQ

2.2.1 Operators

Operators are mathematical functions which arise in quantum mechanics

(see lecture 1); as their name suggest, they operate on functions. In quantum

mechanics operators represent observable quantities, such as energy, angular

momentum and magnetization.

For a single spin-half, the x - y - and z -components of the magnetization

are represented by the spin angular momentum operators I x , I y and I z

respectively. Thus at any time the state of the spin system, in quantum

mechanics the density operator,

, can be represented as a sum of different

amounts of these three operators

2–2

() ()

t atI btI ctI

()

The amounts of the three operators will vary with time during pulses and

delays. This expression of the density operator as a combination of the spin

angular momentum operators is exactly analogous to specifying the three

components of a magnetization vector.

At equilibrium the density operator is proportional to I z (there is only z -

magnetization present). The constant of proportionality is usually

unimportant, so it is usual to write

eq = I z .

2.2.2 Hamiltonians for pulses and delays

In order to work out how the density operator varies with time we need to

know the Hamiltonian (which is also an operator) which is acting during

that time.

The free precession Hamiltonian ( i.e. that for a delay), H free , is

H free =

I z

about

the z -axis; in the quantum mechanical picture the Hamiltonian involves the

z -angular momentum operator, I z – there is a direct correspondence.

The Hamiltonian for a pulse about the x -axis, H pulse , is

H pulse,x =

1 I x

and for a pulse about the y -axis it is

H pulse,y =

1 I y

Again there is a clear connection to the vector model where pulses result in

rotations about the x - or y -axes.

2.2.3 Equation of motion

The density operator at time t , s( t ), is computed from that at time 0, s(0) ,

using the following relationship

()

exp

(

iHt

) ( ) ( )

exp

iHt

are expressed in terms of

the angular momentum operators if turns out that this equation can be solved

easily with the aid of a few rules.

Suppose that an x -pulse, of duration t p , is applied to equilibrium

2–3

In the vector model free precession involves a rotation at frequency

where H is the relevant hamiltonian. If H and

magnetization. In this situation H =

w 1 I x and

(0) = I z ; the equation to be

solved is

() (

exp

i t I I

) ( )

exp

i t I

Such equations involving angular momentum operators are common in

quantum mechanics and the solution to them are already all know. The

identity required here to solve this equation is

exp

(

iI I

)

exp

( )

cos

sin

[2.1]

This is interpreted as a rotation of I z by an angle

about the x -axis. By

putting

1 t p this identity can be used to solve Eqn. [2.1]

()

cos

t I

sin

t I

The result is exactly as expected from the vector model: a pulse about the x -

axis rotates z -magnetization towards the – y -axis, with a sinusoidal

dependence on the flip angle,

2.2.4 Standard rotations

Given that there are only three operators, there are a limited number of

identities of the type of Eqn. [2.1]. They all have the same form

exp

(

{old operator}

)

exp

( )

cos

sin

{new operator}

where {old operator}, {new operator} and I a are determined from the three

possible angular momentum operators according to the following diagrams;

the label in the centre indicates which axis the rotation is about

III

-y

-x

-z

Angle of rotation =

t for offsets and

w 1 t p for pulses

First example: find the result of rotating the operator I y by q about the x -

axis, that is

exp

(

iI I

)

exp

( )

2–4

For rotations about x the middle diagram II is required. The diagram shows

that I y (the "old operator") is rotated to I z (the "new operator"). The required

identity is therefore

exp(- i

I x ) I y exp( i

I x )

cos

I y + sin

I z

Second example: find the result of

exp(- i

I y ) {- I z } exp( i

I y )

This is a rotation about y , so diagram III is required. The diagram shows

that – I z (the "old operator") is rotated to – I x (the "new operator"). The

required identity is therefore

exp

( ) {} ( )

exp

cos

{} {}

sin –

º-

cos

sin

Finally, note that a rotation of an operator about its own axis has no effect

e.g. a rotation of I x about x leaves I x unaltered.

2.2.5 Shorthand notation

To save writing, the arrow notation is often used. In this, the term Ht is

written over an arrow which connects the old and new density operators.

So, for example, the following

() (

exp

i t I

) ()

exp

(

i t I

)

is written

()

¾®

¾¾

1 tI x

()

For the case where

(0) = I z

¾®

¾¾

cos

t I

–sin

t I

2–5

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