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Lecture #2: NMR Operators, Populations, and Coherences

•  Topics–  Quantum description of NMR–  The Density Operator –  NMR in Liouville Space –  Populations and coherences

•  Handouts and Reading assignments–  Kowalewski, Chapter 1–  van de Ven, appendices A-C

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Why Quantum Mechanics for NMR?•  Classical picture of NMR (i.e. Bloch equations) is valid for

collections of non-interacting spins.•  Individual spins don’t have an associated T1 or T2.•  NMR relaxation is described statistically and is driven by

interactions among spins, e.g.–  Nuclear dipolar and J couplings–  Chemical exchange–  Electron-nuclear interactions: shielding, dipolar coupling, J coupling

•  Such interactions are best described using semi-classical or fully quantum mechanical descriptions.

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Wave Functions

- contains all info possible to obtain about the particle

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dΡ(! r ,t)∫ =1

- wave functions typically normalized, i.e.

€

ψ(! r ,t) 2∫ d3! r =1

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1C

= ψ(! r ,t) 2∫ d3! r <<∞square-integrable!

- interpreted as a probability amplitude of the particle’s presence with the probability density given by:

€

dΡ(! r ,t) = Cψ(! r ,t) 2 d3! r , C constant.

•  For the classical concept of a trajectory (succession in time of the state of a classical particle), we substitute the concept of the quantum state of a particle characterized by a wave function, .

€

ψ(! r ,t)

• 

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ψ(! r ,t)

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QM Math: Kets and Bras•  Important to be familiar with the concept of treating functions as

elements in a vector space.•  Example: consider f(n) for 1 ≤ n ≤ N, n integer.

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! f =

f1f2"

fN

"

#

$ $ $

%

&

' ' ' f(n) ...f1

f2

f3

f4fN

n

•  Each wave function is associated with an element of a linear vector space F and is denoted as (called a “ket”).

€

ψ

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ψ(! r )

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ψ(! r )⇔ ψ ∈ F

•  Elements of the dual space of F are denoted as (called a “bra”).

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χ

Dirac notation

•  A linear vector space with a defined metric, in this case the scalar product , is called a Hilbert Space.χ ψ

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QM Math: Bases and Operatorsψ = ci fi

i=1

n

∑•  We can express as a linear combination of other kets:ψ

•  If it follows that: implies

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ci f ii=1

N

∑ = 0 ci = 0 for all iare called linearly independent.

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fi

, then the kets

•  In an N-dimensional vector space, N linearly independent kets constitute a basis, and every element in F can be written as

€

ψ

where ψi is the coefficient of the ket .

€

ψ = ψ i f ii=1

n

∑

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fi

•  We typically deal with orthonormal basis: fif j =

1 if i = j0 if i ≠ j

"#$

%$

•  An operator, , generates one ket from another:

€

ˆ O

€

ˆ O ψ = ξ

Example of a linear operator:

€

ˆ O λψ + µ ξ( ) = λ ˆ O ψ + µ ˆ O ξscalars

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QM Math: Vectors and Matrices•  When a ket, , is expressed as a linear combination of basis kets,

, then a corresponding column vector, , can be constructed.

€

ψ

€

fi

€

! ψ

€

ψ = ψ i f ii=1

N

∑

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! ψ =

ψ1ψ2

"ψn

#

$

% % % %

&

'

( ( ( (

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! ψ † = ψ1

∗,ψ2∗,…,ψn

∗( )

€

ψ

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x = ˆ O y•  Similarly, operators have matrix representations.

Remember: vector and matrix representations

depends on the basis set!

€

! x = O! y

€

Oij = f iˆ O f jwhere is a n x n matrix with elements:

€

O

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! ψ =

x0y0z0

#

$

% % %

&

'

( ( (

€

x , y , z{ }basis:

ψ

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x

€

y€

z

x0y0

z0

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! ψ =

0x02 + y0

2

z0

#

$

% % %

&

'

( ( (

€

" x , " y , z{ }

€

" x

€

" y

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QM Math: Eigenkets and Eigenvalues

then is called an eigenket of with eigenvalue λ.

€

ψ

€

ˆ O

•  Since the trace of a matrix is invariant under a change of basis, we can also talk about the trace of the corresponding operator.

•  The trace of a matrix is defined at the sum of the diagonal elements:

€

Tr O( ) = Oiii∑

€

Tr ˆ O ( ) = λii∑ where λi are the eigenvalues of

€

ˆ O

€

ˆ O ψ = λψ

scalar

•  If

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QM Math: Liouville Space•  Operators defined on an n-dimensional Hilbert space, are

themselves elements of an n2-dimensional vector space known as Liouville space.

•  The trace is the metric in Liouville space: , and in contrast to Hilbert space, the product of two Liouville Space elements is defined.

ˆ A | ˆ B ( ) = Tr ˆ A † ˆ B ( )

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ˆ A , ˆ B ∈ L In general, AB ≠ BA.

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ˆ A B = ˆ C ∈L

a Liouville space

, and we will deal with just one superoperator: the

•  Operators that work on elements of a Liouville space are called superoperators (denoted with a double hat).

€

ˆ ˆ A ˆ O 1 = ˆ O 2

€

ˆ ˆ A B ≡ ˆ A , ˆ B [ ] = ˆ A B − ˆ B A commutator: Do superoperators have associated eigenoperators?

What about supermatrices?

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Summary of Vector Spaces

ˆ ˆ S i! " #

$ % & ,i = 1,..., n4

S Superoperator Algebra

ψ i{ },i = 1,..., nH Hilbert Space

ˆ O i{ },i = 1,..., n2

L Liouville Space(operator algebra)

NMR can be described in Hilbert space ...

…or in Liouville space.

NMR is easiest to understand in

Liouville space!

Key Concept

See R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, 1990.

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Postulates of QM•  The state of a physical system is defined by a ket that

belongs to a linear vector space known as a Hilbert Space.ψ(t)

•  Every measurable quantity A is described by an Hermitian operator whose eigenkets form a basis in state space.ˆ A

•  The only possible result of the measurement of A is an eigenvalue of the , the probability of obtaining eigenvalue an is

where is the associated eigenket, and the state of the system immediately after the measurement is

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P an( ) = un ψ2

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unψ = un .

ˆ A

•  The time evolution of the is governed by the Schrödinger equation:

€

∂∂tψ(t) = −i ˆ H (t)ψ(t)

where , known as the Hamiltonian, is the operator for the observable H(t) associated with the total energy of the system.

ˆ H (t)

ψ(t)

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Spin, Angular Momentum, and Magnetic Moment

•  Spin, angular momentum, and magnetic moment operators are linearly related.

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ˆ µ p = γ ˆ L p = γ!ˆ I p , p = x,y,z{ }magnetic moment angular momentum spin

€

Iz, =121 00 −1#

$ %

&

' (

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Ix, =120 11 0"

# $

%

& '

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Iy, =120 −ii 0#

$ %

&

' (

Iz + = +12+ and Iz − = −

12−•  For a spin ½ particle:

•  Matrix representation in basis:+ , −{ }eigenkets of

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ˆ I z

The Pauli matrices

with the following commutators:

€

ˆ I x, ˆ I y[ ] = iˆ I z

€

ˆ I y, ˆ I z[ ] = iˆ I x

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ˆ I z, ˆ I x[ ] = iˆ I y

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Isolated Spin in a Magnetic Field

•  Procedure:

€

ψ .–  Solve for

–  Find ˆ H (t).

•  Goal: Find the appropriate wavefunction that describes a system consisting of a nucleus (spin = ½) in a uniform magnetic field.

€

ψ(t)

–  Compute quantities of interest: e.g. components of magnetic moment µx , µy , and µz .

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∂∂tψ(t) = −i ˆ H (t)ψ(t)Schrödinger’s Equation:–  Given:

Previously showed these correspond to the familiar quantities Mx, My, and Mz.

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A Collection of Spins

•  To make matters worse, we rarely, if ever, know precisely, e.g. cn may be different for each spin. At best, we have only a statistical model for the state of the system.

€

ψ

•  In a typical experiment, the number of nuclear spins, N, can be very big, e.g. 1024, and the complete quantum state is described by the wavefunction (or state vector):

ψ = cn ψnn=1

N

∑

Rather unwieldly for large N!

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Density Operator for a Pure State•  Consider a spin system in a pure state with normalized state vector

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ψ t( ) = cii∑ t( ) ui where form an orthonormal basis.

€

ui{ }

- Expectation of observable :ˆ A

- Time evolution:

•  We previously showed that we could define an operator:σψ (t) = ψ(t) ψ(t)

A (t) = uji, j∑ σψ (t) ui ui A uj = Tr σψ (t)A{ }

for which

∂∂tσψ (t) =

∂∂t

ψ(t) ψ(t)!" #$

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= −i ˆ H ,ψ(t) ψ(t)[ ]= −i ˆHσψ

•  While there is no “average state vector”, it turns out there is an “average operator” known as the density operator.

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Density Operator: Statistical Mixture

€

ui ui

… in general: P ai( ) = pn

n∑ Pn ai( )= pn

n∑ Tr σ nPi{ }

= Tr pnσ nPin∑"#$

%&'= Tr σ Pi{ }

where is, by definition, the density operator for the system.σ = pnσ nn∑

•  It is easy to show that: A = Tr σ A{ } and ∂∂tσ = −i ˆHσ

Liouville-von Neumann equationensemble average

ˆ A •  Consider a system consisting of a statistical mixture of states

with associated probabilities pn, and ai be an eigenvalue of with associated eigenket

ψn

ui .

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System of independent spin= ½ nuclei•  Most convenient basis set is the eigenkets of

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ˆ H 0 : + , −{ }

€

"

# $

%

& '

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−

€

+

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−€

+

L - “longitudinal magnetization”

I - “transverse magnetization”LI

IL

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σ =c+

2 c+c−*

c+*c− c−

2

$

%

& &

'

(

) )

•  for a one-spin system can be expressed as:

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σ = L11 00 0#

$ %

&

' ( + I1

0 10 0#

$ %

&

' ( + I2

0 01 0#

$ %

&

' ( + L2

0 00 1#

$ %

&

' (

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σ

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ˆ σ = L1ˆ T 11 + I1

ˆ T 12 + I2ˆ T 21 + L2

ˆ T 22In operator form:

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ˆ σ

€

L1I1I2L2

"

#

$ $ $ $

%

&

' ' ' '

“vector” in a 4D Liouville space called

“coherence space”

•  or …

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σ = a11 00 1#

$ %

&

' ( + a2

0 12

12 0

#

$ %

&

' ( + a3

0 − i2

i2 0

#

$ %

&

' ( + a4

12 00 − 12

#

$ %

&

' (

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ˆ σ = a1ˆ E + a2

ˆ I x + a3ˆ I y + a4

ˆ I z

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ˆ E , ˆ I x, ˆ I y, ˆ I z{ }“product operator” basis set

ˆ I zˆ I y

ˆ I x

ˆ σ A 3D subspace of coherence space

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Coupled Two-Spin System•  Eigenkets of H0 : ++ , +− , −+ , −−{ }

Diagonal elements called “populations”

Off-diagonal elements called “coherences”

€

σP1 C1,2 C1,3 C1,4C2,1 P2 C2,3 C2,4C3,1 C3,2 P3 C3,4C4,1 C4,2 C4,3 P4

!

"

###

$

%

&&&

++ +− −+ −−

++

+−

−+

−−

•  As with the single-spin case, the product operators form a convenient orthonormal basis set (16 in total).

€

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ˆ E , ˆ I x, ˆ I y, ˆ I z, ˆ S x, ˆ S y, ˆ S z

€

2ˆ I x ˆ S x, 2ˆ I y ˆ S y, 2ˆ I x ˆ S y, 2ˆ I y ˆ S x

€

2ˆ I z ˆ S z

€

2ˆ I x ˆ S z, 2ˆ I y ˆ S z, 2ˆ I z ˆ S x, 2ˆ I z ˆ S y

In-phase single quantum coherences

Longitudinal two-spin order

Linear combinations of double and zero quantum coherences

Anti-phase single quantum coherences

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Phase Coherence“Coherent superposition of quantum states”

Arrows represents spin polarization axis

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ˆ I x

No net tendency for S spins in any directionNet tendency for I spins to be +x

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ˆ S z IS

Net tendency for S spins to be +zNo net tendency for I spins in any direction

Product OperatorsPictorial Examples (note, we are now considering pairs of spins):

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2ˆ I z ˆ S z

No net tendency for I or S spins to be ±zIf I or S is ±z, increased probability paired spin is ±z

? ?

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Observations about •  Let where are an orthonormal set of Hermitian

basis operators, i.e. ˆ σ = bn

n∑ ˆ B n ˆ B n

€

= bmm∑ Tr ˆ B m ˆ B n( )

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= bn

€

∴ ˆ σ = ˆ B nn∑ ˆ B n

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= Tr ˆ σ B n( )

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ˆ B n

Coefficients of the expansion are the ensemble averages of the expected values of the respective

operators!

€

ˆ B n | ˆ B m( )

€

= Tr ˆ B n† ˆ B m( )

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= Tr ˆ B n ˆ B m( )

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= δnm .

Mx = γ!Tr σ I x{ }= γ! I x

Signals observable with Rf coils:

Bn ∈ {12 E, I x, Sx, I y, Sy,…, 2 IzSz}•  Example: , the 16 two-spin product operators

Some signals not directly observable with Rf coils:

My = γ!Tr σ I y{ }= γ! I y

Mz = γ! Iz Cxz = γ! 2 I xSz“antiphase x”

Czz = γ! 2 IzSz“longitudinal two-spin order”

σ

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Liouville-von Neumann Equation•  Time evolution of :

€

ˆ σ (t)

€

∂∂t

ˆ σ = −i[ ˆ H , ˆ σ ]

- operator corresponding to energy of the system•  Hamiltonian ˆ H

€

= −i ˆ ˆ H ˆ σ

€

ˆ I z

€

ˆ I x

€

ˆ I yNo longer the case that just undergoes simple rotations in operator space.

σ t( )

σ t( )

σ 0( )

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Thermal Equilibrium

•  Using the Boltzmann distribution one can verify:σ B =

1Z e

−!H0 /kT Z = Tr e−!H0 /kT( ).where

•  Using the high temperature approximation:

σ B ≈12 E −

!kT H0( ) (spin 1/2 particles)

remember H defined as E / !

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σ =P1 C1,2 ! C1,nC2,1 P2 " ## " " Cn−1,n

Cn,1 ! Cn,n−1 Pn

$

%

& &

'

(

) )

Diagonal elements called “populations”

Off-diagonal elements called “coherences” (= 0 at thermal equilibrium).

•  The spin density matrix expressed in the eigenkets of the Hamiltonian has the following form:H0

23*I lied in Rad226a, as what I called the “Master Equation” was one that only applied to branch diagrams.

The Master Equation•  As always, finding the Hamiltonian is the key step.

H = H0 + H1 t( )

•  Then look for an equation of the form:

A little foreshadowing…

•  Our ultimate strategy will be to write the Hamiltonian as the sum of a large static component plus a small time-varying perturbation.

This equation is known as the Master Equation* of NMR.

∂∂tσ = −i ˆH0σ − ˆΓ σ − σ B( )

Relaxation superoperator

Rad226a/BioE326a Rad226b/BioE326b

Note the similarity to Bloch’s equations:

€

d!

M dt

= γ!

M × B0 ˆ z −Mx ˆ x + My ˆ y

T2

−Mz − M0( )ˆ z

T1Rotations Relaxation terms

Rotations Relaxation

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Next lecture: The Nuclear Spin Hamiltonian