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Home , Spin (physics)

NMR and Imaging

Department of , FU Berlin

Lecture 5:

Basic Quantum

1 Leibniz-Institut für Molekulare Pharmakologie Lecture 5

1. Some Math Tools

2. QM Concepts

3. Nuclear Spin Hamiltonian

4. Uncoupled Spin-½ Nuclei

2 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools function properties, Dirac notation

functions important to spin are normalized

and orthogonal

which is described for a whole set of (eigen)functions by

short Dirac notation for dealing later with state products etc.

orthonormality

adjoint states

3 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools operators, eigenstates

matrix representation of operators

an returns the same function  this function is an eigenfunction

trace (requires finite )

the commutator is defined to describe the effect of changing the order of two consecutive operators

the complex exponential of an operator is defined as

4 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools adjoint operators

two operators are called adjoint if their matrix elements are related through

and therefore also

an operator that is self-adjoint is called Hermitian

if the adjoint is the inverse of the same operator, it is called unitary

5 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools vector representation of functions

suppose a function is given as linear combination of other functions

this is written as

with the coefficients

6 Leibniz-Institut für Molekulare Pharmakologie 5.1 Some Math Tools eigenbasis of commuting operators

suppose we have eigenvectors and all the eigenvalues are non-degenerate

and for another operator we know

then the eigenbasis of A is also an eigenbasis for B

or: A and B are both diagonalized using the eigenbasis of A

7 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts the eigenstate basis

a system is not necessarily IN one of the eigenstates but in any superposition (provided the eigenstates are normalized)

however, the eigenstates play a special role when dealing with time-independent Hamiltonian

the solution is

and if the system was at the beginning in an eigenstate of the Hamiltonian, i. e.

it will also be an eigenstate at time t, in other words: it is in a

8 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts angular

define the operators as

with etc.

cummutation properties

and eigenstates characterized by the quantum numbers

9 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts angular momentum

an operator commuting with lz is l² with

10 Leibniz-Institut für Molekulare Pharmakologie 5.2 QM Concepts shift operators

define two additional operators

these affect the eigenstates as follows

then we also get

11 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian spin hypothesis

working with the full version of the describing a system, one realizes that this is quite useless  transition to a simplified version

this includes only the terms depending on nuclear spin states and works with an averaged picture of electric and magnetic terms of the

this simplification, the spin Hamiltonian hypothesis, is justified because the electrons move fast on the time scale of nuclear and because the nuclear spin energies are only small compared to spin energies

the Hamiltonian splits into electric and magnetic interactions

12 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian magnetic interactions

suppose we apply an external of inductivity B

then the interaction energy of a with this field is

and the relation between the magnetic moment and the angular is

13 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian interaction magnitudes

the interactions governing the spin Hamiltonian can be of internal or external

for spin ½, we only deal with magnetic interactions

it should be emphasized that under most conditions the of the spin system to the external field is stronger than the internal interactions

this is the opposite situation to other spectroscopy techniques where the system is probed by weak perturbations and governed by internal interactions

14 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian types of external magnetic fields

the total external field contains contributions from the static magnetic field, the RF field to excite the spin system and possibly a gradient field that allows further encoding

the dominant operator will be the one describing the static field

let us assume

then only the z component operator from the magnetic moment remains

15 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian spin operator – a closer look

particular form of angular momentum

with eigenstates for the z- component operator

and shift operators

note: the Zeeman eigenstates can be used to represent any state of the system but contrary to ‘ordinary’ angular momentum there is no visualization possible like the illustrating the electron orbitals

however, one can easily manipulate these states in real physics experiments

16 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

Zeeman eigenstates

spins I represented by a 2I +1 finite basis, i.e. for I = ½

with the following behavior

representation of the z-operator in matrix form is straight forward

for the x- and y-operator, we have a closer look at the shift operators, where we had for their most general form for any angular momentum

17 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

Zeeman eigenstates

then we get for the raising operator

and the other matrix entries vanish because of the orthogonality

and similar for the other operator

hence, we get for the transverse operators in the basis of the Zeeman eigenstates

18 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

RF excitation operators

in its most general form, the RF

field is applied off axis from B0

Q: why do we need RF > 0?

this field acts only for some precisely defined pulse duration

19 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

RF excitation operators

remember decomposing the oscillating transverse component

then we get three components

for the transverse

20 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

RF excitation operators

remember decomposing the oscillating transverse component

then we get three components

and the longitudinal

21 Leibniz-Institut für Molekulare Pharmakologie 5.3 Spin Hamiltonian

RF excitation operators

the longitudinal and off-resonant-components can be safely neglected, so we get the approximation

with the nutation frequency

the RF coil works most efficient

for RF = 90°

since this is a time-dependent operator, we will come back to it later

22 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

Zeeman energy levels

the dominating energy term is given by the external field experienced by the nucleus, i.e. including the correction for the

the energy levels of the Zeeman eigenstates then read

the energy difference is described by the Larmor frequency

23 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins superposition states

the nucleus is NOT restricted to the two eigenstates; it can be in any superposition

with the normalized coefficients

or in vector notation

with the eigenvectors

note: the bra state is a row vector

illustrating the normalization

24 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins some special states

for the following superpostion

we try to measure the x component

here, the x component is sharp whereas y and z would show random results

25 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins some special states

and similar for

where we try measure the y component

this time, the y component is sharp whereas x and z show random results

if B is along z, |x> and |-y> do not have sharp energies and cannot be represented in diagrams

26 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins geometric representations

although we can use the energy levels only for the eigenstates of the operator belonging to the quantization axis, a geometric illustration is possible for these special superpositions

again: |> and |> are still orthonormal!

27 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins arbitrary superposition

also the superposition

follows an eigenequation:

28 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins phase factors

suppose a state is given by multiplying another with just a phase factor

like

is related to

Q: what is the phase factor?

states related through a phase factor obey the same eigenequation

note: the phase cannot be depicted in arrow diagrams -|> is |> with different phase but not |>!

29 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins state evolution

we know that we measure a , so we expect a time-dependent spin state that obeys the Schrödinger equation

assuming only free evolution, i.e. no rf field

this first order differential equation is solved by

with the exponential operator representing a

note: the Hamiltonian is time-independent in this case

30 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins rotation operators

exponentials of the spin operators are called rotation operators – but what do they do?

the complex exponential of an operator is defined as

suppose we have three operators that commute

then they fulfill the so-called sandwich formula

which is like rotating an operator B over A

31 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins rotation operators

for the representation of the rotation vectors in the Zeeman basis

this can be seen from fulfilling the sandwich formula like

with the spin operators in the Zeeman eigenbasis

32 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins state evolution

so back to the time dependent spin state

using the matrix representation

the argument  of the rotation operator obviously depends on the elapsed

time  and the Larmor frequency 0

suppose the elapsed time is given by (>0)

hence the angle for the rotation operator is

33 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins eigenstate evolution

and the state after time  is

using the matrix representation

now assume the initial state was the eigenstate |>

then we see that this is stationary; just a phase is added

again: we still deal with a time-independent Hamiltonian

34 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution of superposition states

now let us assume the starting condition is the superposition

so when this state evolves for the same amount of time, we get

wee see that the superposition state is not stationary

this is the QM description for the precession

35 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution of superposition states

this can be extended for a whole precession cycle

36 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution in the rotating frame

let us now consider the evolution during an RF pulse

problem: the RF field is oscillating, hence the Hamiltonian becomes time dependent!

it is therefore useful to switch to the rotating frame where the operator appears time independent

the frames are related as follows

37 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution in the rotating frame

consider a spin prepared in the |x> state under evolution for some time that yields the angle 

in the rotating frame (~), the state would appear unchanged

this corresponds to the description in the static frame (like time-reversal of the evolved state)

or in generalized form

hence, the change of the state in the rotating frame reads

38 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution in the rotating frame

the time derivative of the rotation operator is

and the first term in the equation yields

the one reads

or (using the back transformation from the rotating frame)

39 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins evolution in the rotating frame

in summary, we have

or

with a transformed Hamiltonian for the rotating frame

this is the central recipe: use the Hamiltonian from the static frame, transform it into the new operator and apply it like in the ordinary time-dependent Schrödinger equation

we still have to set the reference phase for the rotating frame at time zero it will be convenient to choose

40 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins precession in the rotating frame

let us apply this recipe now first to free precession

or with defining the relative Larmor frequency

hence, the superposition states still precess about the z axis, but now with a different evolution speed

41 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins choosing the reference frequency

0 is fixed by nature but 0 is set through the instrument by setting the reference to a

certain chemical shift ref

spins left and right of the center frequency accumulate opposite phase

42 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

RF operator in the rotating frame

coming back to the problem of the time-dependent operator

which is valid for the duration of the pulse

this is like a ‘reduced’ x- operator that contains a y- contribution like in the sandwich formula, i.e. like the x-operator rotated about z

next, we need transformation into the rotating frame the necessary rotation operator just adds up to the existing one

43 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

RF operator in the rotating frame

this is slightly simplified as

and has no time dependence any more

using the convention

we get

using the sandwich formula, this can be written as sum of spin operators

44 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

Hamiltonian for a hard pulse

consider the following pulse, applied exactly on resonance

during the pulse we have

p is the flip angle of the pulse

remember the matrix representation of the operator

45 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

Hamiltonian for a hard pulse

a 90° pulse applied to the spin- up state yields

46 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins

Hamiltonian for a hard pulse

a 180° pulse applied to the spin-up state yields

but applying this pulse to a certain superposition state leaves it unchanged

47 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins view from the lab frame

since we know the evolution in the rotating frame

we can also obtain it in the lab frame by transforming the rotation operator

this describes the much more complicated of two superimposed

48 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins off-resonant effects

for the general case we had

the exponential operators of the three components will cause

• rotations about x and y (dependeing on the pulse phase p) • a rotation about z (depending on the off resonance)

in summary, the rotation axis will tilt out of the x-y plane

we get

49 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins off-resonant effects

the tilt of the axis is given by

and the rotation frequency changes to (faster, but less efficient)

the net effect corresponds to rotating the z-operator first about y and then about z

or as the state evolution

with the total off-resonance

50 Leibniz-Institut für Molekulare Pharmakologie 5.4 Uncoupled Spins transition probabilities

the off-diagonal matrix elements of this operator are linked to the transition probabilities

(starting in the |> state)

applying a  pulse, this is a function of the offset as follows

Q: what does this mean for multinuclear NMR?

51 Leibniz-Institut für Molekulare Pharmakologie Lecture 5 Summary

1. Some Math Tools

2. QM Concepts

3. Nuclear Spin Hamiltonian

52 Leibniz-Institut für Molekulare Pharmakologie Lecture 5 Summary

4. Uncoupled Spin-½ Nuclei

53 Leibniz-Institut für Molekulare Pharmakologie NMR Spectroscopy and Imaging

Department of Physics, FU Berlin

Lecture 6:

Spin Ensemble Description

54 Leibniz-Institut für Molekulare Pharmakologie Lecture 6

1. Spin Ensemble

2. Thermal Equilibrium

3. RF Effect on

4. Free Evolution

5. The One-Pulse Experiment

6. Signal Intensity

55 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble how to describe an ensemble?

we will eventually look at a macroscopic sample, se we need to make the transition from a single spin to the description of an entire ensemble

a single spin in a certain state yields the following expectation value for an operator Q

the important part are the quadratic products of the superposition coefficients

56 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble how to describe an ensemble?

so it would be useful to describe the state of the spin in terms of these products, which can be written in the ket-bra product

and we get the expectation value back as

why is this useful?

consider two spins

or in the general case

57 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble density operator and expectation value

now define an operator where N is the ensemble size

and we can get the macroscopic expectation value for Q as

also written as

this is a serious simplification of the problem

58 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble density matrix: populations and coherences

properties of the matrix for spin-½ nuclei

diagonal elements: populations (using Levitt’s box notation)

off-diagonal elements: coherences

this yields

or using the shift and projection operators as basis for this matrix

59 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble density matrix: populations and coherences

a graphical interpretation

note: coherences are no transitions!

coherences are complex numbers and come in complex conjugate pairs; one cannot have one without the other

for the populations, the normalization of the individual states means also that the sum of all populations is 1

60 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble orders of coherences

what is the meaning of the “+” and “–” in the coherence notation?

general notation for the coherence

and in high magnetic fields the involved states have well defined values for the component along the quantization axis

then the order of a coherence is defined as

and for spin ½ we get a (+1)- quantum coherence as well as a (-1)-quantum coherence

61 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin of populations

the sum is always 1 so the difference tells us more about the system

it represents the net spin and macroscopic magnetization

for >

for <

the population of a state does not say how many spins that are ‘in’ that state there are no individual spins that are polarized exactly along or against the external field majority of spins is always in superpositions of the two energy eigenstates

62 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble interpretation of coherences

a system with non vanishing off-diagonal elements in the density matrix has some transverse magnetization

for example

i.e. no longitudinal net polarization, only transverse contributions

 coherence requires spins with transverse polarization, i.e. spins in superposition states, but this is not enough

the polarization vectors must also be partially aligned

63 Leibniz-Institut für Molekulare Pharmakologie 6.1 Spin Ensemble interpretation of coherences

the coherences have amplitude and phase what does the complex character of the coherences mean?

the phase represents the orientation of the net magnetization in the transverse plane

the (+1)-coherence is just the mirror image of the (-1)

64 Leibniz-Institut für Molekulare Pharmakologie 6.2 Thermal Equilibrium density matrix for thermal equilibrium

coming back to the general form of a system with eigenstates for its Hamiltonian

quantum statistical mechanics yields that the coherences vanish for thermal equilibrium

and the populations are given by the , hence the lower state has the higher population

for spin ½ we have

65 Leibniz-Institut für Molekulare Pharmakologie 6.2 Thermal Equilibrium density matrix for thermal equilibrium

approximation of the Boltzmann factor yields

and therefore

66 Leibniz-Institut für Molekulare Pharmakologie 6.2 Thermal Equilibrium rotating frame density matrix

remember: transformation of the states is given by

and therefore

remember also

note: the populations remain unchanged, the coherences acquire a phase factor

and therefore also

67 Leibniz-Institut für Molekulare Pharmakologie 6.2 Thermal Equilibrium macroscopic magnetization from density matrix

remember vector representation for single spins

this can also be done for the ensemble by defining

with normalization to

hence the density operator may be written as

68 Leibniz-Institut für Molekulare Pharmakologie 6.3 RF Effect on Density Matrix

RF pulses applied to the ensemble

we know for the single state

and therefore (using the unitary properties of the rotation operator)

i.e. the pulses acts as operators from both sides onto the density matrix

69 Leibniz-Institut für Molekulare Pharmakologie 6.3 RF Effect on Density Matrix coherence generation through RF pulses

suppose we start with

(unity operator commutes)

the last term corresponds to a sandwich formula

i.e., applying the 90° pulse changes the spin operator in the density matrix representation from

Îz to Îy

70 Leibniz-Institut für Molekulare Pharmakologie 6.3 RF Effect on Density Matrix coherence generation through RF pulses

keeping in mind that

we get

note: • the populations are equalized • the former population difference appears in the coherences

71 Leibniz-Institut für Molekulare Pharmakologie 6.3 RF Effect on Density Matrix population inversion

now let us apply a  pulse

note: • the populations are inverted • coherences do not appear

72 Leibniz-Institut für Molekulare Pharmakologie 6.4 Free Evolution free precession without relaxation

let us now consider what happens after the /2 pulse (ignoring relaxation)

as before, the rotation operator acting on the state acts on both sides of the density operator

the populations are linked to the eigenstates, hence stationary

for the (-1) coherence we get an additional phase factor linked to the precession speed in the transverse plane

73 Leibniz-Institut für Molekulare Pharmakologie 6.4 Free Evolution free precession without relaxation

in terms of the magnetization we get

74 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment

the one-pulse experiment

let us summarize

75 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment free evolution with relaxation

experimentally we know that after the one-pulse experiment, the magnetization re-approaches the equilibrium value and that the coherences decay to zero

it follows a phenomenological approach for the Bloch equations integrated into the density matrix formalism

decay of the coherences is described by a damping factor 

76 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment

free evolution with relaxation – T2

the decaying coherence yields a shrinking magnetization vector

this can be explained because the presence of coherences does not only require transverse but also net alignment – which gets lost over time

77 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment free evolution with relaxation: inversion-recovery

a similar approach can be done for the recovery of the equilibrium populations

consider application of a 180° pulse

78 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment free evolution with relaxation: saturation-recovery

and similar in case we apply a 90° pulse where we start with

this corresponds to a certain transport of energy in the system

79 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment magnetization vector trajectory

now let us include the coherences

and remember the representation of the spin operators

80 Leibniz-Institut für Molekulare Pharmakologie 6.5 The One-Pulse Experiment magnetization vector trajectory

and using the relationship between the density matrix coefficients and the magnetization we get

which is a superposition of two types of motion yielding a spiral for the tip of the magnetization vector

81 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity density matrix and recorded signal

so how is all this related to the signal that the antenna picks up?

we detect the precessing transverse magnetization

let us assume the coil is aligned along the x axis

keeping in mind that

the coil detects the time derivative of this

and we know the of the coherences including relaxation

82 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity incoming signal treatment

for the derivative we keep in mind that  « 0

Q: what does this tell us for high-field applications compared to low field spectrometers?

remember what happens in quadrature detection

the signal is mixed with references

in two paths

83 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity incoming signal treatment

output of the mixer is the product of the signals

the high-frequency components are removed in a low-pass filter (the cosine term from the reference oscillation is now written as complex exponential)

84 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity incoming signal treatment

hence, the filter output reads

transformation into the rotating frame just adds a phase factor for coherences that is linked to the precession speed

remember notation for freely evolving coherences (this time applied in the rotating frame)

85 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity incoming signal treatment

we obtain the compact notation

and in a similar way for the other signal

then we treat this as complex signal

this contains only contributions from the (-1)-quantum coherence

86 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity phase of the final signal

this signal

can be added a phase from the digitizer

whereas the frame phase shift is removed through phase correction (and this should remove the tilde)

and introducing the total receiver unit phase shift

we get

(“numerical factors are adjusted for convenience”)

87 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity signal intensity and density matrix

so we have

 populations and the (+1)-coherence do not here

and we know for the evolution of the coherence

decomposing this into an amplitude and a phase yields

88 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity signal intensity and density matrix

hence the signal intensity is represented by the value of the coherence at time point zero immediately after the pulse (no surprise)

we can use this for the Lorentzian after FT

example for hard pulse experiment:

using

89 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity signal intensity and density matrix

we get for the (-1)-coherence

(the unity operator does not contribute)

amplitude at the beginning of acquisition

this is then multiplied with the absorptive and dispersive parts of the Lorentzian

90 Leibniz-Institut für Molekulare Pharmakologie 6.6 Signal Intensity multi-pulse experiments

we stop the discussion on pulse-acquire experiments here for now and come back to multi-pulse experiments in lecture 11

91 Leibniz-Institut für Molekulare Pharmakologie Lecture 6 Summary

1. Spin Ensemble

2. Thermal Equilibrium

3. RF Effect on Density Matrix

92 Leibniz-Institut für Molekulare Pharmakologie Lecture 6 Summary

4. Free Evolution

5. The One-Pulse Experiment

6. Signal Intensity

93 Leibniz-Institut für Molekulare Pharmakologie