1. The Quantum Mechanics of MRI Part 1: Basic concepts
David Milstead
Stockholm University

2. Background reading
Fundamentals of Physics, Halliday, Resnick and Walker (Wiley)
The Basics of NMR, J. Hornak ( )

3. Outline What is quantum mechanics Wave-particle duality
Schrödinger’s equation
Bizarreness
uncertainty principle
energy and momentum quantisation
precession of angular momentum

4. What is quantum mechanics ?

5. Basic concepts What is a wave ? What is a particle ?
What is electromagnetic radiation ?
What happens to a magnet in a magnetic field ?

6. What is a wave ? Double slit diffraction
Properties of waves:
Superposition, no localisation (where is a wave??)

7. What is electromagnetic radiation ?
EM radiation is made up of electromagnetic waves of various wavelengths and frequencies
Radio frequency is of most interest to us

8. Circularly polarised light
Rotating magnetic field
Radiation can be produced and filtered to produce a rotating magnetic
field.

9. Light is a wave!

10. Photon

11. Photons

12. Wave-particle Duality

13. We’ve just learned a basic result of quantum mechanics.
Now we move onto some maths.

14. Fundamental equation of quantum mechanics

16. Free particle

17. Wave function of a confined particle
Eg particle trapped in a tiny region of space. How do we model the
wave function ?
Solution to Schrödinger’s equation would be
sum lots of sine waves with different wavelengths/momenta.

18. x x

22. Question
A 12-g bullet leaves a rifle at a speed of 180m/s. a) What is the wavelength of this bullet? b) If the position of the bullet is known to an accuracy of 0.60 cm (radius of barrel), what is the minimum uncertainty in its momentum? c) If the accuracy of the bullet were determined only by the uncertainty principle (an unreasonable assumption), by how much might the bullet miss a pinpoint target 200m away?. 

23. Hydrogen atom
Solve Schrödinger’s equation for an electron around a proton in a hydrogen atom.
The electron is confined due to a Coulomb potential.

25. Where is the electron ?
Wave functions

26. Current loops and magnetic dipoles+ -
Current loop N S
Bar magnet

27. Orbiting electron as a current loop

29. An atom in a magnetic field
l=1 and therefore 2l+1 states E
(1)
(2)
(3)

30. l=1
l=0

32. z y x LZ U X

33. Angular momentum precession
LZ constant and L precesses around z-axis

34. Question

36. The mathematics of spin angular momentum is identical to orbital

38. FN FS
Magnetic field FN FS
Magnetic field
Ignore force not parallel to North-South axis.

39. Magnetic field
Magnetic field

40. Magnetic field

42. Zeeman effect with orbital and spin angular momentum
In the presence of a magnetic field, multiplicities of ”spectral lines” appear

43. Larmor precession for spinz y x SZ S
S sinq

44. Summary Established basic quantum mechanics theory needed for NMR
wave-particle duality
Light is either photons or electromagnetic waves
Schrödinger’s equation and the wave function at the heart of QM predictions
Energy and angular momentum are quantised
Larmor precession
Angular momentum comes in two varieties (orbital and spin)

45. The Quantum Mechanics of MRI Part 2: Understanding MRI
David Milstead
Stockholm University

46. Outline
Spin - reminder
Fermions and bosons
Nuclear energy levels

47. Spin z y x SZ S
S sinq

48. Gyromagnetic ratio
Why do they have different values ?

49. Fermions and bosons Fermions Bosons Spin 1/2, 3/2, 5/2 objects
Electrons, protons and neutrons have spin 1/2
Tricky bit comes when combining their spins to form the spin of, eg, an atom or a nucleus
Bosons
Integer spin objects

51. Similar shell structure for nuclear physics as for atomic physics
Need to fill up, shell by shell
Pairs of protons and neutrons cancel each other’s spins.
Pauli’s exclusion principle ensures that many shells are filled.
Nuclei with uneven (even) atomic number have half-integer (integer) spin
Nuclei with even atomic and mass numbers have zero spin.
Unpaired neutrons/protons provide the spin for MRI.

52. Question
Nature would prefer all electrons to be in the lowest shell and all nucleons protons/neutrons) in the lowest shell ? Why doesn’t this happen ?

53. Usefulness for MRI
Need isotopes with unpaired protons (to produce signal for MRI)
Most elements have isotopes with non-zero nuclear spin
Natural abundance must be high enough for MRI to be performed.

54. Spins of various nuclei

55. Now we can understand MRI

56. No magnetic field

57. Apply an external magnetic fieldB0

58. Another look at the system
Split into ”spin-zones”.
For uniform system we can regard the macroscopic system as giving
a single magnetisation.

59. Putting together what we’ve learned

60. Energy level population

62. Changing the spin populations

63. Question

64. Exciting the nuclei - Rf pulse
Rotating magnetic field B1
Typical pulse duration ~1ms.
Two ways to think about the pulse. Both are needed to understand MRI.

65. Excitation
Rotating magnetic field B1 B0 B1

66. Rotating frame of referenceB0 B1 X’ y’

67. Different types of pulses

68. Question

69. What happens next ?

70. Longitudinal relaxation

71. Relaxation times for different materials

72. Transverse relaxation

73. Free induction decay

74. Free induction decay

75. Summary
Basic quantum mechanics at the heart of nuclear magnetic resonance
Angular momentum quantisation
Energy quantisation
Features of a MRI experiment investigated.