1. Department of Radiology
Problems in MR that really need quantum mechanics: The density matrix approach
Robert V. Mulkern, PhD
Department of Radiology
Children’s Hospital
Boston, MA

2. Nuclear Spin: An inherently Quantum Mechanical (QM) Phenomenon
Angular momentum operators represent spin I

3. But problems in MR that need QM?
Proton imaging? Not really…
Relaxation? Not really…
Radiological interpretations? Sometimes…
Spectroscopy? Absolutely…
Spectroscopic imaging? Yes indeed…
X-nuclei? Why not!

4. Proton Imaging: Our Bread and Butter
T2 Contrast T1 Contrast
Tissue relaxation rates and pulse sequence specifics determine
Tissue contrast – all understood via the classical Bloch equations

5. BPP Theory: Used QM to calculate T1, T2 – 1950’s – rarely used in practice
Fluctuations of Dipolar Hamiltonian

6. QM in Radiological Interpretations?
Magic angle effect (3cos2 – 1) = 0
Bright fat effect (quenching of J-coupling with multiple 180’s)

7. “When molecules lie at 54.74° there is
lengthening of T2 times (don't understand why, but it involves 'bipolar coupling')”

8. “Dipolar Coupling” - Magnetic energy between two dipoles

9. The Dipolar Hamiltonian

10. Bright Fat Phenomenon

11. Where QM Really Rules: Coupled Spin Systems and Spectroscopy

12. “Shut up and Calculate”
Richard Feynman
The real beauty of the Density
Matrix Formalism – no thinking…

13. Spin ½ Rules of the Road Iz|+> = ½ |+> Iz|-> = -1/2 |->
Ix = (I+ + I-)/2
Iy = (I+ - I-)/2i
I+|+> = 0
I+|-> = |+>
I-|-> = 0
I-|+> = |->
h = 1, let’s be friends
Commutation Relations
[I,S] = 0 (two spins)
[Ii,Ij] = ijkIk

14. Typical Hamiltonians of Interest
1) H = woIz
2) H = (wo + /2)Iz + (wo – /2)Sz + JIzSz
3) H = (wo + /2)Iz + (wo – /2)Sz + JIxSx + J IySy + JIzSz
4) H = w1Iy or w1Ix RF pulses
Weak vs strong and “secular” terms: J <<  means weak and no secular terms

15. Density Matrix Example: Free Precession1 2
H = woIz
H|+> = (1/2)wo|+>
H|-> = -(1/2)wo|->
 = exp(-iHt)exp(-iIy)Izexp(iIy)exp(iHt)
Calculate the Signal as Tr{(Ix+iIy)} = Tr{I+}

16. The Matrix and its Trace
Tr{(Ix+iIy)} = Tr{I+}
<+|I+|+> <+|I+|->
<-|I+|+> <-|I+|->
<-|I+|-> = only nonvanishing diagonal element
<-|exp(-iHt)exp(-iIy)Izexp(iIy)exp(iHt)|+> =
exp(iwot/2) <-|exp(-iHt)exp(-iIy)Izexp(iIy)|+> = ?
How to handle the RF pulses?

17. The Pauli Spin Matrices
Wolfgang Pauli

18. Matrix Representations of Angular Momentum Operators
0 1
= The Identity Matrix

19. So…keep on trucking to get the classical FID result
exp(iwot/2) <-|exp(-iHt)exp(-iIy)Izexp(iIy)|+> =
exp(iwot) <-|exp(-iIy) Iz (cos/2 + sin/2 (I+-I-))|+>
= …
exp(iwot) cos/2 sin/2 = (1/2) exp(iwot) sin y t 1 2

20. The general approach Identify pulse sequence, Hamiltonian(s)
Construct density matrix operator 
Calculate Tr({ I+} to get time domain signal – the diagonal elements
Multiply by exp(-R2t) and Fourier transform for spectrum

21. The citrate molecule
AB System

22. Citrate quantitation and prostate cancer

24. Projection Operator: Sum over States (when you get stuck)

25. Two Spin Hard Pulse RF Operators
Fy = Iy + Sy
[I,S] = 0, I and S commute

26. So…shut up and calculate!

27. Localization with PRESS sequence

28. The Best Day of My Life?
Theory Experiment

29. Joining the Greats!

30. Inverted lactate at TE = 140 ms

31. The lactate molecule
AX3 system

32. Lactate (AX3) Calculation

33. Why is lactate inverted at TE = 140 ms and up again at 240 ms?

34. Ethanol Detection with brain MRS
270 ms TE

35. An A2X3 Calculation…Optimize Ethanol detection in the Brain

36. 6 minute scans
18 minute scan

37. 31P MRI of ATP

38. RARE Sequence and Density Matrix

39. With J = J = J and J = 0

40. J-Coupled modulation of k-space lines

41. Hey you great guys and girl - Thanks for the QM!
…and we still have a lot
to calculate…

42. Be careful what you say in print…
Magn Reson Med 1993;29:38-33 “ “
Be careful what you say in print…

43. Every Pulse Sequence has a Density Matrix Operator1 2
Gradient Echo
90y t
180x 
Spin Echo