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This page contains copies of the class handouts, and otherpertinent items ofinterest for the U.C. Santa Cruz Physics 216graduate quantum mechanics class.

I have put the electricity file online as a web page that allows you to jump to different sections if you want to skip ahead or go back to something quickly – just use the left hand navigation column for this. If you want to read through one page at a time, just use the arrows at the bottom of the screen.

SPECIAL ANNOUNCEMENTS

The solutions to the last problem set and the solutions to the finalexam are available for download. An access problem preventing thedownload has been rectified.

Nova Scotia Physics 12 Curriculum Guide Quiz: Make your own quiz with an answer key HERE Guest Presentation: Logan Coffill ( Operational Meteorologist-Aviation and Defence Services, Meteorological Service of Canada Environment and Climate Change Canada/Govt of Canada.
The book by Stephen Gasiorowicz, Quantum Physics, 3rd Edition (John Wiley & Sons, Inc., Hoboken, NJ, 2003) includes free supplemental material on the Wiley website. Here, I provide Gasiorowicz's Supplement 8A, which provides some clever techniques for evaluating the expectation values 1/r n (n=1,2,3) with respect to the hydrogen atom radial.
CO + is significant because it can tell us about the physics and chemistry in NGC 7027 when observed together with more common neutral and ionized molecules such as H 2 and HCO +. Learn more in our AAS Nova article and the Rochester Institute of Technology press release.
Fermilab’s Theoretical Physics Department is inviting applications for one or more postdoctoral Research Associate positions in the area of theoretical quantum information science (QIS), working at its interface with High Energy Physics, quantum field theory, and fundamental physics. This position is for a three (3) year term.

The final exam produces a pretty wide spread of results. Out of 120possible points, the grades were 103, 91, 79, 74, 74, 69, 56, 55, 51, 46, 43.I suspect the results might have been better had you had a little moretime between final exams (and perhaps a little more sleep!). AlthoughI had hoped for a somewhat better result, taking into account theeffort over the spring quarter, everyone has earned a passing gradefor the course. Grades will be posted before I leave for Germany onSaturday.

The graded homework and final exams will be left in the ISB mailbox ofthe grader, Eddie Santos. Feel free to retrieve your work the nexttime you stop by the Physics Department.

Final exam solutions have been posted to Section III of this website.

Have a great summer break---you earned it!!!

[ I. General Information and Syllabus | II. Problem Sets and Exams | III. Solutions to Problem Sets and Exams | IV. Other Class Handouts | References on special functions of mathematical physics | VI. Free Textbooks on Quantum Mechanics | VII. Other books of interest | VIII. Articles of interest | IX. Web pages of interest ]

I. General Information and Syllabus

The General Information and Syllabus handout is availablein either PDF or Postscript format [PDF | Postscript]
Some of the information in this handout is reproduced here.

General Information
Instructor	Howard Haber
Office	ISB 326
Phone	459-4228
Office Hours	Mondays 2--4 pm
e-mail	haber@scipp.ucsc.edu
web page	http://scipp.ucsc.edu/~haber/index.html

Class Hours

Lectures: Tuesdays and Thursdays, 12:00--1:45 pm, ISB 235

Required Textbooks

Principles of Quantum Mechanics, 2nd Edition,by Ramamurti Shankar [Errata forthe 13th printing (dated 2006) can be found here inPDF format.]
Lectures on Quantum Mechanics, by Gordon Baym

Pre-requisites

It is assumed that you are familiar with the material fromShankar, Chapters 1,4,5,6,7,9,11,12,13,14,15.1 and 15.2;and Baym, Chapters 3,4,5,6,7,14,15.

Course Grading and Requirements

45% Homework (5 problem sets)
20% Midterm Exam (take-home exam; Monday May 14--Wednesday May 16, 2012, due at 5 pm)
35% Final Exam (Wednesday June 13, 2012, 8--11 am)

Homework assignments will be due on every second Thursday of theacademic quarter, with the exception of the firstassignment, which is due on Tuesday April 17, 2012.The homework problem sets are not optional.You are encouraged to discuss the class material and homeworkproblems with your classmates and to work in groups, but allsubmitted problems should represent your own work and understanding.

The final exam will be held in ISB 235.This exam will be three hours long and cover the complete coursematerial. You must take the final exam to pass the course.You will be permitted to consult the class textbook, your ownhandwritten notes, and any class handout during the final exam.

Course Syllabus

Advanced topics in angular momentum theory [Shankar, Chapter 15.3; Baym, Chapter 17]
Path integral formulation of quantum theory [Shankar, Chapter 8]
The WKB approximation [Shankar, section 16.2]
The variational method [Shankar, section 16.1]
Time-independent perturbation theory [Shankar, Chapter 17; Baym, Chapter 11]
Quantum theory of scattering [Shankar, Chapter 19; Baym, Chapters 9--10]
Time-dependent perturbation theory [Shankar, sections 18.1--18.3; Baym, Chapter 12]
Quantum theory of radiation [Shankar, section 18.5; Baym Chapter 13]
Identical particles [Shankar, Chapter 10; Baym, Chapter 18]
Second quantization [Baym, Chapter 19]

II. Problem Sets and Exams

Problem sets and exams are available in either PDF or Postscript formats.

Problem Set #1--due: Tuesday, April 17, 2012 [PDF | Postscript]
Problem Set #2--due: Friday, April 27, 2012 [PDF | Postscript]
Problem Set #3--due: Monday, May 14, 2012 [PDF | Postscript]
Midterm exam--due: 5 pm Wednesday, May 16, 2012 [PDF| Postscript]
Problem Set #4--due: Tuesday, May 29, 2012 [PDF | Postscript]
Problem Set #5--due: Monday, June 11, 2012 [PDF | Postscript]
Final exam--8--11 am on Wednesday, June 13, 2012 [PDF| Postscript]

III. Solutions to Problem Sets and Exams

The problem set and exam solutionsare available in either PDF or postscript formats.

Solution Set #1 [PDF | Postscript]
Solution Set #2 [PDF | Postscript]
Solution Set #3 [PDF |Postscript]
Midterm Exam Solutions [PDF | Postscript]
Solution Set #4 [PDF |Postscript]
Solution Set #5 [PDF |Postscript]
Final Exam Solutions [PDF | Postscript]

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IV. Other Class Handouts

Class handouts are available in either PDF or postscript formats.
1. This handout examines the properties andderives an explicit form for the mostgeneral 3x3 proper rotation matrix R(n,θ)corresponding to a rotation by θ about an axis n.Using these results one can easily determine therotation axis and the angle of rotation given an arbitrary real 3x3orthogonal matrix of determinant +1. An arbitrary rotation canalso be parameterized by three Euler angles.The relations between the Euler angles and (n,θ) arederived in these notes. [PDF | Postscript].

2. This handout defines the tensor spherical harmonics as thesimultaneous eigenfunctions of J²,J_z , L² and S²in the coordinate representation. The cases of s=1/2 and s=1 aretreated explicitly. A useful table of Clebsch-Gordon coefficients isprovided for the analysis. The Clebsch-Gordon series and itsapplications are treated in Appendix A, and thevector spherical harmonics are treated explicitly in Appendix B. [PDF | Postscript].

3. The section on time reversal in Shankar does not provide muchdetail. So, I will take advantage of a wonderful collection ofnotesthat were prepared for the graduate quantum mechanicscourse at Berkeley for the 2011--2012 academic year by Professor RobertLittlejohn. Links to his notes on time reversal in quantum mechanicsare provided here: [PDF | Postscript]

4. For a very nice treatment of the mathematics of antilinearand antiunitary operators, have a look at the followingset of notes by C.M. Caves: [PDF].

5. A summary of the important properties of the Airy functions can befound in this handout. [PDF | Postscript].

6. A nice set of notes by Frank Porter of Caltechdiscusses the fundamental role of the electromagnetic vector potentialand the Aharonov-Bohm effect. These note can be found here:[PDF | Postscript]
Solutions to the problems given at the end of Porter's notes can befound here:[PDF | Postscript]

7. In these notes, the calculation of the ground stateenergy of the helium atom using the variational principle,is given. These will fill in some of the steps omitted by Shankar.Details of all thesteps in the calculation are provided (along with a number ofintegration tricks) here: [PDF | Postscript]

8.These notes review the Schrodinger equation fora charged particle in an external electromagnetic field. In order toobtain the relevant equation, we first examine the classicalHamiltonian of a charged particle in an electromagnetic field.We then use this result to obtain the Schrodinger equation using the principle of minimalsubstitution. As an example, the special case of a uniform magnetic field is exhibited. Finally,we demonstrate the origin of the coupling of the spin operator to the external magnetic fieldin the case of a charged spin-1/2 particle. [PDF | Postscript]

9. The book by Stephen Gasiorowicz, Quantum Physics, 3rd Edition(John Wiley & Sons, Inc., Hoboken, NJ, 2003) includes free supplemental material on the Wileywebsite. Here, I provide Gasiorowicz's Supplement 8A, which providessome clever techniques for evaluating the expectation values <1/rⁿ> (n=1,2,3) with respect to the hydrogen atom radial wave functions. These are needed in computation of the fine structure of hydrogen. [PDF].

10. In quantum mechanics, one often deals with two-state systems. Tosolve such a system requires one to diagonalize a general 2x2 hermitian matrix. Having done so once, one can then apply these results in manydifferent circumstances. The derivation of the eigenvalues and thediagonalizing unitary matrix can be found in this handout. [PDF | Postscript]

11. A handout entitled TheRiemann-Lebesgue Lemma is a very important result ofFourier analysis. It has many applications in mathematical physics.In this course, we employ the Riemann-Lebesgue lemma in our derivationof the optical theorem. A simple proof of the Riemann-Lebesgue lemmais given in this handout. [PDF| Postscript].

12. A handout entitled The Optical Theoremprovides two derivations of the celebrated optical theorem ofscattering theory. The first proof employs an interesting mathematicalidentity that interpretslim_r→∞e^ik·x as a distribution.The second proof makes use of the abstract scattering theoryformalism. [PDF| Postscript].

V. References on special functions of mathematical physics

1. A superb resource for both the elementary functions and thespecial functions of mathematical physics is theHandbookof Mathematical Functions by Milton Abramowitz and Irene A. Stegun,which is freely available on-line. The home page for thisresource can be foundhere. There, youwill find links to aframesinterface of the book. Another scan of the book can be foundhere.A third independent link to the book can be foundhere.

2. The NISTHandbook of Mathematical Functions (publishedby Cambridge University Press), together with its Webcounterpart, the NIST Digital Libraryof Mathematical Functions(DLMF), is the culmination of a project that was conceived in 1996 atthe National Institute of Standards and Technology (NIST). The projecthad two equally important goals: to develop an authoritativereplacement for the highly successful Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables, published in1964 by the National Bureau of Standards (M. Abramowitz andI. A. Stegun, editors); and to disseminate essentially the sameinformation from a public Web site operated by NIST. The new Handbookand DLMF are the work of many hands: editors, associate editors,authors, validators, and numerous technical experts.The NISTHandbook coversthe properties of mathematical functions, from elementarytrigonometric functions to the multitude of special functions.All of the mathematical information contained in the Handbook is alsocontained in the DLMF,along with additional features such as moregraphics, expanded tables, and higher members of some families offormulas. A PDF copy of the handbook is provided here: [PDF]

3. One of the classic references to special functions is a threevolume set entitled Higher Transcendental Functions(edited by A. Erdelyi), which wascompiled in 1953 and is based in part on notes left by Harry Bateman.This was the primary reference for a generation of physicists andapplied mathematicians, which is colloquially referred to as theBateman Manuscripts. This esteemed reference workcontinues to be a valuable resources for students and professionals.PDF versions of the three volumes are now available free of charge.Check out the three volumes by clicking on the relevant links here: [Volume 1 |Volume 2 |Volume 3].

4. Another very useful reference for both the elementary functions and thespecial functions of mathematical physics isAn Atlasof Functions (2nd edition) by Keith B. Oldham, Jan Myland and JeromeSpanier, published by Springer Science in 2009.This resource is freely available on-line to studentsat the University of California at thislink.

5. Yet another excellent website for both the elementary functions and thespecial functions of mathematical physics isthe Wolfram Functionssite. This site was created withMathematica and is developed and maintainedby Wolfram Research with partial support from theNational Science Foundation.

6. One of my favorite books on special functions isSpecial Functions and Their Applications by N.N. Lebedev(Dover Publications, Inc., Mineola, NY, 1972). Itprovides invaluable information on special functions, while beingextremely cheap to buy (and even cheaper to peruse onGoogle Books).

7. In your problem sets, you will encounter many difficultintegrals. Although you may be tempted to use Mathematica or Maple(which sometimes is the easiest approach), I cannot overestimate thevalue of a good table of integrals. Professionals always choose firstto consult Table of Integrals, Series and Products, 7th edition,by I.S. Gradshteyn and Ryzhik, edited by Alan Jeffrey andDaniel Zwillinger (Elsevier, Inc., Amsterdam,2007).You can preview quite a few pages usingGoogle Books.

VI. Free Textbooks on Quantum Mechanics

1. Quantum Mechanics:Fundamental Principles and Applications by John F. Dawsonis a compilation of notes for a first year graduate course innon-relativistic quantum mechanics which Professor Dawsontaught at the University of New Hampshire for a number of years. [PDF]

2. Lecture Notes inQuantum Mechanics by Doron Cohen is based on a course givenby Professor Cohen at Ben-Gurion University. [PDF | Postscript]

3. The first four chapters of Advanced Modern Physics by John Dirk Walecka is provided free ofcharge by WorldScientific. Included are chapters on quantum mechanics, angularmomentum and scattering theory.For further details, check out the links on the WorldScientific website. Additional pages of this book can bepreviewed on the google books website. [PDF]

VII. Other books of interest

1. There are many classic books on angular momentum in quantummechanics. These include book by M.E. Rose; A.R. Edmonds;D.M. Brink and D.R. Sachler; and L.C. Biedenharn and J.D. Louck.However, for a modern introductory treatment of this subject,I canrecommend Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems by William J. Thompson published byWiley-VCH in 2004. Unfortunately, the University of California does notprovide free access to the Wiley Online Library. However, if youhappen to be at Stanford University, you can access the chaptersof this book from a Stanford computer. Otherwise, you can perusethis bookherecourtesy of google books.

2. Another modern book on angular momentum in quantum mechanics,entitled Angular Momentum Techniques in Quantum Mechanics, byVaradarajan Devanathan (Kluwer Academic Publishers, New York, 2002),provides excellent reference for formulae and identities forrotations, Clebsch-Gordon, Wigner-Eckart, coupling of angular momenta,spherical tensors, etc. (as noted by an enthusiastic reviewer onAmazon). In fact, this book can be viewed online atScribd,although some patience is required.

3. A very nice treatment of the semi-classical approximation toquantum mechanics can be found in a book entitled QualitativeMethods in Quantum Theory, by A.B. Migdal (W.A. Benjamin, Inc.,Reading, MA, 1977), In this book, the connection formulae of theWKB approximation are derived by a very clever technique that involvesanalytic continuation into the complex plane. This book can be partiallypreviewed online courtesy ofat google books.

4. For the more traditional approach to the WKB approximation and thederivation of the connection formulae, check out Chapter 14 ofQuantum Mechanics:Fundamental Principles and Applications by John F. Dawson,previously referenced in Section VI of this website.Another good source for this material can be found in Chapter 7 ofEugen Merzbacher, Quantum Mechanics, Third Edition(John Wiley & Sons, Inc., New York, NY, 1998). In fact, theSecond Edition of this book can be viewed online atScribd,although it takes a while to download.

VIII. Articles of interest

1. For a nice review of time reversal, have a look atTimereversal in classical and quantum mechanics by J.M.Domingos, International Journal of Theoretical Physics 18,213--230 (1979). [PDF]

2. For a very nice introduction to the technique of path integralsand their applications in physics, I highly recommendPathIntegral Methods and Applications by Richard MacKenzie. [PDF | Postscript]

3. A slightly more sophisticated treatment of path integralsin quantum mechanics can befound in LectureNotes on the Path Integral Approach to Quantum Mechanicsby Matthias Blau. [PDF |gzipped Postscript]

4. For a pedagogical treatment of the WKB approximation tothree-dimensional spherically symmetric potentials, with applicationsto the analysis of the energy levels of quark-antiquark (quarkonium)bound states, check out C. Quigg and Jonathan L. Rosner,Quantum mechanics with applications to quarkonium, PhysicsReports 56 (1979) 167-235. [PDF]

5. Following Chapter 11 of Baym, I showed in class thatto second order in perturbation theory, the wave functionrenormalization constant was equal to the partial derivative of the perturbedenergy eigenvalue with respect to the unperturbed energy, keepingfixed the matrix elements of the perturbation. In fact, this result istrue to all orders. For a general proof, seeNormalization of states in perturbation theories by D.R. Bès, G.G. Dussel, and H. M. Sofía, American Journal of Physics 45 (1977) 191--192. [PDF]

IX. Web pages of interest

1. At UC Berkeley, the graduate quantum mechanics course during thepast two academics years has been given by Professor RobertLittlejohn. He maintains a comprehensivewebpagefor this year's course (given over the 2011--2012 academic year).On this webpage, you can find an extensive set of notes that are quitewell written, along with many problems and solutions. Feel freeto explore if you have a chance.

2. You should learn how to read atable of Clebsch-Gordoncoefficients. The classic table that provides all Clebsch-Gordoncoefficients for the addition of angular momentum with all possiblevalues of j up to and including j=2 can be found inPDFformatcourtesy of the Particle Data Group.To obtain the Clebsch-Gordon coefficients from this table, it isessential that you take note of the following instruction near the topof the table that reads:Note: A square-root sign is to be understoodover every coefficient, e.g., for -8/15 read-√ 8/15 .

3. In 1977, J.D. Jackson gave a colloquium in which he explained theconnection between the fundamental intrinsic magnetic dipole momentand the hyperfine structure of the s-states of the hydrogen atom.Jackson provided a writeup ofhis colloquium as a CERN Yellow Report (CERN 77-17), which you canfind here: [PDF]

haber@scipp.ucsc.edu
Last Updated: January 24, 2019

Born	21 February 1942 (age 78)
Nationality	British
Awards	Balzan Prize(1981) Wollaston Medal(1983) Japan Prize(1990) Royal Medal(1991) Copley Medal(2011) William Bowie Medal(2001) Crafoord Prize(2002)
Scientific career
Fields	Geophysics
Institutions	University of Cambridge
Thesis	The shape of the earth(1967)
Doctoral advisor	Teddy Bullard
Influences	Walter Munk, Don L. Anderson

Dan Peter McKenzieCHFRS (born 21 February 1942) is a Professor of Geophysics at the University of Cambridge, and one-time head of the Bullard Laboratories of the Cambridge Department of Earth Sciences. He wrote the first paper defining the mathematical principles of plate tectonics on a sphere, and his early work on mantle convection created the modern discussion of planetary interiors.

Early life[edit]

Born in Cheltenham, the son of an ear, nose, and throat surgeon,^[1] he first attended Westminster Under School and later Westminster School, London.

Education and career[edit]

McKenzie attended King's College, Cambridge where he read physics, obtaining a 2:1 in his final degree.

As a graduate student, he worked with Edward 'Teddy' Bullard who suggested he work on the subject of thermodynamic variables. He was awarded a Research Fellowship at King's College at the beginning of his second year which enabled him to study anything he wanted. As such, he gave up doing what Teddy had suggested and became interested in how the interior of the earth convects, something completely speculative at that time. McKenzie taught himself fluid mechanics and then went to the Scripps Institution of Oceanography at the University of California, San Diego, on the invitation of Freeman Gilbert and Walter Munk. After eight months he returned to Cambridge, submitting his PhD in 1966. He has since said that nothing in his early life as a scientist had such a profound effect on him as those eight months in California.^[2]

Plate tectonics[edit]

Spending time between Cambridge and a Fellowship held in Caltech, McKenzie was invited, along with Teddy Bullard, to a conference in New York which initiated his revolutionary work on plate tectonics. After listening to separate talks from Fred Vine on plate tectonics,^[3] looking at the thermal structure of oceanic plates as they formed and cooled.^[1]

Following this, he published a seminal paper with Bob Parker,^[4] which employed Euler's Fixed Point Theorem, in conjunction with magnetic anomalies and earthquakes to determine a precise mathematical theory on plate tectonics. Option 2 may 1st payment. This work was published some 3–4 months after the same work had been carried out by Jason Morgan at Princeton. Allegations were subsequently made suggesting that McKenzie was at Morgan's spring AGU talk where he presented his plate tectonics work.^[1] Later in 1968 he went to Princeton where he found that he and Morgan had solved two or three problems using identical mathematics in exactly the same way – plate tectonics was one, another was the thermal structure of the oceans and another was looking at earthquake mechanisms in a different way to seismologists.^[1]

Working with John Sclater, McKenzie determined the entire geological history of the Indian Ocean, the publication^[5] of which eventually resulted in them both receiving Fellowships at the Royal Society.

Mantle convection and sedimentary basins[edit]

McKenzie was awarded a University position and took it up in 1969. At this point he decided to move away from plate tectonics, choosing instead to focus on the behavior of fluids below the plates. He studied cellular convection and motions in the mantle whilst at the same time pursuing yet another new avenue of research; the development of sedimentary basins. It was from this work that he produced a classic paper^[6] that has been widely accepted by oil companies as the 'McKenzie Model of Sedimentary Basins.'^[1]

McKenzie was elected a Fellow of the Royal Society in 1976 aged just 34, and by 1978 was awarded a University Readership position.

Later career[edit]

McKenzie continues to work at the Bullard Laboratories in Cambridge where he is Professor of Earth Science. Most recently his research has provided new insights into the tectonic evolution of Mars and Venus. In 2002 he was awarded the prestigious Crafoord Prize from the Royal Swedish Academy of Sciences for his contributions to research in the field of plate tectonics, sedimentary basin formation and mantle melting. With his appointment as a Companion of Honour in 2003, he brought the then current Cambridge membership of this elite group to four: Brenner, McKenzie, Hobsbawm and Hawking.

Selected bibliography[edit]

McKenzie, D., Nimmo, F., Jackson, J., Gans, P. B. & Miller, E. L. 2000 Characteristics and consequences of flow in the crust. J. geophys. Res. 105, 11029-11046.
McKenzie, D. & Fairhead, D. 1997 Estimates of the effective elastic thickness of the continental lithosphere from Bouguer and free air gravity anomalies. J. geophys. Res. 102 27523-27552.
White, R.; McKenzie, D. (1989). 'Magmatism at rift zones: The generation of volcanic continental margins and flood basalts'. Journal of Geophysical Research. 94: 7685–7729. Bibcode:1989JGR..94.7685W. doi:10.1029/JB094iB06p07685. S2CID140181589.
Jackson, J. A. & McKenzie, D., 1988 The relationship between plate motions and seismic moment tensors, and the rates of active deformation in the Mediterranean and Middle East. Geophys. J. R. astr. Soc. 93, 45–73.
Bickle, M. J. & McKenzie, D., 1987 The transport of heat and matter by fluids during metamorphism. Contrib. Mineral. Petrol. 95, 384–92.
McKenzie, D. & O'Nions, R. K., 1983 Mantle reservoirs and ocean island basalts. Nature 301 229–231.
England, P. & McKenzie, D., 1982 A thin viscous sheet model for continental deformation. Geophys. J. R. astr. Soc. 70, 295–321.
McKenzie, D. (1978). 'Some remarks on the development of sedimentary basins'. Earth and Planetary Science Letters. 40 (1): 25–32. Bibcode:1978E&PSL.40..25M. CiteSeerX10.1.1.459.4779. doi:10.1016/0012-821X(78)90071-7.
Parsons, B. & McKenzie, D., 1978 Mantle convection and the thermal structure of plates. J. geophys. Res. 83, 4485–96.
McKenzie, D., Roberts, J. & Weiss, N. O., 1974 Convection in the Earth's mantle: towards a numerical simulation. J. Fluid Mech., 62, 465–538.
McKenzie, D., Molnar, P. & Davies, D., 1970 Plate tectonics of the Red Sea and East Africa. Nature 226, 243–8.
McKenzie, D.; Sclater, J. G. (1971). 'The Evolution of the Indian Ocean since the Late Cretaceous'. Geophysical Journal International. 24 (5): 437. Bibcode:1971GeoJ..24.437M. doi:10.1111/j.1365-246X.1971.tb02190.x.
McKenzie, D.; Parker, R. L. (1967). 'The North Pacific: An Example of Tectonics on a Sphere'. Nature. 216 (5122): 1276. Bibcode:1967Natur.216.1276M. doi:10.1038/2161276a0. S2CID4193218.
McKenzie, D. (1966). 'The viscosity of the lower mantle'. Journal of Geophysical Research. 71 (16): 3995–4010. Bibcode:1966JGR..71.3995M. doi:10.1029/JZ071i016p03995.

Awards[edit]