**Derek F. Lawden; Dover Publishing Books, Edition 1995.**

Commented by: Hassan El-Saghir, PE; M. Eng (Env)

I have read the whole seven chapters of this book and enjoyed reading it. Personally, I found it good as a mean to recycle my know how in this field of science after an interruption of about eight years.

Now I will start with my commentaries chapter by chapter on this book.

The author is a mathematics teacher and writer in physics hence his care about the mathematical rigidity of the theory as compared to the scientific deductions in physics.

This doesn?t mean that the whole theory behind quantum mechanics is exposed in its completeness, since I found that the reader has to do an additional effort of calculation (with the use of a pencil and paper) in some intermediate mathematical steps leading to a quantum mechanics important formula. This is essentially in those chapters concerned with Hermit, Legendre and Laguerre polynomials although some of their theory is well exposed in the Appendix.

Chapter 1: The Vector Representation of States

This is mainly about vector space in relation to Quantum Mechanics i.e. in short QM.

This is well exposed operator algebra around state functions from the statistical mechanics perspective using the condensed Dirac like notation of statistics pertaining to Hilbert Space logic. Every thing from matrices to conjugate and hermitian forms is exposed in an attractive way. Also, the important Heisenberg principle is exposed here since it is mainly a statistical formalism as compared to classical Newtonian mechanics. An important aspect worth to elucidate is that this book preserves the old school type of approach to state functions with no mention of Dirac Bra and Ket notations (like the addition of salt and spices to a meal, that can give it taste to some while to others: Harm to Health!) which personally are more confusing than necessary in physics. In that manner, the author converges to the methods exposed by Landau and Lifshitz of the sixties! Let us not forget that Nobel Laureate (1962) Lev Landau was a prominent Soviet figure in addition to be considered as the hero of former USSR. I have found this book by Lawden very close to The Principles of Quantum Mechanics by Bloukinstev another Russian expert that publishes on Mir editions of Moscow. Something attracted my mind that is worth to mention is that the Russian approach to Atomic physics is very close to the British one rather than to the American or French ones.

Chapter 2: Spin

This is a very important topic in QM especially for those concerned with nuclear chemistry and Relativistic QM. This chapter in fairly well exposed but contains some holes inside the waving of the theory that made me go to Mc Gill Library to check with some books whom among them just An Introduction to QM by W. Greiner on Springer gave me the answer and clarification on some spinor matrix equations.

Lawden approach to spin matrix writings is through geometric frame transformations as compared to Greiner who is more analytic in his calculations and hence easier to grasp and retain than Lawden.

Chapter 3: Observables having Continuous Spectra

This is an extraordinary clear chapter that shows the reader how to transit from the discrete representations to continuous ones with examples around the particle in the well and the harmonic oscillator. In this entire chapter the Hamiltonian operator is used extensively and hence, the theoretical methods of this book converge with others on the market of scientific academy.

Chapter 4: Time variations of State

In this chapter, Lawden starts to approach QM with a variation in time and elaborates statistical and Hamiltonian equations of phenomena explanation using equation containing the time as a variable. Again spin variation in time is elaborated in addition to certain case studies like particle movement over a potential barrier and the consequence tunnel effect. Lawden derive the differential equations using Hamiltonian operators and solve them to reach an exponential form in addition to the determination of the boundary conditions in order to evaluate the constants. It is only in this chapter that the author approaches the concept of Heisenberg picture versus Schr?dinger representation.

Chapter 5: Angular Momentum

This is a very important chapter that completes the spin picture in QM. It is very well exposed based on operator algebra in this chapter; Lawden derives the Hamiltonian operator in spherical coordinates. Case studies are about central force field as a potential,

The hydrogen atom and last Zeeman Effect. In this chapter the author couples the spin with angular momentum in a central potential atomic field. The characteristic equations and eigenfunctions with their eigenvalues are derived in detail in this book. Another case

Study is the classical situation of a charged particle in an electromagnetic field that gives the complete Hamiltonian operator equation.

Chapter 6: Perturbation Methods

This is the classical exposure of variational calculus leading to the understanding of Zeemann effect, Stark effect, splitting of rays and hyperfine structures of the atom.

Chapter 7: Dirac?s Relativistic Equation

This is an outstanding chapter on this modern field of study and research. Lawden brings us an innovative original method of the derivation of Dirac?s equation which differs from author to author and from one book to the other. Dirac equation is derived using Hamiltonian operator techniques applied to the characteristic equation of the wave to reach the intermediate Klein-Gordon equation then the final Dirac form.

As an example, the most notorious book of Quantum Mechanics by Schiff who is based on Dr. R. Oppenheimer lecture notes start with the Einstein relativistic equation:

** E = C [P**2 + (MC) **2] ** (1/2) **

Whereby ** denotes raising to the power

E is energy of the particle

C is speed of light

P is particle momentum

M is particle mass

From it through some manipulation of the Hamiltonian, they reach the final Dirac form.

I have to mention that the Author Lawden claims that his derivation is identical to the one of its original finder i.e. the Nobel Laureate P.A.M. Dirac of Great Britain.

Again, Lawden bring fourth a complete package of operator methods that has to do with the derivation of additional equations of spin and others.

Case studies are about coupling of Angular momentum and Spin, Eigenfunctions of Dirac Hamiltonian, Motion in a Central Field and last, the Hydrogen Atom.

Last, the book contains good exercises but unsolved. I encourage professional in Physics to purchase it at budget price since it might be useful in their carriers.

This is my personal commentary on the work of the past

Dr. D. Lawden that with his unfortunate death last february 2008, resulted in a big loss to the world scientific community.

My God bless his soul. It was today only that I knew about this sad event.

The most important remark that I wish to raise concernning this article is that: It is not through the use of the complicated Bra and Ket Dirac notations in QM equations that we mean to be modern and up to date in QM and Physics, since usage of these notations won’t mean that we are dealing with Hilbert Space (a 19th century theory!) which was created many years before Dirac representation of Vector Spaces. Hilbert space is a subject of mathematical functional analysis and has to do with orthogonal functions and space normalisation that enables Hilbert Space to be approached also by Riemanian Geometry that deals a lot with metric tensor algebra.

In what follows is a brief biography of the late professor Lawden obtained from his friends and collegues.

Derek Frank Lawden

ScD Cantab FRSNZ FIMA

1919- 2008

DEREK LAWDEN was born in Birmingham, UK, on 15th September 1919 and died in Warwick, UK, on 15th February 2008. He attended King Edward VI Grammar School, Aston and won a scholarship to Cambridge to study Mathematics. He graduated in 1947 as a Wrangler in the Tripos examinations. He served as a captain in the Royal Artillery from 1939 until 1946, for a time being in charge of coast artillery radar at Gibraltar. After leaving the army he was appointed a lecturer in mathematics at the Royal Military College of Science, researching in control systems. Later, he moved to the College of Advanced Technology Birmingham, where he began work on the theory of optimal rocket trajectories.

In 1956 he was appointed Professor of Mathematics at the University of Canterbury, New Zealand. At that time the department had a total staff of just five, including one chair. Appointment to this chair automatically meant being Head of Mathematics, including pure and applied mathematics, and statistics.

Derek’s arrival had an immediate and stimulating effect on the department, through his energy and his refreshing ideas on teaching and research. He reformed and modernized the prescriptions of most of the courses. He was keen to foster a commitment to research and was already strongly involved himself, having published several papers on the optimization of rocket trajectories, particularly in applications to interplanetary travel. He made significant contributions in this field, which had become the subject of intense research in the United States, that country having already committed itself to sending a man to the moon. This resulted in appointments as a consultant to such companies as Boeing and Lockheed, which led to several visits to the USA. These contacts were eventually terminated because of his opposition to the war in Vietnam. All this work culminated in a text-book, ‘Optimal Trajectories for Space Navigation’ published in 1963.

The quality of his research led to several high awards: the Sc.D. degree by Cambridge University (1962), a Fellowship of the Royal Society of New Zealand (1962), the Society’s Hector Medal (1964) and the Mechanics and Control of Flight Award of the American Institute of Aeronautics and Astronautics (1967). He was unable to receive the last of these in person, his visa having been cancelled by the US government.

Derek was certainly inclined towards applied mathematics, rather than to the more abstract topics of pure mathematics. This shows in his choice of subject matter for his books. He believed in the importance of establishing a sound, coherent mathematical framework for any applied topic, based on a careful interpretation of known observational facts. However, he was very ready to make use of advanced results from pure mathematics if need be, such as in the field of differential equations.

Socially Derek was a witty, entertaining and provocative conversationalist and public speaker, often expressing an unconventional viewpoint. For example, early in life he espoused the views of the British Interplanetary Society, at a time when these were frequently derided, and he became a member of the New Zealand Rationalist Association.

He enjoyed playing bridge, and his founding of the University bridge club gave a real lift to the social life of the university.

In his eleven years in New Zealand he contributed much, in various ways. When he announced that he would be returning to England with his family in 1967, the news was received with a feeling of real regret by the mathematical community throughout New Zealand and by many others, especially in Christchurch. He made it clear that he had enjoyed his time in New Zealand, and that it was for family reasons that he was returning to England.

He took up the position of Professor of Mathematical Physics at the University of Aston in Birmingham and was appointed Head of Mathematics in 1977. He retired from this post in 1983 and, after a short spell as a visiting professor at the University of Natal in South Africa, moved to the country, where he wrote a series of books on topics from mathematical physics, several of which have been reprinted as Dover editions.

Apart from mathematics, he had a life-long interest in psychical research, believing that consciousness was the missing link in our scientific understanding of the universe. He had a strong moral sense which, in his early life, drew him towards left-wing idealism. However, he eventually became deeply disillusioned with it because of its toxic practical effects, particularly with what he regarded as the harmful effect of egalitarianism on standards in education. His views on this were influenced by his gratitude for his grammar school education, which he considered had given him his chance in life. Although his views changed, he always expressed them with force and clarity, and scorned the obscurantist, opaque language used in modern academia.

He leaves three sons and a widow.

Bibliography

Books:

* Lawden, D. F. 1954: Mathematics of Engineering Systems, Methuen. 404pp.

* Lawden, D. F. A. 1960: A Course in Applied Mathematics, English Universities Press, London. 655pp.

* Lawden, D. F. 1962: An Introduction to Tensor Calculus and Relativity, Methuen. 186pp.

* Lawden, D. F. 1963: Optimal Trajectories for Space Navigation, Butterworths. 126pp.

* Lawden, D. F. 1967: The Mathematical Principles of Quantum Mechanics, Methuen. 280pp.

* Lawden, D. F. 1972: Analytical Mechanics, Allen & Unwin. 78pp.

* Lawden, D. F. 1973: Electromagnetism, Allen & Unwin. 96pp.

* Lawden, D. F. 1975: Analytical Methods of Optimization, Scottish Academic Press, Edinburgh. 157pp.

* Lawden, D. F. 1985: Elements of Relativity Theory, Wiley. 108pp.

* Lawden, D. F. 1987: Principles of Thermodynamics and Statistical Mechanics, Wiley. 154pp.

* Lawden, D. F. 1989: Elliptic Functions and Applications, Springer-Verlag. 334pp.

Published Papers:

Psychical Research:

* Lawden, D. F. 1979: On a Poltergeist Case. Journal of The Society for Psychical Research 50. No. 780.

* Lawden, D. F. 1989: Some Thoughts on Birth and Death. Journal of Psychophysical Systems, October.

Philosophical Research:

* Lawden, D. F. 1964: Chemical Evolution + Origin of Life. Nature 202: 412

* Lawden, D. F. 1965: The Rise of the Concept of a Physical Model. Transactions of the Royal Society of New Zealand 1: 161–174.

* Lawden, D. F. 1968: Modelling Physical Reality. Philosophical Journal 5, No 2.

* Lawden, D. F. 1969: Are Robots Conscious? New Scientist 43: No 6.

* Lawden, D. F. 1972: Towards a Non-Behavioural Psychology. Philosophical Journal 9: 116–124.

* Lawden, D. F. 1983: Alternate Culture. Nature 301: 9.

Mathematical Research:

* Lawden, D. F. 1951a: A General Theory of Sampling Servo Systems. Institution of Electrical Engineers 98: 31–36.

* Lawden, D. F. 1951b: Entry into Circular Orbits. 1. Journal of the British Interplanetary Society 10, No. 1.

* Lawden, D. F. 1951c: The Function Infinity-igma-n=1nrzn and Associated Polynomials. Proceedings of the Cambridge Philosophical Society 47: 309–314.

* Lawden, D. F. 1952a: Inter-Orbital Transfer of a Rocket. Annual Report of The British Interplanetary Society.

* Lawden, D. F. 1952b: The Determination of Minimal Orbits. Journal of The British Interplanetary Society 11, No. 5.

* Lawden, D. F. 1952c: Orbital Transfer via Tangential Ellipses. Journal of The British Interplanetary Society 11, No. 6.

* Lawden, D. F. 1952d: On the Solution of Linear Difference Equations. Mathematical Gazette 36, No. 317.

* Lawden, D. F. 1953a: Minimal Rocket Trajectories. Journal of the American Rocket Society 23: 360–382.

* Lawden, D. F. 1953b: Escape to Infinity from Circular Orbits. Journal of The British Interplanetary Society 12: No 2.

* Lawden, D. F. 1954a: Entry into Circular Orbits – 2. Journal of The British Interplanetary Society 13, No. 1.

* Lawden, D. F. 1954b: Fundamentals of Space Navigation. Journal of The British Interplanetary Society 13, No. 2.

* Lawden, D. F. 1954c: Correction of Interplanetary Orbits. Journal of The British Interplanetary Society 13, No. 4.

* Lawden, D. F. 1954d: Perturbation Manoeuvres. Journal of The British Interplanetary Society 13, No. 6

* Lawden, D. F. 1954e: Stationary Rocket Trajectories. Quarterly Journal of Mechanics and Applied Mathematics 7: 488–504.

* Lawden, D. F. 1954f: Comment on Satellite Orbits for Interplanetary Flight. Jet Propulsion 24: 382–382.

* Lawden, D. F. 1955a: Dynamic Problems of Interplanetary Flight. Aeronautical Quarterly 6: 165–180.

* Lawden, D. F. 1955b: Optimal Transfer Between Circular Orbits About Two Planets. Astronautica Acta 1, No. 2.

* Lawden, D. F. 1955c: Optimum Launching of a Rocket into an Orbit about the Earth. Astronautica Acta 1, No. 4.

* Lawden, D. F. 1955d: Optimal Programming of Rocket Thrust Direction. Astronautica Acta 1, No. 6.

* Lawden, D. F. 1956a: Maximum Ranges of Intercontinental Missiles. Aeronautical Quarterly 8: 269–278.

* Lawden, D. F. 1956b: Transfer Between Circular Orbits. Jet Propulsion 26: 555–558.

* Lawden, D. F. 1957a: The Simulation of Gravity. Journal of The British Interplanetary Society 6, No. 3.

* Lawden, D. F. 1957b: Mathematical Problems of Astronautics. Mathematical Gazette 41, No. 337.

* Lawden, D. F. 1957c: Optimal Rocket Trajectories. Jet Propulsion 27: 1263–1263.

* Lawden, D. F. 1958a: Optimal Escape from a Circular Orbit. Astronautica Acta 4, No. 3.

* Lawden, D. F. 1958b: Escape from a Circular Orbit Using Tangential Thrust. Jet Propulsion, March.

* Lawden, D. F. 1958c: The Employment of Aerodynamic Forces to obtain Maximum Range of a Rocket Missile. Aeronautical Quarterly 9: 97–109.

* Lawden, D. F. 1959a: Discontinuous Solutions of Variational Problems. Journal of the Australian Mathematical Society 1, No 1.

* Lawden, D. F. 1959b: Necessary Conditions for Optimal Rocket Trajectories. Quarterly Journal of Mechanics and Applied Mathematics 12, No. 4.

* Lawden, D. F. 1960a: Optimal Transfer Between Circular Orbits About Two Planets. Astronautica Acta 1, No. 2.

* Lawden, D. F.; Long, R. S. 1960: The Theory of Correctional Manoeuvres in Interplanetary Space. Astronautica Acta 6, No. 1.

* Lawden, D. F. 1960b: Optimal Programme for Correctional Manoeuvres. Astronautica Acta 6, No. 4.

* Lawden, D. F. 1961a: Optimal Powered Arcs in an Inverse Square Law Field. American Rocket Society 31: 566–568.

* Lawden, D. F. 1961b: Optimal Intermediate-Thrust Arcs in a Gravitational Field. Astronautica Acta 8, No. 2–3.

* Lawden, D. F. 1963: Analytical Techniques for the Optimization of Rocket Trajectories. Aeronautical Quarterly 14: 105–124.

* Lawden, D. F. 1965: Celestial Mechanics and Astrodynamics: Journal of the Royal Aeronautical Society 69.

* Lawden, D. F. 1968: Coordinate and Momentum Representations in Quantum Mechanics. Journal of the Australian Mathematical Society 8, No. 2.

* Lawden, D. F. 1969: Electro-Chemical Dissolution of an Anode. Journal of the Australian Mathematical Society 10, No. 34.

* Lawden, D. F. 1970: The Phenomenon of Time Dilation. Spaceflight. 12: 178–179.

* Lawden, D. F. 1971: The Phenomenon of Time Dilation. Spaceflight. 13.

* Lawden, D. F. 1984: Pseudo-Random Sequence Loops. Mathematical Gazette 68: 39–41.

* Lawden, D. F. 1991: Rocket Trajectory Optimization: 1950-1963. Journal of Guidance, Control, and Dynamics 14: 705–711.

* Lawden, D. F. 1992a: Optimal Transfers Between Coplanar Elliptic Orbits. Journal of Guidance, Control, and Dynamics 15: 788–791.

* Lawden, D. F. 1992b: Calculation of Singular Extremal Rocket Trajectories. Journal of Guidance, Control, and Dynamics 15: 1361–1365.

* Lawden, D. F. 1993: Time-Closed Optimal Transfer by 2 Impulses Between Coplanar Elliptic Orbits. Journal of Guidance, Control, and Dynamics 16: 585–587.

* Lawden, D. F. 1999: Families of ovals and their orthogonal trajectories. Mathematical Gazette 83, No. 498.

* Lawden, D. F. 2000: Touching hyperspheres. Mathematical Gazette 84, No. 499.

Michael Lawden

Senior Scientific Officer (Retired)

Starlink Project

Rutherford Appleton Laboratory

UK

Robert Long

Reader in Mathematics (Retired)

University of Canterbury

Christchurch

New Zealand