# Commentaries on the Mathematical Principles of Quantum Mechanics

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.

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