Quirky quantum concepts. Physical, conceptual, geometric, and pictorial physics that didn't fit in your textbook (Q2637866)

This book aims at helping students of Quantum Mechanics (QM) learn the subject by providing explanations that, in the author's opinion, generally lack in standard texts of this level (``When standard presentations do not make sense, come here.'' (p.~viii)). The book is meant to be read in parallel with the texts listed in the references, and is therefore not self-contained; coverage is selective (``almost no relativistic QM'' (p.~ix)). It is meant to be read, after the first two chapters, ``in any order that suits you.'' These two chapters respectively deal with ``basic wave mechanics concepts'' and with the Hilbert space formalism in ``bra-ket'' notation. Chapter 3 deals with scattering. Chapter 4 is devoted to ``matrix mechanics;'' the title of this chapter does not refer to the early form of QM, but to ``the QM of systems which can be represented by finite dimensional vectors and matrices'' (p.~159). Chapter 5 deals with angular momentum, Chapter 6 with``multiparticle'' QM, and Chapter 7 develops some aspects of ``quantum electromagnetic radiation.'' The last chapter discusses ``desultory topics,'' such as whether \(\nabla^2\) is the square of an operator (oddly enough, spinors are not mentioned in this connection). The appendix contains a list of formulae with page references to several texts, a glossary, 28 references (mostly textbooks) and an index. Here is an example of the author's explanations: ``A wave-function is some unknown `thing' that oscillates. That thing can be thought of as a `quantum field', similar to an electric or magnetic field, but with some important differences'' (p.~8). The two stumbling blocks that the author stresses are the counter-intuitive aspects of the predictions of QM, and its mathematical aspects. The work claims not to indulge in `` `popularizations' or oversimplifications'', adding that ``[w]hen mathematical or physical words have precise meanings, we adhere to those meanings'' (p.~vii). Unfortunately, this is not the case. Here are two typical examples. (1) The glossary defines ``continuous'' as follows: ``Having the property that between any two elements there are an infinite number of other elements, e.g., real numbers are continuous.'' With this definition, some countable sets, such as the rationals are also ``continuous''. The standard meaning of the word is not mentioned. Now, the title of section 2.4.4 (p.~77) is ``Continuous (or Uncountable) Basis Sets.'' It is not clear whether the author means such sets to be necessarily uncountable. On p.~81, we further read that ``Countable or Uncountable does not mean much,'' because the same Hilbert space may have a ``continuous basis set'' as well as a ``countably infinite basis set''. The notion of separable Hilbert space is not introduced. (2) We read on p.~119 that ``A particle in an angular momentum eigenstate is also stationary, but `moving': it is revolving around the center.'' By contrast, we are told p.~242 that ``the Bohr model of the atom is completely wrong, and probably should not be taught any more.'' It is not clear what meaning is assigned here to the word ``revolving'': are quantum ``particles'' supposed to be associated with classical trajectories, or should on the contrary such classical pictures be carefully avoided in teaching? Two errors of a different nature should be mentioned. We again give two typical examples. First, we read on p.~xi: {\parindent=1em\narrower There is not always a clear mathematical distinction between \(d/dx\) and \(\partial/\partial x\). When the function arguments are independent, they are both the same thing. [\dots] However, in some cases, it's not clear what arguments a function has, and it is not important. [\dots] Also, for the record, derivatives \textit{are} fractions, despite what you might have been told in calculus. [\dots] All of physics treats them like fractions, [\dots] because they \textit{are} fractions.'' } No reference is given for these surprising statements. It seems worthwhile to point out that Physics, far from justifying them, provides strong arguments against them. Contemporary usage does not seem to leave room for ambiguity: \(d/dx\) refers to the derivative of a function of \(x\) alone, while \(\partial/\partial x\) refers to the partial derivative, with respect to \(x\), of a function that also depends on other variables, that are kept fixed. Especially in Thermodynamics, it is customary to list as subscripts those of the variables that are fixed, as in \((\partial U/\partial S)_V\). The total derivative -- that the author seems to have in mind -- is a special case of the derivative of a function of a single variable; thus, in Mechanics, if \(F=F(p,q,t)\), \({d\over dt}F(p(t),q(t),t)\) is merely the derivative of the function of one variable \(t\mapsto F(p(t),q(t),t)\). Many branches of Physics also require the consideration of total \textit{differentials} (not considered by the author), such as \(dU=TdS-pdV\), where \(T\) is of course not the quotient of the differentials \(dU\) and \(dS\). Regarding the identification of derivatives with fractions, while it is true that computing directly with infinitesimal increments of variables is extremely useful, students of Thermodynamics know well that derivatives do not behave like fractions, even formally, as soon as there is more than one independent variable: the classical example is provided by the relation between the thermoelastic coefficients; in mathematical terms, if three variables \(x\), \(y\) and \(z\) satisfy a relation \(f(x,y,z)=0\), such that the implicit function theorem may be applied to solve for \(x\), \(y\) or \(z\) in terms of the other two variables; one has \[ \bigg({\partial x\over\partial y}\bigg)_z\bigg({\partial y\over\partial z}\bigg)_x\bigg({\partial z\over\partial x}\bigg)_y=-1, \] and not \(+1\). This stems from relations such as \(({\partial x/\partial y})_z=-({\partial f/\partial y})/({\partial f/\partial x})\). In all of these relations, operating on derivatives as if they were fractions would lead to an incorrect result. The importance of Thermodynamics and other branches of Physics in the development of calculus in several variables cannot be overrated. The second example refers to the experimental confirmation of de Broglie's suggestion (Thesis, 1924) that material particles may be associated with waves, and illustrates the mistaken notion that experiments provide ``facts'' that are independent of any theoretical framework. On p.~6, we read that ``[The de Broglie relation] was demonstrated for electrons serendipitously by Davisson and Germer in 1927, after their equipment failed.'' This suggests that Davisson and Germer immediately interpreted their results as evidence of electron diffraction. This is not what happened. In 1925, measures taken to repair their apparatus had led to patterns they did not expect; but their research program ``was not, at its inception, a test of the wave theory. Only in the summer of 1926, after [Davisson] had discussed the investigation in England with Richardson, Born, Franck and others, did it take on this character'' (Davisson, Nobel Lecture). In other words, ``[o]nly in late 1926 did [Davisson and Germer] understand what was going on'' [Weinert, in Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, D. Greenberger at al.\ (eds.) Springer, 2009, p.~150]. Far from showing the serendipitous character of Davisson and Germer's achievement, the actual sequence of events illustrates that a given experiment may receive several interpretations depending on one's theoretical framework. This neglect of the intermingling of theory and experiment affects the entire orientation of the book. Thus, this work, by its lack of consistency, and its exaggerated or incorrect statements, is likely to be confusing to its readers.