Abstract
The first chapter presents the physical and mathematical fundamentals of quantum mechanics that provide the basis for the philosophical considerations in the subsequent chapters.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Quantum physics, like classical physics, encompasses more than just mechanics, in particular also quantum field theory. Insofar as these other subfields are included, we will use the term “quantum physics”. As a rule, however, this introductory chapter will deal with quantum mechanics.
- 2.
That is, treating such problems as the relationship of a single object to its properties, or of a whole to its parts, or the relation between cause and effect and the questionable nature of persistence and temporal change in quantum systems.
- 3.
The momentum of an object is its velocity multiplied by its mass.
- 4.
Spin is an angular momentum which can be intuitively imagined as a rotation of the object around itself (which indeed for point-like objects becomes highly non-intuitive).
- 5.
A note for readers with a physics background: Mathematically, the spin should also behave as an angular momentum; that is, its operators should obey the same commutation relations as those for the orbital angular momentum. That it behaves not only analogously to an angular momentum, but indeed is one, is shown by the fact that, in general, the quantum-mechanical orbital angular momentum is a conserved quantity only when it is coupled together with the spin.
- 6.
This is of course only possible if one does not collect the particles on a screen.
- 7.
Here and in the following, we use the word “measurement” with as few preconceptions as possible. We neither presume that a macroscopic detector (screen) has to register events irreversibly, nor even that the quantum-mechanical system in some way “collapses”.
- 8.
But how do the particles “exhibit” a particular spin value if they are not registered by a detector (e. g. a screen), showing that they were deflected upwards or downwards?—Those who are already familiar with this problem may well ask this question. We, however, follow the particle in thought along its “trajectories” and show that this leads inevitably to inconsistencies.
- 9.
At least in the sense that we, in thought, can follow the “paths” of the particles along which it is supposed to have certain spin values in the x direction.
- 10.
In physics, superpositions are also well known from the classical field theory in the addition of wave phenomena. Historically, the superposition principle was therefore introduced into quantum mechanics in order to describe the wave character of the particles. Here, however, it serves as a very abstract principle: Spin states are not waves.
- 11.
The concept of “states” will be discussed in more detail in the various sections on the interpretation of quantum mechanics. Intuitively, it somehow maps the set of properties which a quantum system possesses at a particular moment.
- 12.
One more comment: We have chosen the more abstract algebraic approach to quantum mechanics which, in particular, is suited for generalization to QFT. It is computationally less demanding; for example, no differential equations must be solved. For the (perhaps more intuitive, but computationally more difficult) calculus approach, we suggest that students of philosophy consult the book by Nortmann (2008).
- 13.
Complex numbers are an extension of the real numbers. The idea here is that quadratic equations can always be made solvable, i. e. even for negative numbers, by setting the imaginary quantity i as the solution to \(x^2 = -1\) (this simplifies many computations). With the real numbers a and b, complex numbers in general take on the form \(a + ib\), thus with a real part and an imaginary part, and they can be graphically represented in a plane, where one coordinate axis is taken to be imaginary.
- 14.
Instead of \(\vec {c}\), it is convenient in quantum mechanics to write \(|C\rangle \) for a state vector. The dual vector is written as \(\langle C|\), so that, as we shall soon see, the inner product of two vectors is expressed simply as \(\langle A|B\rangle \) (bra-ket notation due to P.A.M. Dirac).
- 15.
By addition and extension/contraction, one always obtains new vectors which are also elements of the same space as the original vectors. The superposition of two possible states of a particular single physical system is likewise always an additional possible state of the same system. For many particles of similar type, however, there is an important limitation (keyword: superselection rules; cf. Chap. 3).
- 16.
A set of vectors forms a basis of a vector space when all the other vectors in the space can be generated as linear combinations of them. We consider only orthonormal bases. It is important to note that a given vector space has an infinite number of such bases.
- 17.
In the extreme case, this dimensionality can be (countably) infinite.
- 18.
One can readily verify that the basis vectors given above are indeed normalized to length \(=\) 1 and are orthogonal (mutually perpendicular) to each other, i. e. orthonormal.
- 19.
In every interpretation of the formalism, parallel vectors correspond to physically indistinguishable states, so that linear operations bring no physical differences into play where none were present before.
- 20.
The complex conjugate of \(z = a + ib\) is \(z^* = a - ib\), and \(z^2 = z^*z\).
- 21.
For single systems, we can say that every self-adjoint operator represents some sort of quantum-mechanical observable quantity such as spin, energy, even if it is not always easy to identify the concrete physical realization corresponding to a given mathematical operator. For many-particle systems, however, not every self-adjoint operator corresponds to an observable quantity (keyword: superselection rules; cf. Chap. 3).
- 22.
The following calculation is intended only to be illustrative. There is a standard mathematical procedure for calculating the eigenvalues and eigenvectors of self-adjoint matrices. More details on the mathematics of physics can be found in Räsch (2011), for those who lack experience in this area.
- 23.
For electrons, the spin projection values are of course \(\frac{1}{2}\) and \(-\frac{1}{2}\), but numerical details are irrelevant here.
- 24.
Caution: Normally, each eigenvector has exactly one eigenvalue; while conversely, each eigenvalue belongs to exactly one eigenvector only when it (the eigenvalue) is simple, i. e. it occurs only once—which is in fact the case here. The problem of multiple eigenvalues will be treated in the following Sect. 1.2.3.
- 25.
Also, this holds for many-particle systems only with some limitations.
- 26.
Note that two operators can only be different if they have some different eigenvectors, because an operator is completely characterized by its eigenvectors. In order to have a common basis of eigenvectors they thus must have (at least) N eigenvectors in common and must have more than N eigenvectors in total, from which (at least) some are different. For more details see the following section.
- 27.
In the two-dimensional case, this statement is not very rich in content. We will see, however, that there are very interesting examples of this principle in higher-dimensional vector spaces.
- 28.
Casting a glance backwards: Sequential spin measurements are mathematically represented by successive application of (self-adjoint or Hermitian) operators.
- 29.
Consider, for example, two successive rotations in space around two different rotational axes.
- 30.
Note that the resulting matrices in this example are not Hermitian; i. e., they do not represent observable quantities.
- 31.
Therefore, in the following, we will speak simply of (non-)commuting operators, when the matrices which represent them (do not) commute.
- 32.
Note that the commutation relation is not transitive: It can happen that A in fact commutes with B, and B with C, but not necessarily A with C.
- 33.
In quantum mechanics, the unitary vector space of at most countably infinite dimensionality which underlies the computations is denoted as the “Hilbert space”.
- 34.
The expectation value of an operator is mathematically derived from the usual mean value. Geometrically, the vector that results from the application of \(\hat{O}\) to \(|A\rangle \) is projected back onto \(|A\rangle \) Physically, this corresponds (uncontroversially) to just the mean value of numerous measurements of the observable quantity associated with \(\hat{O}\).
- 35.
The variance (here: the standard deviation) is calculated from \(\triangle \hat{O} = \langle A|\hat{O^2}|A\rangle - |\langle A|\hat{O}|A\rangle |^2\), which here yields zero.
- 36.
This representation is not unique; there are infinitely many such different representations.
- 37.
This has as its consequence that in every physical interpretation of the formalism, two vectors which differ only in their lengths correspond to physically indistinguishable states.
- 38.
As already stated, this last formulation is indeed standard, but nevertheless subject to different interpretations.
- 39.
If we wished to take these additional variables into account at the same time, we would have to base our calculations on a space with still more dimensions than the four-dimensional vector space considered here.
- 40.
Energy and spin values are left out of consideration here.
- 41.
Keep complex conjugation in mind!
- 42.
This property should not be confused with the requirement of repeatability of a measurement.
- 43.
Note that, as always, the vector \(|\Psi \rangle \) is normalized to unit length, so that for its components, we find \(\sum _{i} |c_i|^2 = 1\).
- 44.
“One-dimensional” refers to projection operators which project onto just one (normalized) vector, whose eigenvalue 1 is thus single.
- 45.
The “trace” (symbol: Tr) of a matrix is the sum of its diagonal elements. It is invariant under basis transformations, so that with a basis composed of eigenvectors, the trace is just the sum of the eigenvalues.
- 46.
With Eq. 1.24, we have \(\hat{O} = \sum _{i} \lambda _{i}|\Psi _{i}\rangle \langle \Psi _{i}|\).
- 47.
For a discussion of these problems, see in particular van Fraassen (1991) (pp. 157ff. and pp. 206/7).
- 48.
In the bra-ket notation, \(\hat{O}^{*}\) is always the operator which “acts” to the left, so that for unitary operators, we have: \(\langle \Psi |\hat{U}^{*}\hat{U}|\Psi \rangle = \langle \Psi |\Psi \rangle \) – as desired.
- 49.
The complex exponential function is periodic, so that \(\hat{U}\) is analogous to a rotation matrix. \(\hbar \) is the universal quantum-mechanical constant, the (reduced) Planck’s constant .
- 50.
In non-standard interpretations of quantum mechanics, however, one disagrees: Thus, for example, in the GRW variation, the spatial behaviour is eminently important, and in Bohm’s mechanics, particles (again) move along trajectories.
- 51.
Strictly speaking, the position and the momentum operators of single particles naturally each have three components (which mutually commute). We neglect this fact here and in the following.
- 52.
Since the time of Cantor, one distinguishes (at least) two types of infinite sets: those whose elements can be counted, which are thus no larger than the infinite set of the natural numbers or the set of the rational numbers; and those which are no longer countable, which thus appear to be greater, of higher cardinality, such as the real numbers. A continuum then forms a set of points which are more than countably infinite, and are in addition dense. More details can be found in the appropriate textbooks on calculus.
- 53.
In physics, such quantities are called “\(\delta \) functions”.
- 54.
Note that this “position space” corresponds to the ordinary three-dimensional intuitive space only for a single-particle system. In many-particle systems, we must in contrast operate in a 3N-dimensional configuration space , corresponding to the particle number N; and it is no longer intuitively comprehensible. But our conclusion remains (for the time being): Only the eigenvalues (here: of the position operator) correspond to real observable values or to real properties of real quantum-physical systems.
- 55.
In the position representation, the eigenvectors of the momentum operator are found (after a short computation) to be plane waves , which could correspond in an intuitive manner to a complete spatial delocalization of the particle accompanying a precise value for its momentum.
- 56.
In the position representation, the commutation relation is found from \([x,-i\frac{d}{dx}]f(x) = -i(xf'(x)) - \frac{d}{dx}(xf(x)) = if(x)\).
- 57.
It holds furthermore that \(\hat{H} = \hat{E}_{kin} + \hat{E}_{pot}\), where V(x) is a potential that depends only on the position, such as the Coulomb potential in a hydrogen atom (and m is the mass of the particle).
References
Albert, David Z. 1992. Quantum Mechanics and Experience. Cambridge: Harvard University Press.
Aristotle. 1988. Physics. Books V–VIII, English translation.
van Fraassen, Bas C. 1991. Quantum Mechanics. An Empiricist View. Oxford: Clarendon Press.
Kant, Immanuel (1781/87). Kritik der reinen Vernunft. Cited according to the A- and B- editions. English translation: Critique of Pure Reason (available online at various sites).
Mellor, Hugh D. 1998. Real Time II. London: Routledge.
Nortmann, Ulrich. 2008. Unscharfe Welt? Was Philosophen über Quantenmechanik wissen möchten. Darmstadt: Wissenschaftliche Buchgesellschaft.
von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer (Reprint 1968). English translation (1996): Mathematical Foundations of Quantum Mechanics. Princeton Landmarks in Mathematics.
Räsch, Thoralf (2011). Mathematik der Physik für Dummies. Weinheim: Wiley-VCH. English: See e. g. the “For Dummies” series, http://physicsdatabase.com/2012/03/08/for-dummies-series-for-maths/.
Author information
Authors and Affiliations
Exercises
Exercises
-
1.
Niels Bohr introduced the concept of “complementarity” into the interpretation of quantum mechanics. Distinguish two readings of how it is to be understood.
-
2.
In sequential spin measurements, we apparently distinguished finally between two effects which could be reversed by mixing the particles. Describe these two ostensible effects and explain why they are in fact only a single effect. What can we deduce from this?
-
3.
Consider the expectation values of operators in regard to whether the physical system is represented by an eigenvector of the given operator or not. Compute the expectation values of the spin operators discussed earlier, relative to the various vectors described there. Explain the results by referring to the figures showing a repeated and a destructive measurement.
-
4.
What does von Neumann’s projection postulate state? Explain in particular to what extent this postulate goes beyond what we have considered to be well established in connection with expectation values.
-
5.
In contrast to the general opinion of many philosophical schools of thought, and also of some alternative physical interpretations (e. g. GRW, Bohm), (intuitive or physical) space is not at the centre of standard quantum mechanics. Discuss this hypothesis, first informally and then by referring to the particular mathematical characteristics of the position operator.
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Friebe, C. (2018). Physical and Mathematical Foundations. In: The Philosophy of Quantum Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78356-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-78356-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-78354-3
Online ISBN: 978-3-319-78356-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)