Notes on Quantum Mechanics. John R. Boccio Professor of Physics Swarthmore College

Transcription

1 Notes on Quantum Mechanics John R. Boccio Professor of Physics Swarthmore College Septemer 14, 2012

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3 Contents 1 Aout Projectors and Measurement Introduction System at One Time Phase Space and Hilert Space Frameworks Again with More Details Classical and Quantum Properties Toy Model Toy Model and Spin 1/ Continuous Quantum Systems Negation Properties(NOT) Conjunction and Disjunction (AND, OR) Incompatile Properties Proailities and Physical Variales Quantum sample space and event algera Refinement, coarsening, and compatiility Proailities and ensemles Random variales and physical variales Averages Composite Systems and Tensor Products Introduction Definition of tensor products Examples of composite quantum systems Product operators General operators, matrix elements, partial traces Product properties and product of sample spaces Unitary Dynamics Example #1: Toy Model - Particle and Detector Example #2: Toy Model- Alpha Decay Dependent(contextual) events An Example Classical analogy Contextual properties and conditional proailities Dependent events i

4 ii CONTENTS 1.8 Preparation and Measurement Classical Limit Quasiclassical frameworks

5 Chapter 1 Aout Projectors and Measurement 1.1 Introduction Scientific advances can significantly change our view what the world is like, and one of the tasks of the philosophy of science is to take successful theories and tease out of them their roader implications for the nature of reality. Quantum mechanics, one of the most significant advances of twentieth century physics, is an ovious candidate for this task, ut up till now efforts to understand its roader implications have een less successful than might have een hoped. The interpretation of quantum theory found in textooks, which comes as close as anything to defining standard quantum mechanics, is widely regarded as quite unsatisfactory. Among philosophers of science this opinion is almost universal, and among practicing physicists it is widespread. It is ut a slight exaggeration to say that the only physicists who are content with quantum theory as found in current textooks are those who have never given the matter much thought, or at least have never had to teach the introductory course to questioning students who have not yet learned to shut up and calculate! On all sides it is acknowledged that the major difficulty is the quantum measurement prolem. Significantly, it occupies the very last chapter of von Neumanns Mathematical Foundations of Quantum Mechanics, and forms what many regard as the least satisfactory feature of this monumental work, the great-grandfather of current textooks. The difficulties in the way of using measurement as a fundamental component of quantum theory were summed up y Wigner in The Prolem of Measurement - AJP(1963), and confirmed y much later work; see, e.g., the careful analysis y Mittelstaedt in The Interpretation of Quantum Mechanics and the Measurement Process. A more recent review y Wallace The Quantum Measurement Prolem: State of Play(2007) testifies oth to the continuing centrality of the measurement prolem for the philosophy of quantum mechanics, and to the continued lack of progress in resolving it; all that has changed is the numer and variety of unsatisfactory solutions. 1

6 Actually there are two distinct measurement prolems. The first measurement prolem, widely studied in quantum foundations, comes aout ecause if the time development of the measurement apparatus (and its environment, etc.) is treated quantum mechanically y integrating Schrodingers equation, the result will typically e a macroscopic quantum superposition or Schrodinger cat in which the apparatus pointer we shall continue to use this outdated ut picturesque languagedoes not have a definite position, so the experiment has no definite outcome. Contrary to the elief of experimental physicists. If this first prolem can e solved y getting the wiggling pointer to collapse down into some particular direction, one arrives at the second measurement prolem: how is the pointer position related to the earlier microscopic situation which the apparatus was designed to measure, and which the experimental physicist elieves actually caused the pointer to point this way and not that way? When one hears experimental particle physicists give talks, it sounds as if they elieve their detectors are triggered y the passage of real particles zipping along at enormous speed and producing electrical pulses y ionizing the matter through which they pass. No mention of the sudden collapse of (the modern electronic counterpart of) the apparatus pointer at the end of the measurement process. Instead, these physicists elieve not only that the apparatus is in a well-defined macroscopic stateeach it in the memory is either 0 or 1after the measurement, ut in addition that this outcome is well correlated with a prior state of affairs: one can e quite confident that a negative muon was moving along some specified path at a particular moment in time. Have they forgotten what they learned in their first course in quantum theory? Here we must note that laoratory measurements in particle physics are, in the vast majority of cases, not appropriately modeled y the scheme proposed y von Neumann in which a seemingly aritrary wave function collapse leads to a correlation etween the measurement outcome and a property of the measured system at a time after the measurement has taken place. In a typical particle physics experiment the particle will either have disintegrated, escaped from the apparatus, or een asored long efore the measuring apparatus has registered its ehavior, and in any case the experimentalist is interested in what was going on efore detection rather than afterwards. The asence of a satisfactory solution to this second measurement prolem has led to the development of a certain quantum orthodoxy which affirms that the only task of quantum theory is to relate the outcome of a macroscopic measurement to an earlier and equally macroscopic (i.e., descriale in ordinary language) preparation. The quantum wave function ψ(t) associated with the prepared system, regarded as isolated from its environment until it interacts with the measuring apparatus does not have any ontological reference (ontology deals with questions concerning what entities exist or can e said to exist, and how such entities can e grouped, related within a hierarchy, and sudivided according to similarities and differences); it is simply a mathematical device for 2

7 calculating the proailities of measurement outcomes in terms of the earlier preparation procedure. Stated slightly differently, it provides an astract representation of the preparation process, or at least as much of, or that part of, the preparation process which is needed to calculate proailities of outcomes of future measurements. In this approach preparation and measurement can e thought of as part of the real world, ut what happens in etween occurs inside a lack ox which cannot e opened for further analysis and is completely outside the purview of quantum theoryalthough some future theory, as yet unknown, might allow a more precise description. One can e sympathetic with strict orthodoxy in that it is intended to keep the unwary out of troule. Careless thinkers who dare open the lack ox will fall into the quantum foundations Swamp, where they risk eing consumed y the Great Smoky Dragon, driven insane y the Paradoxes, or allured y the siren call of Passion at a Distance into suservience to Nonlocal Influences. Young scientists and philosophers who do not heed the admonitions of their elders will, like the children in one of Grimms fairy tales, have to learn the truth y itter experience. The measurement prolems and the associated lack of a clear conceptual foundation for quantum theory have not only een a stumling lock in quantum foundations work. They have also slowed down, though fortunately not stopped, mainstream physics research. In older fields such as scattering theory, the pioneers spent a significant amount of time working through conceptual issues. But once the accepted formulas are in place and yield results consistent with experiment, their intellectual descendents have the luxury of calculating without having to rethink the issues which confused their predecessors. In fields in which quantum techniques are applied in fresh ways to new prolems, such as quantum information, conceptual issues that have not een resolved give rise to confusion and wasted time. Both students and researchers would enefit from having the rather formal approach to measurements found in, e.g., Nielsen and Chuang (Quantum Computation and Quantum Information), to mention one of the est known ooks on the suject, replaced y something which is clearer and more closely tied to the physical intuition needed to guide good research, even in a field heavily larded with mathematical formalism. Black oxes can e a useful approach to a prolem, ut can also stand in the way of a good physical understanding. 3

8 1.2 System at One Time Phase Space and Hilert Space Starting with classical phase space and classical dynamics, what changes are needed to arrive at the corresponding concepts for quantum theory? The phase space (position x, momentum p) of a particle moving in one dimension is shown in part (a) of the figure elow, while part () is a somewhat schematic representation of a two-dimensional (complex) Hilert space(we follow the convention of modern quantum information theory, where the term Hilert space is not restricted to infinite- dimensional spaces, ut also includes any finite-dimensional complex linear vector space equipped with an inner product. By restricting the following discussion to finite-dimensional Hilert spaces we avoid various technical issues that are not pertinent to the conceptual issues we are concerned with), the closest quantum counterpart to phase space, for the simplest of quantum systems: the spin angular momentum of a spin-half particle (a single quit in the jargon of quantum information theory). Figure 1.1: (a) Classical phase space; () Hilert space Following von Neumann we assume that a single point in classical phase space corresponds to a ray or one-dimensional suspace in the Hilert space: all multiples of a nonzero ψ y an aritrary (complex) numer. A ray is a single line through the origin in part () of the figure. In addition, the counterpart of a classical property P, such as the energy is etween 1.9 and 2.0 Joules, represented y some (measurale) collection of points P in the classical phase space, is a quantum property represented y a (closed) suspace of the quantum Hilert space. The classical property can e represented y an indicator function P (γ), where γ denotes a point in the classical phase space, and P (γ) is 1 at all points where the property holds or is true, and 0 at all other points. The quantum counterpart of an indicator function is a projector, a Hermitian operator on the Hilert space that is equal to its square, so its eigenvalues are 4

9 0 and 1. It projects onto the suspace corresponding to the quantum property; any ket in this suspace is an eigenvector with eigenvalue 1. The negation P, NOT P, of a classical property corresponds to the set-theoretic complement P c of P, with indicator I P ; I is the function whose value is 1 everywhere on phase space. Following von Neumann we assume that the negation of a quantum property is the orthogonal complement P of the corresponding suspace P, with projector I P, where I is the identity operator of the Hilert space. Von Neumanns proposal for associating the negation of a quantum property with the complementary suspace is a very important idea, and hence it is appropriate to discuss why this is reasonale and to e preferred to the alternative of letting the negation of a property consist of the set-theoretic complement of the ray or suspace. First, the von Neumann approach has een widely accepted and is not contested y physicists, though it is not always accepted in the quantum foundations community. (We are dealing with an example of the eigenvalue-eigenvector link.) It is consistent with the way students are taught in textooks to make calculations, even though the idea itself is (alas) not always included in textook discussions. Second, it is mathematically natural in that it makes use of a central property, the inner product, of the Hilert space, which is what distinguishes such a space from an ordinary complex linear vector space. (The inner product defines what one means y orthogonal.) Third, the orthogonal complement of a suspace is a suspace, whereas the set-theoretic complement is not a suspace; see, e.g., part () of the figure aove. Fourth, in the case of a spin-half particle the von Neumann proposal asserts that the complement or negation of the property S z = +1/2 (in units of ) is S z =?1/2, which is to say one of these properties is the negation of the other, in accord with the Stern-Gerlach experimental result, which showed, somewhat to the surprise of the physics community at the time, that silver atoms passing through a magnetic field gradient come out in two distinct eams, not the infinite numer which would have een expected for a classical spinning particle or might still e expected for a spin-half quantum particle, were one to assume that all the other rays in the two-dimensional Hilert space represent alternative possiilities to Sz = +1/2. However, the Stern-Gerlach result is in complete accord with von Neumanns approach: the negation of the property corresponding to a particular ray in a two-dimensional Hilert space is the unique property defined y the orthogonal ray, and a measurement determines which of these two properties is correct in a given case. Fifth there is a sense in which all of chemistry is ased on the idea that the electron, a spin-half particle, has only two possile spin states: one up and one down. Granted, students of chemistry find this confusing, since it is not clear how up and down are to e defined, though they eventually are ludgeoned into shutting up and calculating. Among the calculations which lead to the wrong 5

10 answer if 2 is replaced y some other numer is that of finding the entropy of a partially ionized gas, where it is important to take into account all degrees of freedom, and log 2 is the correct contriution from the electron spin. In addition, the modern theory of quantum information is consistent with the idea that a single quit with its two-dimensional Hilert space can contain at most one it of information. It is important to stress this point, for if one accepts von Neumanns negation a wide gap opens etween classical phase space and quantum Hilert space, one clearly visile in the aove figure. Any point in classical phase space, any possile mechanical state of the system, is either inside or outside the set that defines a physical property, which is therefore either true or false. In the quantum case there are vast numers of rays in the Hilert space that lie neither inside the ray or, more generally in higher dimensions, the suspace corresponding to a property, nor in the complementary suspace that corresponds to its negation. For these, the property seems to e neither true nor false, ut somehow undefined. How are we to think aout this situation? This is in a sense the central issue for any ontology that uses the quantum Hilert space as the physicists fundamental mathematical tool for representing the quantum world. One approach, typical of textooks, is to ignore the prolem and instead take refuge in measurements. But prolems do not go away y simply eing ignored, and ignoring this particular prolem makes it re-emerge under a different name: the measurement prolem. Before going further, let us restate the matter in the language of indicator functions and projectors introduced earlier. The product of two (classical) indicator functions P and Q is itself an indicator function for the property P AND Q, the intersection of the two sets of points P and Q in the phase spaces of the earlier figure. However, the product of two (quantum) projectors P and Q is itself a projector if and only if PQ = QP, that is if the projectors commute. Otherwise, when PQ is not equal to QP, neither product is a projector, and thus neither can correspond to a quantum property. A ray Q in a two-dimensional Hilert space, part () of the earlier figure, that coincides neither with a ray P nor its orthogonal complement P is represented y a projector Q that does not commute with P, and thus it is unclear how to define the conjunction of the corresponding properties. Von Neumann was not unaware of this prolem, and he and Birkhoff had an idea of how to deal with it. Instead of using the product of two noncommuting projectors it seemed to them sensile to employ the set theoretic intersection of the corresponding suspaces of the Hilert space, which is itself a suspace and thus corresponds to some property, as a sort of quantum counterpart of AND. The corresponding OR can then e associated with the direct sum of the two suspaces. When projectors for the two suspaces commute these geometrical constructions lead to spaces whose quantum projectors, PQ and P + Q?PQ, coincide precisely with what one would expect ased on the analogy with clas- 6

11 sical indicators. The resulting structure, known as quantum logic, oeys some ut not all of the rules of ordinary propositional logic, as Birkhoff and von Neumann pointed out. If one naively goes ahead and applies the usual rules of reasoning with these definitions of AND and OR it is easy to construct a contradiction. While quantum logic has een fairly extensively studied it seems fair to say that this route has not made much if any progress in resolving the conceptual difficulties of quantum theory. It is not mentioned in most textooks, and is given negligile space in papers. Perhaps the difficulty is that physicists are simply not smart enough, and the resolution of quantum mysteries y this route will have to await the construction of superintelligent roots. (But if the roots succeed, will they e ale to, or even interested in, explaining it to us?) von Neumann lets suspaces represent properties, with the negation of a property represented y the orthogonal complement of the suspace. What aout the case of properties P and Q represented y noncommuting projectors? Since neither P Q nor QP is a projector, let us not try and talk aout their conjunction. Let us adopt a language for quantum theory in which P Q makes sense and represents the conjunction of the two properties in those cases in which QP = P Q. But if P Q QP the statement P AND Q is meaningless, not in a pejorative sense ut in the precise sense that this interpretation of quantum mechanics can assign it no meaning. In much the same way that in ordinary logic the proposition P Q, P AND; OR Q, is meaningless even if P and Q make sense: the comination P Q has not een put together using the rules for constructing meaningful statements. Likewise, in quantum mechanics oth the conjunction and also the disjunction P OR Q, must e excluded from meaningful discussion if the projectors do not commute. The two-dimensional Hilert space of spin-half particle can serve as an illustration. We let z + z + and z z e projectors for the properties S z = +1/2 and S z = 1/2, respectively; similarly x + x + and x x are the corresponding properties for S x. The product of z + z + and z z is zero (i.e., the zero operator) in either order, so they commute and their conjunction is meaningful: it is the quantum property that is always false (the counterpart of the empty set in the case of a phase space). However, neither z + z + nor z z commutes with either x + x + or x x, so the conjunction S x = +1/2 AND S z = +1/2 is meaningless. In support of the claim that it lacks meaning one can note that every ray in the spin-half Hilert space has the interpretation S w = +1/2 for some direction in space w. Thus thus there are no spare rays availale to represent S x = +1/2 AND S z = +1/2; there is no room in the Hilert space for such conjunctions. It is very important to distinguish meaningless from false. In ordinary logic if a statement is false then its negation is true. But the negation of a meaningless statement is equally meaningless. Thus the statement S z = +1/2 AND S z =?1/2 is meaningful and always false, whence its negation S z =?1/2 OR S z = +1/2 is meaningful 7

12 and always true, and is consistent with the Stern-Gerlach experiment. On the other hand, S x = +1/2 AND S z =?1/2 is meaningless, and its formal negation, S x =?1/2 OR S z = +1/2 is equally meaningless. The student who has learned quantum theory from the usual courses and textooks may well go along with the idea that S x = +1/2 AND S z = +1/2 lacks meaning, since he can think of no way of measuring it (and has proaly een told that it cannot e measured). However he will e less likely to go along with the equally important idea that the disjunction S x = +1/2 OR S z = +1/2 is similarly meaningless. Granted, there is no measurement which can distinguish the two, ut he has a mental image of a spin-half particle in the state S z = +1/2 as a little gyroscope with its axis of spin coinciding with the z axis. Such mental images are very useful to the physicist, and perhaps indispensale, for they help organize our picture of the world in terms that are easily rememered; they provide physical intuition. But this particular mental image can e quite misleading in suggesting that when S z = +1/2 the orthogonal components of angular momentum are zero. However, this cannot e the case since, since as the student has een taught, S 2 x = S 2 y = 1/4. Now any classical picture is ound to mislead to some extent when one is trying to think aout the quantum world, ut in this case a slight modification is less misleading. Imagine a gyroscope whose axis is oriented at random, except one knows that the z component of angular momentum is positive rather than eing negative. Among other things this modified image helps guard against the error that the spin degree of freedom of a spin-half particle can carry a large amount of information, when in fact the limit is one it (log 2 2) Frameworks Ordinary proaility theory uses the concept of a sample space: a collection of mutually exclusive alternatives or events, one and only one of which occurs or is true in a particular realization of some process or experiment. One way of introducing proailities in classical statistical mechanics is to imagine the phase space divided up into a collection of nonoverlapping cells, a coarse graining in which each cell represents one of the mutually exclusive alternatives one has in mind. Let P j (γ) e the indicator function for the j th cell: equal to 1 if γ lies within the cell and 0 otherwise. Oviously the product P j P k of the indicator functions for two different cells is 0, and the sum of all the indicator functions is the identity function I(γ) equal to 1 for all γ: P j P k = δ jk P j, P j = I Next, proailities are assigned to the events making up a Boolean event algera. If one coarse grains the phase space in the manner just indicated, an event algera can e constructed in which each event is represented y the union of some of the cells in the coarse graining; equivalently, the event algera is the collection of all indicator functions which are sums of some of the indicators in the collection {P j }, including the functions and Iwhich are everywhere 0 8 j

13 and 1, respectively. The proailities themselves can e specified y a collection {p j } of nonnegative real numers that sum to 1, with the proaility of an event E eing the sum of those p j for which the corresponding indicator functions P j appear in the sum defining E. To e sure, classical statistical mechanics is usually constructed without using a coarse graining, employing the Borel sets as an event algera, and then introducing an additive positive measure to define proailities. There is nothing wrong with this, ut for our purposes a coarse graining provides a more useful classical analog. Let us call quantum counterpart of a classical sample space a framework. It is a projective decomposition (P D) of the identity operator I: a collection {P j } of mutually orthogonal projectors which sum to the identity operator I, and thus formally satisfy exactly the same conditions as a collection of classical indicators. The fact that P j P k vanishes for j k means that the corresponding quantum properties (events is the customary term in proaility theory) are mutually exclusive: if one is true the other must e false, and the fact that they sum to the identity operator I means that at least one, and therefore only one, is true or real or actual. The corresponding event algera is the Boolean event algera of all projectors which can e formed y taking sums of projectors in {P j } along with the 0 operator, which plays the same role as the empty set in ordinary proaility theory. As long as the quantum event algera and the sample space are related in this way, there is no harm in using the somewhat loose term framework to refer to either one, as we shall do in what follows. Two frameworks {P j } and {Q k } are compatile if all the projectors in one commute with all of those in the other: P j Q k = Q k P j for all values of j and k. Otherwise they are incompatile. One says that {P j } is a refinement {Q k } if the projectors in the P D of the latter are included in the event algera of the former; equivalently, {Q k } is a coarsening of {P j }. Oviously two frameworks must e compatile if one is to e a refining or coarsening of the other. In addition, two compatile frameworks always possess a common refinement using the P D consisting of all the nonzero products P j Q k ; its event algera includes the union of the event algeras of the separate frameworks which it refines. Note that a given framework {P j } may have various different refinements, and two refinements need not e compatile with each other. Therefore when discussing quantum systems one must keep track of the framework eing employed in a particular argument. This is not important in classical physics, where one can either adopt the finest framework possile at the outset, or else refine it as one goes along without needing to call attention to this fact. In quantum mechanics one does not have this freedom, and carelessness can lead to paradoxes. 9

14 1.3 Again with More Details Classical and Quantum Properties The term physical property refers to something which can e said to e either true or false for a particular physical system.the statement the particle is etween x 1 and x 2 is an example of a physical property. One must distinguish etween a physical property and a physical variale, such as the position or energy or momentum of a particle. A physical variale can take on different numerical values, depending on the state of the system, whereas, a physical property is either a true or false description of a particular physical system at a particular time. A physical variale taking on a particular value, or lying in some range of values, is an example of a physical property. In the case of a classical mechanical system, a physical property is always associated with some suset of points in its phase space. Consider, for example, a harmonic oscillator whose phase space is the x, p plane shown elow. p x 1 x 2 x Figure 1.2: Phase space x, p for a particle in one dimension. The ellipse is a possile orit for a harmonic oscillator. The cross-hatched regions corresponds to x 1 x x 2 The property that its energy is equal to some value E 0 > 0 is associated with a set of points on an ellipse centered at the origin. The property that the energy is less than E 0 is associated with the set of points inside the ellipse. The property that the position x lies etween x 1 and x 2 corresponds to the set of points inside a vertical and cross-hatched in the figure, and so forth. Given a property P associated with a set of points P in the phase space, its is convenient, as we said earlier, to introduce an indicator function, or indicator 10

15 for short, P (γ), where γ is a point in the phase space, defined y { 1 if γ P P (γ) = 0 otherwise Thus, if at some instant of time the phase point γ 0 associated with a particular physical system is inside the set P, then P (γ 0 ) = 1, meaning that the systems possesses this property, or the property is true. Similarly, if P (γ 0 ) = 1, then the system does not possess this property, so for this system the property is false. A physical property of a quantum system is associated with a suspace P of the quantum Hilert space H in much the same way as a physical property of a classical system is associated with a suset of points in its phase space. A Digression on Projectors and Suspaces A particular type of Hermitian operator called a projector plays a central role in quantum theory, as we have seen. A projector is any operator P which satisfies the two conditions P 2 = P, P = P The first of these equations, P 2 = P, defines a projection operator which need not e Hermitian. Hermitian projection operators are sometimes called orthogonal projection operators. Associated with a projector P is a linear suspace P of H consisting of all kets which are left unchanged y P, that is, those ψ for which P ψ = ψ. We say that P projects onto P, or is the projector onto P. The projector P acts like an identity operator on the suspace P. The identity operator I is a projector, and it projects onto the entire Hilert space H. The zero operator ) is a projector which projects onto the suspace consisting of nothing ut the zero(null) vector. Any nonzero ket φ generates a one-dimensional suspace P, often called a ray or (y quantum physicists) a pure state, consisting of all scalar multiples of φ, that is to say, the collection of kets of the form {α φ }, where α is any complex numer. The projector onto P is then(assuming normalized vectors) P = φ φ The symol φ φ for the projector is the projector onto the ray generated y φ. It is straightforward to show that the symol φ φ satisfies the standard conditions for it to e a projector and that P (α φ ) = φ φ (α φ ) = α φ φ φ = α φ so that P leaves elements of P unchanged. When it acts on any vector χ orthogonal to φ, i.e., where φ χ = 0, P produces the null vector: P ( χ ) = φ φ χ = 0 11

16 These last two properties of P can e given a geometrical interpretation, or at least one can construct a geometrical analogy using real numers instead of complex numers. Consider the two-dimensional plane shown elow ω > Ƥ ω > η > Ƥ χ > ϕ > ϕ > α ϕ > Q η > Q ω > (a) () Figure 1.3: Illustrating (a) an orthogonal (perpendicular) projection onto P; () a nonorthogonal projection represented y Q with vectors laeled using Dirac kets. The line passing through φ is the suspace P. Let ω e some vector in the plane, and suppose that its projection onto P, along a direction perpendicular to P, part (a) of the figure, falls at the point α φ. Then ω = α φ + χ where χ is a vector perpendicular (orthogonal) to φ, indicated y the dashed line. Applying P to oth sides, one finds that P ω = αp φ + P χ = α φ i.e., P acting on any point ω in the plane projects onto P along a line perpendicular to P, as indicated y the arrow in part (a) of the figure. Of course, such a projection applied to a point already on P leaves it unchanged, corresponding to the fact that P acts as the identity operation for elements of this suspace.. For this reason P (P ( ω )) is always the same as P ( ω ), which is equivalent to the statement that P 2 = P. It is also possile to imagine projecting points onto P along a fixed direction which is not perpendicular to P, as in part () of the aove figure. This defines a linear operator Q which is again a projection operator, since elements of P are mapped onto themselves, and thus Q 2 = Q. However, this operator is not Hermitian (in the terminology of real vector spaces, it is not symmetrical), so it is not an orthogonal ( perpendicular ) projection operator. Given a projector P, we define its complement, written as P or P, also called 12

17 the negation of P, to e the projector defined y P = I P It is easy to show that P satisfies the conditions for a projector and that P P = 0 = P P Clearly, P is, in turn, the complement (or negation) of P. Let P and P e the suspaces of H onto which P and P project. Any element ω of P is orthogonal to any element φ of P: ω φ = ( ω ) φ = ( P ω ) (P φ ) = ω P P φ = 0 ecause P P = 0. Here we have used the fact that P and P acts as identity operators on their respective suspaces. As a consequence, any element ψ of H can e written as the sum of two orthogonal kets, one elonging to P and one to P : ψ = I ψ = P ψ + P ψ Using this result one can show that P is the orthogonal complementof P, the collection of all elements of H which are orthogonal to every ket in P. Similarly, P is the orthogonal complement of P. Digression on Orthogonal Projectors and Orthonormal Bases Two projectors P and Q are said to e (mutually) orthogonal if P Q = 0 By taking the adjoint of this equation we also have that QP == 0, so that the order of the operators in the product does not matter. An orthogonal collection of projectors, or a collection of (mutually) orthogonal projectors is a set of nonzero projectors {P 1, P 2,...} with the property that P j P k = 0 for j k The zero operator never plays a useful role in such collections and excluding it simplifies various definitions. One can now show that the sum P + Q of two orthogonal projectors P andq is a projector, and more generally, the sum R = j P j of the the memers of an orthogonal collection of projectors is a projector. When a projector R is written as the sum of projectors in an orthogonal collection, one says that this collection constitutes a decomposition or refinement of 13

18 R. In particular, if R is the identity operator I, the collection is a decomposition(refinement) of the identity I = j P j One often writes down a sum of this form and calls it a decomposition of the identity. However, it is important to note that the decomposition is not the sum itself, ut rather it is the set of summands, the collection of projectors which enter the sum. Whenever a projector R can e written as a sum of projectors as aove, it is necessarily the case that these projectors form an orthogonal collection, meaning that the orthogonal projector definition is satisfied. If two nonzero kets ω and φ are orthogonal, the same is true of the corresponding projectors ω ω and φ φ. An orthogonal collection of kets is a set { φ 1, φ 2,...} of nonzero elements of H such that φ j φ k = 0 when j k. If in addition the kets in such a collection are normalized, so that φ j φ k = δ jk ones says that it is anorthonormal collection. The corresponding projectors form an orthogonal collection, and { φ 1 φ 1, φ 2 φ 2,...} ( φ j φ j ) φ k = φ j φ j φ k = δ jk φ j Let R e the suspace of H consisting of all linear cominations of kets elonging to an orthonormal collection { φ j }, that is, all elements of the form ψ = sum j σ j φ j where the σ j are complex numers. The the projector R onto R is the sum of the orthogonal projectors, i.e., R = j φ j φ j This follows directly from the previous relations. If every element of H can e written in the form ψ = sum j σ j φ j, the orthonormal collection is said to form an orthonormal asis of H, and the corresponding decomposition of the identity is I = j φ j φ j A asis of H is a collection of linearly independent kets which span H in the sense that any element of H can e written as a linear comination of kets in the collection. 14

19 1.3.2 Toy Model The Hilert space H for a quantum particle in one dimension is extremely large; viewed as a linear space it is infinite-dimensional. Infinite-dimensional spaces provide headaches for physicists and employment for mathematicians. Most of the conceptual issues in quantum theory have nothing to do with the fact that the Hilert space is infinite-dimensional, and therefore it is useful, in order to simplify the mathematics, to replace infinite dimensional variales(continuous or discrete) with discrete variales which take on only a finite numer of integer values. This means that we will assume that the quantum particle is located at one of a finite collection of sites arranged in a straight line, or, it is located in a finite numer of oxes or cells. It is convenient to think of this system of sites having periodic oundary conditions or as placed on a circle, so that the last site is adjacent to (just in front of) the first site. If one were representing a wave function numerically on a computer, it would e sensile to employ a discretization of this type. However, our goal is not numerical computations, ut physical insight. Temporarily shunting mathematical difficulties out of the way is part of a useful divide and conquer strategy for attacking difficult prolems. Our aim will not e realistic descriptions, ut instead simple descriptions which still contain the essential features of quantum theory. For this reason, the term toy model seems appropriate. Let us suppose that the quantum wave function is of the form ψ(m), with m an integer in the range M a m M where M a and M are fixed integers, so m can take on M = M a + M + 1 different values. Such wave functions form an M dimensional Hilert space. The inner product of two wave functions is given y φ ψ = m φ (m)ψ(m) where the sum is over the allowed values of m, and the norm of ψ is the positive square root of ψ 2 = ψ(m) 2 m The toy wave function χ n, defined y χ n (m) = δ mn { 1 m = n 0 m n has the significance that the particle is at site n (or in cell n). Now suppose that M a = 3 = M, and consider the wave function It is sketched elow ψ(m) = χ 1 (m) + 1.5χ 0 (m) + χ 1 (m) 15

20 Figure 1.4: The toy wave packet and one can think of it as a relatively coarse approximation to a continuous function. What can one say aout the location of the particle with this quantum wave function? It seems sensile (ased on our earlier discussions) to interpret ψ(m) as signifying that the position of the quantum particle is not outside the interval [ 1, +1], where y [ 1, +1] we mean the three values 1,0, and +1. The circumlocution not outside the interval can e replaced with the more natural inside the interval provided the latter is not interpreted to mean at a particular site inside this interval, since the particle with the aove wave function cannot e said to e at m = 1 or at m = 0 or at m = +1. Instead it is delocalized, and its position cannot e specified any more precisely than y giving the interval [ 1, +1]. There is no concise way of stating this in English, which is the reason we need a mathematical notation in which quantum properties can e expressed in a precise way - this is the Dirac language we have een discussing. It is important not to look at a wave function written out as a sum of different pieces whose physical significance one understands, and interpret it in physical terms as meaning the quantum system has one or the other of the properties corresponding to the different pieces. In particular, one should not interpret the aove wave function to mean that the particle is at m = 1 or at m = 0 or at m = +1. A simple example which illustrates how such an interpretation can lead one astray is otained y writing χ 0 in the form χ 0 (m) = 1 2 [χ 0(m) + iχ 2 (m)] [χ 0(m) + ( i)χ 2 (m)] If we carelessly interpret + to mean or, then oth of the functions in the square rackets and therefore also their sum, have the interpretation that the particle is at 0 or 2, whereas, in fact, χ 0 means that the particle is at 0 and not at 2. The correct quantum mechanical way to use or will e discussed later. 16

21 1.3.3 Toy Model and Spin 1/2 Now let n e the Dirac ket that correspond to the particle eing at a particular site n. Then we have k n = δ kn or that k n corresponds to χ k (n). This implies that the kets { n } form an orthonormal asis of the Hilert space. Any scalar multiple α n of n, where α is a nonzero complex numer, has precisely the same physical significance as n. The set of all kets of this form together with the zero ket, that is, the set of all multiples of n, form a onedimensional suspace of H, and the projector onto this suspace is n n. Thus, it is natural to associate the property that the particle is at site n (something which can e true or false) with this suspace, or, equivalently, with the corresponding projector, since there is a one-to-one correspondence etween suspaces and projectors. Since the projectors 0 0 and 1 1 for sites 0 and 1 are orthogonal to each other, their sum is also a projector R = The suspace R onto which R projects is two-dimensional and consists of all linear cominations of 0 and 1, that is, all states of the form φ = α 0 + β 1 Equivalently, it corresponds to all wave function ψ(m) which vanish when m is unequal to 0 or 1. The physical significance of R is that the toy particle is not outside the the interval [0, 1], where, since we are using a discrete model, the interval [0, 1] consists of two sites m = 0 and m = 1. One can interpret not outside as meaning inside, provided that is not understood to mean at one or the other of two sites m = 0 or m = 1. The reason one needs to e cautious is that a typical state in R will have the form φ = α 0 + β 1 with oth α and β unequal to zero. Such a state does not have the property that it is at m = 0, for all states with this property are scalar multiples of 0, and φ is not of this form. Indeed, φ is not an eigenstate of the projector 0 0 representing the property m = 0, and hence according to earlier discussions, the property m = 0 is undefined. The same comments apply to the property m = 1. Thus, it is certainly incorrect to say that the particle is either at 0 or 1. Instead, the particle is represented y a delocalized wave(an entangled state). There are some states in R which are localized at 0 or localized at!, ut since R also contains other, delocalized states, the property corresponding to R or its projector R, which holds for all states in this suspace, needs to e expressed y some English phrase other than at 0 or 1. The phrases not outside the interval [0, 1] or no place other than 0 17

22 or 1, while they are a it awkward, come closer to saying what one wants to say. The way to e perfectly precise is to use the projector R itself, since it is a precisely defined mathematical quantity. But, of course, one needs to uild up an intuitive picture of what it is that R means(if that turns out to e possile!). The process of uilding up one s intuition aout the meaning of R is aided y noting that the form R = is not the only way of writing it as the sum of two orthogonal projectors. Another possiility is(simple algera show it to e identical to the original) R = σ σ + τ τ where σ = 1 2 ( 0 + i 1 ), τ = 1 2 ( 0 i 1 ) are two normalized states in R which are mutually orthogonal. There are many other way to do this. In fact, given any normalized state in R, one can always find another normalized state orthogonal to it, and the sum of the projectors corresponding to the two states is equal to R. The fact that R can e written as a sum P +Q of two orthogonal projectors P and Q in many different ways is one reason to e cautious in assigning R a physical interpretation of property P and property Q, although there are occasions(see later) when such an interpretation is appropriate. The simplest nontrivial toy model has only M = 2 sites and it is convenient to discuss it using language appropriate to the spin degree of freedom of a particle of spin 1/2. Its Hilert space H consists of all linear cominations of two mutually orthogonal and normalized states which will e denoted y z + and z, and which can e thought of as the counterparts of 0 and 1 in the toy model. In this ook they are often denoted y + and, or y and. The normalization and orthogonality conditions are z + z + = 1 = z z, z + z = 0 The physical significance of z + is that the z component S z of the internal or spin angular momentum has the value +1/2 in units of, while z means that S z = 1/2 in the same units. One sometimes refers to z + and z as spin up and spin down. The two-dimensional Hilert space H consist of all linear cominations of the form α z + + β z where α and β are any complex numers. It is convenient to parameterize these state in the following way. Let w denote a direction in space corresponding to θ, ϕ in spherical polar coordinates. For example, θ = 0 is the +z direction, while 18

23 θ = π/2 and ϕ = π is the x direction. Then define the two states w + = + cos θ/2e iϕ/2 z + + sin θ/2e iϕ/2 z w = sin θ/2e iϕ/2 z + + cos θ/2e iϕ/2 z These are normalized and mutually orthogonal, w + w + = 1 = w w, w + w = 0 as can e easily proved. The physical significance of w + is that S w, the component of spin angular momentum in the w direction, has a value of 1/2, whereas for w, S w has the value 1/2. For θ = 0, w + and w are the same as z + and z., respectively, apart from a phase factor, e iϕ, which does not change their physical significance. For θ = π, w + and w are the same as z and z + respectively, apart from phase factors. Suppose thatw is a direction which is neither along nor opposite to the z axis, for example w = x. Then oth α and β are nonzero and w + does not have the property S z = +1/2 nor does it have the property s Z = 1/2. The same is true if S z is replaced y S v, where v is any direction which is not the same as w or opposite to w. The situation is analogous to that discussed earlier for the toy model: think of z + and z as corresponding to the states m = 0 and m = 1. Any nonzero wave function α z + + β z can e written as a complex numer times w + for a suitale choice of the direction w. For β = 0, the choice w = z is ovious, while for α = 0 it is W = z. For other cases we can write that state as β[(α/β) z + + z A comparison with the expression aove for w + shows that θ and ϕ are determined y the equation e iϕ cot θ/2 = α β which, since α/β is finite (neither 0 nor ), always has a unique solution in the range 0 ϕ < 2π, 0 < θ < π Continuous Quantum Systems We now deal with the quantum properties of a particle in one dimension descried y a wave function ψ(x) depending on the continuous variale x. Quantum properties are again associated with suspaces of the Hilert space H, and since H is infinite-dimensional, these suspaces can e either finite- or infinitedimensional - we will consider oth. As a first example, consider the property that the particle lies inside (which is to say not outside) the interval x 1 x x 2 19

24 with x 1 < x 2. The (infinite-dimensional) suspace X which corresponds to this property consist of all wave functions which vanish for x outside the interval x x x x 2. The projector X associated with X is defined as follows. Acting on some wave function ψ(x), X produces a new function ψ X (x) which is identical to ψ(x) for x inside the interval x x x x 2, and zero outside the interval x x x x 2 : { ψ(x) for x x x x 2 ψ X (x) = Xψ(x) = 0 for x < x 1 or x > x 2 Note that X leaves unchanged any function which elongs to the suspace X, so it acts as the identity operator on this suspace. If a wave function ω(x) vanishes throughout the interval x x x x 2, it will e orthogonal to all of the functions in X, and X applied to ω(x) will yield a function which is everywhere equal to 0, Thus X has the properties one would expect for a projector as discussed earlier. In Dirac language we have Similar results hold for momentum. ψ X = C ψ Another example is a one-dimensional harmonic oscillator. In the text it has een shown that the energy E of an oscillator with angular frequency ω takes on a discrete set of values E = n + 1 2, n = 0, 1, 2,... in units of ω. Let φ n (x) e the normalized wave function for energy E = n + 1/2, and φ n the corresponding ket. The one-dimensional suspace of H consisting of all scalar multiples of φ n represents the property that the energy is n + 1/2. The corresponding projector is φ n φ n. When this projector acts on some ψ in H, the result is ψ = ( φ n φ n ) ψ = φ n ψ φ n that is, φ n multiplied y the scalar φ n ψ. Since the states φ n for different n are mutually orthogonal, the same is true of the corresponding projectors φ n φ n. Using this fact makes it easy to write down projectors for the energy to lie in some interval which includes two or more energy eigenvalues. For example, the projector Q = φ 1 φ 1 + φ 2 φ 2 onto the two-dimensional suspace of H consisting of linear cominations of φ 1 and φ 2 expresses the property that the energy E (in units of ω) lies inside some interval such that 1 < E < 3 20

25 where the choice of endpoints of the interval is somewhat aritrary, given that the energy is quantized and takes on only discrete values; any other interval which includes 1.5 and 2.5, ut excludes 0.5 and 3.5, would e just as good. Once again, it is important not to interpret energy lying inside the interval 1 < E < 3 as meaning that either it has the value 1.5 or that it has the value 2.5. The suspace onto which Q projects also contains states such as φ 1 + φ 2, for which the energy cannot e defined more precisely than y saying that it does not lie outside the interval, and thus the physical property expressed y Q cannot have a meaning which is more precise than this Negation Properties(NOT) A physical property can e true or false in the sense that the statement that a particular physical system at a particular time possesses a physical property can e either true or false. Books on logic present simple logical operations y which statements which are true or false can e transformed into other statements which are true or false. Let us consider the operation called negation. A classical property P, as we said earlier, is associated with a suset P consisting of those points in the classical phase space for which the property is true. The points of the phase space which do not elong to P form the complementary set P, and this complementary set defines the negation NOT P of the property P. We will write it as P or as P. Alternatively, one can define P as the property which is true if and only if P is false, and false if and only if P is true. From this as well as the other definition it is ovious that the negation of the negation of a property is the same as the original property: P or ( P ) is the same property as P. The indicator P (γ) of the property P is given y P = I P or P (γ) = I(γ) P (γ), where I(γ), the indicator of the identity property, is equal to 1 for all values of γ. Thus, P is equal to 1 (true) if P is 0 (false) and P = 0 when P = 1. Consider once again Figure 1.2, the phase space of a one-dimensional harmonic oscillator, where the ellipse corresponds to an energy E 0. The property P that the energy is less than or equal to E 0 corresponds to the set P of points inside and on the ellipse. Its negation P is the property that the energy is greater than E 0, and the corresponding region P is all the points outside the ellipse. The vertical and Q corresponds to the property Q that the position of the particle is in the interval x 1 x x 2. The negation of Q is the property Q that the particle lies outside this interval, and the corresponding set of points Q in the phase space consists of the half planes to the left of x = x 1 and to the right of x = x 2. A property of a quantum system is associated with a suspace of the Hilert 21

26 space, and thus the negation of this property will also e associated with some suspace of the Hilert space. Consider, for example, a toy model with M a = 2 = M. Its Hilert space consists of all linear cominations of the states 2, 1, 0, 1, and 2. Suppose that P is the property associated with the projector P = projecting onto the suspace P of all linear cominations of 0 and 1. Its physical interpretation is that the quantum particle is confined to these two sites, that is, it is not at some location apart from these two sites. The negation P of P is the property that the particle is not confined to these two sites, ut is instead someplace else, so the corresponding projector is P = This projects onto the orthogonal complement P of P, consisting of all linear cominations of 2, 1, and 2. Since the identity operator for this Hilert space is given y 2 I = m m and it is clear that m= 2 P = I P This is the same as the classical formula except that the symols now refer to quantum projectors rather than classical indicators. As a second example, consider a one-dimensional harmonic oscillator and suppose that P is the property that the energy is less than or equal to 2 in units of ω. The corresponding projector is P = φ 0 φ 0 + φ 1 φ 1 The negation of P is the property that the energy is greater than 2, and its projector is P = φ 2 φ 2 + φ 3 φ 3 + φ 4 φ = I P In this case, P projects onto a finite dimensional suspace and P projects onto an infinite-dimensional suspace. As another example, consider the property X that a particle in one dimension is located in (that is, not outside) the interval x 1 x x 2 ; the corresponding projector X was defined earlier. Using the fact that Iψ(x) = ψ)x), it is easy to show that the projector X = I X, corresponding to the property that the particle is located outside (not inside) the interval is given y { 0 for x x x x 2 Xψ(x) = ψ(x) for all other x values 22

27 Note that in this case the action of the projectors X and X is to multiply ψ(x) y the indicator function for the corresponding classical property. As a final example, consider a spin-1/2 particle, and let P e the property S z = +1/2 (in units of ) corresponding to the projector z + z +. One can think of this as analogous to a toy model with M = 2 sites m = 0, 1, where z + z + corresponds to 0 0. Then it is evident from earlier discussion that the negation P of P will e the projector z z, the counter part of 1 1 in the toy model, corresponding to the property S z = 1/2. Of course, the same reasoning can e applied with z replace y an aritrary direction ω: The property S ω = 1/2 is the negation of S ω = +1/2, and vice versa. The relationship etween the projector for a quantum property and the projector for its negation is formally the same as the relationship etween the corresponding indicators for a classical property. Despite this close analogy, there is actually an important difference. In the classical case, the suset corresponding to p is the complement of the suset corresponding to P : any point in the phase space is in one or the other, and the two susets do not overlap. In the quantum case, the suspaces P and P have one element in common, the zero vector. This is different from classical phase space, ut is not important, for the zero vector y itself stands for the property which is always false, corresponding to the empty suset of the classical phase space. Much more significant is the fact that H contains many nonzero elements which elong neither to P nor to P. In particular, the sum of a nonzero vector from P and a nonzero vector from P elongs to H, ut does not elong to either of these suspaces. For example, the ket x + for a spin-1/2 particle corresponding to S x = +1/2 elongs neither to the suspace associated with S z = +1/2 nor to that of its negation S z = 1/2. Thus, despite the formal paralle, the difference etween the mathematics of Hilert space and that of classical phase space means that negation is not quite the same thing in quantum physics as it is in classical physics Conjunction and Disjunction (AND, OR) Consider two different properties P and Q of a classical system, corresponding to susets P and mathcalq of its phase space. The system will possess oth properties simultaneously if its phase point γ lies in the intersection P Q of the sets P and Q or, using indicators, if P (γ) = 1 = Q(γ). Se the Venn diagram in part (a) of the figure elow. 23

28 P Q P Q (a) () Figure 1.5: The circles represent the properties P and Q, In (a) the grey region is P Q and in () it is P Q. In this case we can say that the system possesses the property P AND Q, the conjunction of P and Q, which can e written compactly as P Q. The corresponding indicator function is P Q = P Q that is, (P Q)(γ) is the function P (γ) times the function Q(γ). In the case of a one-dimensional harmonic oscillator, let P e the property that the energy is less than E 0, and Q the property that x lies etween x 1 and x 2. Then the indicator P Q for the comined property P Q, energy less than E 0 AND x etween x 1 and x 2, is 1 at those points in the cross-hatched and in Figure 1.2 which lie inside the ellipse, and 0 everywhere else. Given the close correspondence etween classical indicators and quantum projectors, on might expect that the projector for the quantum property P Q (P AND Q) would e the product of the projectors for the separate properties, as in the classical case aove. This is indeed the case if P and Q commute with each other, this is, if QP = P Q. In this case it is easy to show that the product P Q is a projector satisfying the two required conditions. On the other hand, if P and Q do not commute(p Q QP ), then P Q will not e a Hermitian operator, so it cannot e a projector. We now discuss the conjunction and disjunction properties of P and Q assuming that the two projectors commute. We will discuss the non-commuting case later. As a first example, consider the case of a one-dimensional harmonic oscillator in which P is the property that the energy E is less than 3 (in units of ω), and Q is the property that E is greater than 2. The two projectors are P = φ 0 φ 0 + φ 1 φ 1 + φ 2 φ 2 Q = φ 2 φ 2 + φ 3 φ 3 + φ 4 φ