Quantum Mechanics (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeQuantum MechanicsFirst published Wed Nov 29, 2000Quantum mechanics is, at least at first glance and at least in part, amathematical machine for predicting the behaviors of microscopicparticles — or, at least, of the measuring instruments we use toexplore those behaviors — and in that capacity, it is spectacularlysuccessful: in terms of power and precision, head and shoulders aboveany theory we have ever had. Mathematically, the theory is wellunderstood; we know what its parts are, how they are put together, andwhy, in the mechanical sense (i.e., in a sense that can be answered bydescribing the internal grinding of gear against gear), the whole thingperforms the way it does, how the information that gets fed in at oneend is converted into what comes out the other. The question of whatkind of a world it describes, however, is controversial; there is verylittle agreement, among physicists and among philosophers, about whatthe world is like according to quantum mechanics. Minimallyinterpreted, the theory describes a set of facts about the way themicroscopic world impinges on the macroscopic one, how it affects ourmeasuring instruments, described in everyday language or the languageof classical mechanics. Disagreement centers on the question of what amicroscopic world, which affects our apparatuses in the prescribedmanner, is, or even could be, like intrinsically; or how thoseapparatuses could themselves be built out of microscopic parts of thesort the theory describes.[1] That is what an interpretation of the theory would provide: a properaccount of what the world is like according to quantum mechanics,intrinsically and from the bottom up. The problems with giving aninterpretation (not just a comforting, homey sort of interpretation,i.e., not just an interpretation according to which the world isn't toodifferent from the familiar world of common sense, but anyinterpretation at all) are dealt with in other sections of thisencyclopedia. Here, we are concerned only with the mathematical heartof the theory, the theory in its capacity as a mathematical machine,and — whatever is true of the rest of it — this part of thetheory makes exquisitely good sense.1. Terminology2. Mathematics Vectors and vector spaces Operators 3. Quantum Mechanics 3.1 Physical States 3.2 Physical Quantities 3.3 Composition 3.4 Dynamics 4. Structures on Hilbert Space Loose Ends BibliographyOther Internet ResourcesRelated Entries1. TerminologyPhysical systems are divided into types according totheir unchanging (or ‘state-independent’) properties, andthe state of a system at a time consists of a completespecification of those of its properties that change with time (its‘state-dependent’ properties). To give a completedescription of a system, then, we need to say what type of system it isand what its state is at each moment in its history. A physical quantity is a mutually exclusive andjointly exhaustive family of physical properties (for those who knowthis way of talking, it is a family of properties with the structure ofthe cells in a partition). Knowing what kinds of values a quantitytakes can tell us a great deal about the relations among the propertiesof which it is composed. The values of a bivalent quantity, forinstance, form a set with two members; the values of a real-valuedquantity form a set with the structure of the real numbers. This is aspecial case of something we will see again and again, viz.,that knowing what kind of mathematical objects represent the elementsin some set (here, the values of a physical quantity; later, the statesthat a system can assume, or the quantities pertaining to it) tells usa very great deal (indeed, arguably, all there is to know) about therelations among them.In quantum mechanical contexts, the term‘observable’ is used interchangeably with‘physical quantity’, and should be treated as a technicalterm with the same meaning. It is no accident that the early developersof the theory chose the term, but the choice was made for reasons thatare not, nowadays, generally accepted. The state-spaceof a system is the space formed by the set of its possible states,[2] i.e., the physically possible ways ofcombining the values of quantities that characterize it internally. Inclassical theories, a set of quantities which forms a superveniencebasis for the rest is typically designated as ‘basic’ or‘fundamental’, and, since any mathematically possible wayof combining their values is a physical possibility, the state-spacecan be obtained by simply taking these as coordinates.[3] So, for instance, the state-space of a classical mechanical systemcomposed of n particles, obtained by specifying the values of6n real-valued quantities — three components of position, andthree of momentum for each particle in the system — is a6n-dimensional coordinate space. Each possible state of sucha system corresponds to a point in the space, and each point in thespace corresponds to a possible state of such a system. The situationis a little different in quantum mechanics, where there aremathematically describable ways of combining the values of thequantities that don't represent physically possible states. As we willsee, the state-spaces of quantum mechanics are special kinds of vectorspaces, known as Hilbert spaces, and they have more internal structurethan their classical counterparts.A structure is a set of elements on which certainoperations and relations are defined, a mathematicalstructure is just a structure in which the elements aremathematical objects (numbers, sets, vectors) and the operationsmathematical ones, and a model is a mathematicalstructure used to represent some physically significant structure inthe world.The heart and soul of quantum mechanics is contained in the Hilbertspaces that represent the state-spaces of quantum mechanical systems.The internal relations among states and quantities, and everything thisentails about the ways quantum mechanical systems behave, are all woveninto the structure of these spaces, embodied in the relations among themathematical objects which represent them.[4] This means thatunderstanding what a system is like according to quantum mechanics isinseparable from familiarity with the internal structure of thosespaces. Know your way around Hilbert space, and become familiar withthe dynamical laws that describe the paths that vectors travel throughit, and you know everything there is to know, in the terms provided bythe theory, about the systems that it describes.By ‘know your way around’ Hilbert space, I meansomething more than possess a description or a map of it; anybody whohas a quantum mechanics textbook on their shelf has that. I mean knowyour way around it in the way you know your way around the city inwhich you live. This is a practical kind of knowledge that comes indegrees and it is best acquired by learning to solve problems of theform: How do I get from A to B? Can I get there without passing throughC? And what is the shortest route? Graduate students in physics spendlong years gaining familiarity with the nooks and crannies of Hilbertspace, locating familiar landmarks, treading its beaten paths, learningwhere secret passages and dead ends lie, and developing a sense of theoverall lay of the land. They learn how to navigate Hilbert space inthe way a cab driver learns to navigate his city.How much of this kind of knowledge is needed to approach thephilosophical problems associated with the theory? In the beginning,not very much: just the most general facts about the geometry of thelandscape (which is, in any case, unlike that of most cities,beautifully organized), and the paths that (the vectors representingthe states of) systems travel through them. That is what will beintroduced here: first a bit of easy math, and then, in a nutshell, thetheory.2. MathematicsVectors and vector spacesA vector A, written ‘|A>’, is amathematical object characterized by a length, |A|, and a direction. Anormalized vector is a vector of length 1; i.e., |A| = 1. Vectors canbe added together, multiplied by constants (including complex numbers),and multiplied together. Vector addition maps any pair of vectors ontoanother vector, specifically, the one you get by moving the secondvector so that it's tail coincides with the tip of the first, withoutaltering the length or direction of either, and then joining the tailof the first to the tip of the second. This addition rule is known asthe parallelogram law. So, for example, adding vectors |A> and |B> yields vector |C> (= |A> + |B>) as in Figure 1:  Figure 1: Vector AdditionMultiplying a vector |A> by n, where n is aconstant, gives a vector which is the same direction as |A> butwhose length is n times |A>'s length. In a real vector space, the (inner or dot) product of a pair of vectors |A> and |B>, written‘<A|B>’ is a scalar equal to the product of theirlengths (or ‘norms’) times the cosine of the angle,θ, between them:<A|B> = |A| |B| cos θLet |A1> and |A2> be vectors of length 1("unit vectors") such that <A1|A2> = 0. (Sothe angle between these two unit vectors must be 90 degrees.) Then wecan represent an arbitrary vector |B> in terms of our unit vectorsas follows: |B> = b1|A1> +b2|A2>For example, here is a graph which shows how |B> can be representedas the sum of the two unit vectors |A1> and |A2>:  Figure 2: Representing |B> by Vector Addition of UnitVectorsNow the definition of the inner product <A|B> has to bemodified to apply to complex spaces. Let c* be the complexconjugate of c. (When c is a complex number of theform a ± bi, then the complex conjugatec* of c is defined as follows:[a + bi]* = a − bi [a −bi]* = a + biSo, for all complex numbers c, [c*]* = c,but c* = c just in case c is real.) Nowdefinition of the inner product of |A> and |B> for complex spacescan be given in terms of the conjugates of complex coefficients asfollows. Where |A1> and |A2> are the unitvectors described earlier, |A> =a1|A1> +a2|A2> and |B> =b1|A1> +b2|A2>, then <A|B> =(a1*)(b1) +(a2*)(b2)The most general and abstract notion of an inner product, of whichwe've now defined two special cases, is as follows. <A|B> is aninner product on a vector space V just in case<A|A> = |A|2, and <A|A>=0 if andonly if A=0<B|A> = <A|B>*<B|A+C> = <B|A> + <B|C>.It follows from this that the length of |A> is the square root of innerproduct of |A> with itself, i.e., |A| = √<A|A>, and |A> and |B> are mutually perpendicular, ororthogonal, if, and only if, <A|B> = 0.A vector space is a set of vectors closed underaddition, and multiplication by constants, an inner productspace is a vector space on which the operation of vectormultiplication has been defined, and the dimension ofsuch a space is the maximum number of nonzero, mutually orthogonalvectors it contains. Any collection of N mutually orthogonal vectors of length 1 in anN-dimensional vector space constitutes an orthonormalbasis for that space. Let |A1>, … ,|AN> be such a collection of unit vectors. Then everyvector in the space can be expressed as a sum of the form:|B> = b1|A1> +b2|A2> + … +bN|AN>,where bi =<B|Ai>. The bi's here areknown as B's expansion coefficients in the A-basis.[5] Notice that:for all vectors A, B, and C in a given space, <A|B+C> = <A|B> + <A|C> for any vectors M and Q, expressed in terms of the A-basis, |M> + |Q> = (m1 +q1)|A1> + (m2 +q2)|A2> + ... +(mN + qN)|AN>, and <M|Q> = m1q1 +m2q2 + ... +mnqn There is another way of writing vectors, namely by writing theirexpansion coefficients (relative to a given basis) in a column, likeso: |Q> =[q1q2]where qi =<Q|Ai> and the Ai are thechosen basis vectors. When we are dealing with vector spaces of infinite dimension, sincewe can't write the whole column of expansion coefficients needed topick out a vector since it would have to be infinitely long, so insteadwe write down the function (called the ‘wave function’ forQ, usually represented ψ(i))which has those coefficients as values. We write down, that is, thefunction:ψ(i) =qi =<Q|Ai>Given any vector in, and any basis for, a vector space, we can obtainthe wave-function of the vector in that basis; and given awave-function for a vector, in a particular basis, we can construct thevector whose wave-function it is. Since it turns out that most of theimportant operations on vectors correspond to simple algebraicoperations on their wave-functions, this is the usual way to representstate-vectors. When a pair of physical systems interact, they form a compositesystem, and, in quantum mechanics as in classical mechanics, there is arule for constructing the state-space of a composite system from thoseof its components, a rule that tells us how to obtain, from thestate-spaces, HA and HB for A and B,respectively, the state-space — called the ‘tensorproduct’ of HA and HB, and writtenHA HB — of the pair. There are two importantthings about the rule; first, so long as HA andHB are Hilbert spaces, HAHB willbe as well, and second, there are some facts about the wayHAHB relates to HA andHB, that have surprising consequences for the relationsbetween the complex system and its parts. In particular, it turns outthat the state of a composite system is not uniquely defined by thoseof its components. What this means, or at least what it appears tomean, is that there are, according to quantum mechanics, facts aboutcomposite systems (and not just facts about their spatialconfiguration) that don't supervene on facts about their components; itmeans that there are facts about systems as wholes that don't superveneon facts about their parts and the way those parts are arranged inspace. The significance of this feature of the theory cannot beoverplayed; it is, in one way or another, implicated in most of itsmost difficult problems.In a little more detail: if{viA} is an orthonormal basisfor HA and {ujB} isan orthonormal basis for HB, then the set of pairs(viA,ujB) is taken to form anorthonormal basis for the tensor product space HAHB. Thenotation viAujB is usedfor the pair(viA,ujB), and inner product on HAHB isdefined as:[6]<viAumB |vjAunB> =<viA |vjA><umB |unB>It is a result of this construction that although every vector inHAHB is a linear sum of vectors expressible in the form vAuB, not every vector in the spaceis itself expressible in that form, and it turns out that any composite state defines uniquely the states of itscomponents.if the states of A and B are pure (i.e., representable byvectors vA and uB,respectively), then the state of (A+B) is pure and represented byvAuB, andif the state of (A+B) is pure and expressible in the form vAuB, then the states of A and B are pure, butif the states of A and B are not pure, i.e., if they are mixedstates (these are defined below), they do not uniquely define the stateof (A+B); in particular, it may be a pure state not expressible in theform vAuB.OperatorsAn operator O is a mapping of a vector space ontoitself; it takes any vector |B> in a space onto another vector |B′> also inthe space; O|B> = |B′>. Linear operators are operatorsthat have the following properties: O(|A> + |B>) = O|A> + O|B>, and O(c|A>) = c(O|A>).Just as any vector in an N-dimensional space can be represented by acolumn of N numbers, relative to a choice of basis for the space, anylinear operator on the space can be represented in a column notation byN2 numbers: O =[O11O21O12O22]where Oij = < Ai |O|Aj> and the |AN> are the basisvectors of the space. The effect of the linear operator O on the vectorB is, then, given by O|B>== [ O11 O21 O12 O22 ] × [ b1 b2 ] = [ (O11b1 + O12b2) (O21b1 + O22b2) ] = (O11b1 +O12b2)|A1> + (O21b1 + O22b2)|A2> =|B′>Two more definitions before we can say what Hilbert spaces are, andthen we can turn to quantum mechanics. |B> is an eigenvector of O witheigenvalue a if, and only if, O|B> = a|B>.Different operators can have different eigenvectors, but theeigenvector/operator relation depends only on the operator and vectorsin question, and not on the particular basis in which they areexpressed; the eigenvector/operator relation is, that is to say,invariant under change of basis. Hermitean operatorsare linear operators, which have only real eigenvalues. A Hilbert space, finally, is a vector space onwhich an inner product is defined, and which is complete, i.e., whichis such that any Cauchy sequence of vectors in the space converges to avector in the space. All finite-dimensional inner product spaces arecomplete, and I will restrict myself to these. The infinite caseinvolves some complications that are not fruitfully entered into atthis stage.3. Quantum MechanicsFour basic principles of quantum mechanics are: 3.1 Physical StatesEvery physical system is associated with a Hilbert Space, every unitvector in the space corresponds to a possible pure state of the system,and every possible pure state, to some vector in the space.[7] In standardtexts on quantum mechanics, the vector is represented by a functionknown as the wave-function, or ψ-function. 3.2 Physical QuantitiesHermitian operators in the Hilbert space associated with a systemrepresent physical quantities, and their eigenvalues represent thepossible results of measurements of those quantities. 3.3 CompositionThe Hilbert space associated with a complex system is the tensorproduct of those associated with the simple systems (in the standard,non-relativistic, theory: the individual particles) of which it iscomposed. 3.4 Dynamics Contexts of type 1: Given the state of a system at t andthe forces and constraints to which it is subject, there is anequation, ‘Schrödinger's equation’,that gives the state at any other time U|vt> → |vt′>.[8] The important properties of U for ourpurposes are that it is deterministic, which is to saythat it takes the state of a system at one time into a unique state atany other, and it is linear, which is to say that ifit takes a state |A> onto the state |A′>, and it takesthe state |B> onto the state |B′>, then it takesany state of the form α|A> + β|B> onto the state α|A′> + β|B′>. Contexts of type 2 ("Measurement Contexts"):[9] Carrying outa "measurement" of an observable B on a system in a state |A> hasthe effect of collapsing the system into a B-eigenstate correspondingto the eigenvalue observed. This is known as the CollapsePostulate. Which particular B-eigenstate it collapsesinto is a matter of probability, and the probabilities are given by arule known as Born's Rule:prob(bi) = |<A|B=bi>|2.There are two important points to note about these two kinds ofcontexts:The distinction between contexts of type 1 and 2 remains to be madeout in quantum mechanical terms; nobody has managed to say in acompletely satisfactory way, in the terms provided by the theory, whichcontexts are measurement contexts, andEven if the distinction is made out, it is an open interpretivequestion whether there are contexts of type 2; i.e., it is anopen interpretive question whether there are any contexts in whichsystems are governed by a dynamical rule other thanSchrödinger's equation.4. Structures on Hilbert SpaceI remarked above that in the same way that all the information we haveabout the relations between locations in a city is embodied in thespatial relations between the points on a map which represent them, allof the information that we have about the internal relations among (andbetween) states and quantities in quantum mechanics is embodied in themathematical relations among the vectors and operators which represent them.[10] From a mathematical point of view, whatreally distinguishes quantum mechanics from its classical predecessorsis that states and quantities have a richer structure; they formfamilies with a more interesting network of relations among theirmembers. All of the physically consequential features of the behaviors ofquantum mechanical systems are consequences of mathematical propertiesof those relations, and the most important of them are easilysummarized:(P1) Any way of adding vectors in a Hilbert space ormultiplying them by scalars will yield a vector that is also in thespace. In the case that the vector is normalized, it will, from (3.1),represent a possible state of the system, and in the event that it isthe sum of a pair of eigenvectors of an observable B with distincteigenvalues, it will not itself be an eigenvector of B, but will beassociated, from (3.4b), with a set of probabilities for showing one oranother result in B-measurements. (P2) For any Hermitian operator on a Hilbert space, there areothers, on the same space, with which it doesn't share a full set ofeigenvectors; indeed, it is easy to show that there are other suchoperators with which it has no eigenvectors in common.If we make a couple of additional interpretive assumptions, we can saymore. Assume, for instance, that (4.1) Every Hermitian operator on the Hilbert spaceassociated with a system represents a distinct observable, and (hence)every normalized vector, a distinct state, and (4.2) A system has a value for observable A if, and only if, thevector representing its state is an eigenstate of the A-operator. Thevalue it has, in such a case, is just the eigenvalue associated withthat eigenstate.[11]It follows from (P2), by (3.1), that no quantum mechanical state is aneigenstate of all observables (and indeed that there are observableswhich have no eigenstates in common), and so, by (3.2), thatno quantum mechanical system ever has simultaneous values for all ofthe quantities pertaining to it (and indeed that there are pairs ofquantities to which no state assigns simultaneous values). There are Hermitian operators on the tensor productH1H2 of a pair of Hilbert spacesH1 and H2 ... In the event that H1 andH2 are the state spaces of systems S1 and S2,H1H2 is the state-space of the complexsystem (S1+S2). It follows from this by (4.1) that there areobservables pertaining to (S1+S2) whose values are not determined bythe values of observables pertaining to the two individually.These are all straightforward consequences of taking vectors andoperators in Hilbert space to represent, respectively, states andobservables, and applying Born's Rule (and later (4.1) and (4.2)), togive empirical meaning to state assignments. That much is perfectlywell understood; the real difficulty in understanding quantum mechanicslies in coming to grips with their implications — physical,metaphysical, and epistemological.There is one remaining fact about the mathematical structure of thetheory that anyone trying to come to an understanding about what itsays about the world has to grapple with. It is not a property ofHilbert spaces, this time, but of the dynamics, the rules that describethe trajectories that systems follow through the space. From a physicalpoint of view, it is far more worrisome than anything that haspreceded. For, it does much more than present difficulties to someonetrying to provide an interpretation of the theory, it seems topoint either to a logical inconsistency in the theory'sfoundations.Suppose that we have a system S and a device S* which measures anobservable A on S with values {a1,a2, a3...}. Then there is somestate of S* (the ‘ground state’), and some observable Bwith values {b1, b2,b3...} pertaining to S* (its ‘pointerobservable’, so called because it is whatever plays the role ofthe pointer on a dial on the front of a schematic measuring instrumentin registering the result of the experiment), which are such that, ifS* is started in its ground state and interacts in an appropriate waywith S, and if the value of A immediately before the interaction isa1, then B's value immediately thereafter isb1. If, however, A's value immediately before theinteraction is a2, then B's value afterwards isb2; if the value of A immediately before theinteraction is a3, then B's value immediately afteris b3, and so on. That is just what itmeans to say that S* measures A. So, if we represent thejoint, partial state of S and S* (just the part of it which specifiesthe value of [A on S & B on S*], the observable whose valuescorrespond to joint assignments of values to the measured observable onS and the pointer observable on S*) by the vector |A=ai>s|B=bi>s*, and let "→" stand in for thedynamical description of the interaction between the two, to say thatS* is a measuring instrument for A is to say that the dynamical lawsentail that,|A=a1>s|B=groundstate>s* → |A=a1>s|B=b1>s* |A=a2>s|B=groundstate>s* → |A=a2>s|B=b2>s* |A=a3>s|B=groundstate>s* → |A=a3>s|B=b3>s*and so on.[12]Intuitively, S* is a measuring instrument for an observable A just incase there is some observable feature of S* (it doesn't matter what,just something whose values can be ascertained by looking at thedevice), which is correlated with the A-values of systems fed into itin such a way that we can read those values off of S*'s observablestate after the interaction. In philosophical parlance, S* is ameasuring instrument for A just in case there is some observablefeature of S* which tracks or indicates the A-valuesof systems with which it interacts in an appropriate way. Now, it follows from (3.1), above, that there are states of S (toomany to count) which are not eigenstates of A, and if we consider whatSchrödinger's equation tells us about the joint evolution of S andS* when S is started out in one of these, we find that the state of thepair after interaction is a superposition of eigenstates of [A on S& B on S*]. It doesn't matter what observable on S is beingmeasured, and it doesn't matter what particular superposition S startsout in; when it is fed into a measuring instrument for that observable,if the interaction is correctly described by Schrödinger'sequation, it follows just from the linearity of the U in that equation,the operator that effects the transformation from the earlier to thelater state of the pair, that the joint state of S and the apparatusafter the interaction is a superposition of eigenstates of thisobservable on the joint system.Suppose, for example, that we start S* in its ground state, and S inthe state1/√2|A=a1>s| + 1/√2|A=a2>sIt is a consequence of the rules for obtaining the state-space of thecomposite system that the combined state of the pair is 1/√2|A=a1>s|B=groundstate>s* + 1/√2|A=a2>s|B=groundstate>s* and it follows from the fact that S* is a measuring instrument for A,and the linearity of U that their combined state afterinteraction, is 1/√2|A=a1>s|B=b1>s* + 1/√2|A=a2>s|B=b2>s* This, however, is inconsistent with the dynamical rule for contexts oftype 2, for the dynamical rule for contexts of type 2 (and if there areany such contexts, this is one) entails that the state of thepair after interaction is either |A=a1>s|B=b1>s* or |A=a2>s|B=b2>s*Indeed, it entails that there is a precise probability of 1/2 that itwill end up in the former, and a probability of 1/2 that it will end upin the latter. We can try to restore logical consistency by giving up the dynamicalrule for contexts of type 2 (or, what amounts to the same thing, bydenying that there are any such contexts), but then we havethe problem of consistency with experience. For it was no mere blunderthat that rule was included in the theory; we know what asystem looks like when it is in an eigenstate of a given observable,and we know from looking that the measuring apparatus aftermeasurement is in an eigenstate of the pointer observable. And so weknow from the outset that if a theory tells us something elseabout the post-measurement states of measuring apparatuses, whateverthat something else is, it is wrong.That, in a nutshell, is the Measurement Problem in quantummechanics; any interpretation of the theory, any detailed story aboutwhat the world is like according to quantum mechanics, and inparticular those bits of the world in which measurements are going on,has to grapple with it.Loose EndsMixed states are weighted sums of pure states, andthey can be used to represent the states of ensembles whose componentsare in different pure states, or states of individual systems aboutwhich we have only partial knowledge. In the first case, the weightattached to a given pure state reflects the size of the component ofthe ensemble which is in that state (and hence the objectiveprobability that an arbitrary member of the ensemble is); in the secondcase, they reflect the epistemic probability that the system inquestion to which the state is assigned is in that state. If we don't want to lose the distinction between pure and mixedstates, we need a way of representing the weighted sum of a set of purestates (equivalently, of the probability functions associated withthem) that is different from adding the (suitably weighted) vectorsthat represent them, and that means that we need either an alternativeway of representing mixed states, or a uniform way of representing bothpure and mixed states that preserves the distinction between them.There is a kind of operator in Hilbert spaces, called a densityoperator, that serves well in the latter capacity, and itturns out not to be hard to restate everything that has been said aboutstate vectors in terms of density operators. So, even though it iscommon to speak as though pure states are represented by vectors, theofficial rule is that states – pure and mixed, alike - arerepresented in quantum mechanics by density operators.Although mixed states can, as I said, be used to representour ignorance of the states of systems that are actually in one oranother pure state, and although this has seemed to many to be anadequate way of interpreting mixtures in classical contexts, there areserious obstacles to applying it generally to quantum mechanicalmixtures. These are left for detailed discussion in the other entrieson quantum mechanics in the Encyclopedia.Everything that has been said about observables, strictly speaking,applies only to the case in which the values of the observable form adiscrete set; the mathematical niceties that are needed to generalizeit to the case of continuous observables arecomplicated, and raise problems of a more technical nature. These, too,are best left for detailed discussion.This should be all the initial preparation one needs toapproach the philosophical discussion of quantum mechanics,but it is only a first step. The more one learns about therelationships among and between vectors and operators in Hilbert space,about how the spaces of simple systems relate to those of complex ones,and about the equation which describes how state-vectors move throughthe space, the better will be one's appreciation of both the nature andthe difficulty of the problems associated with the theory. The funnybackwards thing about quantum mechanics, the thing that makes itendlessly absorbing to a philosopher, is that the more one learns, theharder the problems get.BibliographyAlbert, D., 1992, Quantum Mechanics and Experience,Cambridge, MA: Harvard University PressHalmos, P., 1957, Introduction to Hilbert Space, 2ndedition, Providence: AMS Chelsea PublishingOther Internet ResourcesPreskill, J., 1998, Quantum Computation (Lecture Notes for Physics 219,California Institute of Technology)Related Entries quantum mechanics: Bohmian mechanics | quantum mechanics: collapse theories | quantum mechanics: Copenhagen interpretation of | quantum mechanics: Everett's relative-state formulation of | quantum mechanics: Kochen-Specker theorem | quantum mechanics: many-worlds interpretation of | quantum mechanics: modal interpretations of | quantum mechanics: relational | quantum mechanics: the role of decoherence in Copyright © 2000 byJenann Ismael<jtismael@U.Arizona.EDU> |
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