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That is what an interpretation of the theory would provide: a proper account of what the world is like according to quantum mechanics, intrinsically and from the bottom up. The problems with giving an interpretation (not just a comforting, homey sort of interpretation, i.e., not just an interpretation according to which the world isn't too different from the familiar world of common sense, but any interpretation at all) are dealt with in other sections of this encyclopedia. Here, we are concerned only with the mathematical heart of the theory, the theory in its capacity as a mathematical machine, and  whatever is true of the rest of it  this part of the theory makes exquisitely good sense.
A physical quantity is a mutually exclusive and jointly exhaustive family of physical properties (for those who know this way of talking, it is a family of properties with the structure of the cells in a partition). Knowing what kinds of values a quantity takes can tell us a great deal about the relations among the properties of which it is composed. The values of a bivalent quantity, for instance, form a set with two members; the values of a realvalued quantity form a set with the structure of the real numbers. This is a special case of something we will see again and again, viz., that knowing what kind of mathematical objects represent the elements in some set (here, the values of a physical quantity; later, the states that a system can assume, or the quantities pertaining to it) tells us a very great deal (indeed, arguably, all there is to know) about the relations among them.
In quantum mechanical contexts, the term ‘observable’ is used interchangeably with ‘physical quantity’, and should be treated as a technical term with the same meaning. It is no accident that the early developers of the theory chose the term, but the choice was made for reasons that are not, nowadays, generally accepted. The statespace of a system is the space formed by the set of its possible states,^{[2]} i.e., the physically possible ways of combining the values of quantities that characterize it internally. In classical theories, a set of quantities which forms a supervenience basis for the rest is typically designated as ‘basic’ or ‘fundamental’, and, since any mathematically possible way of combining their values is a physical possibility, the statespace can be obtained by simply taking these as coordinates.^{[3]} So, for instance, the statespace of a classical mechanical system composed of n particles, obtained by specifying the values of 6n realvalued quantities  three components of position, and three of momentum for each particle in the system  is a 6ndimensional coordinate space. Each possible state of such a system corresponds to a point in the space, and each point in the space corresponds to a possible state of such a system. The situation is a little different in quantum mechanics, where there are mathematically describable ways of combining the values of the quantities that don't represent physically possible states. As we will see, the statespaces of quantum mechanics are special kinds of vector spaces, known as Hilbert spaces, and they have more internal structure than their classical counterparts.
A structure is a set of elements on which certain operations and relations are defined, a mathematical structure is just a structure in which the elements are mathematical objects (numbers, sets, vectors) and the operations mathematical ones, and a model is a mathematical structure used to represent some physically significant structure in the world.
The heart and soul of quantum mechanics is contained in the Hilbert spaces that represent the statespaces of quantum mechanical systems. The internal relations among states and quantities, and everything this entails about the ways quantum mechanical systems behave, are all woven into the structure of these spaces, embodied in the relations among the mathematical objects which represent them.^{[4]} This means that understanding what a system is like according to quantum mechanics is inseparable from familiarity with the internal structure of those spaces. Know your way around Hilbert space, and become familiar with the dynamical laws that describe the paths that vectors travel through it, and you know everything there is to know, in the terms provided by the theory, about the systems that it describes.
By ‘know your way around’ Hilbert space, I mean something more than possess a description or a map of it; anybody who has a quantum mechanics textbook on their shelf has that. I mean know your way around it in the way you know your way around the city in which you live. This is a practical kind of knowledge that comes in degrees and it is best acquired by learning to solve problems of the form: How do I get from A to B? Can I get there without passing through C? And what is the shortest route? Graduate students in physics spend long years gaining familiarity with the nooks and crannies of Hilbert space, locating familiar landmarks, treading its beaten paths, learning where secret passages and dead ends lie, and developing a sense of the overall lay of the land. They learn how to navigate Hilbert space in the way a cab driver learns to navigate his city.
How much of this kind of knowledge is needed to approach the philosophical problems associated with the theory? In the beginning, not very much: just the most general facts about the geometry of the landscape (which is, in any case, unlike that of most cities, beautifully organized), and the paths that (the vectors representing the states of) systems travel through them. That is what will be introduced here: first a bit of easy math, and then, in a nutshell, the theory.
Multiplying a vector A> by n, where n is a constant, gives a vector which is the same direction as A> but whose length is n times A>'s length.
Figure 1: Vector Addition
In a real vector space, the (inner or dot) product of a pair of vectors A> and B>, written ‘<AB>’ is a scalar equal to the product of their lengths (or ‘norms’) times the cosine of the angle, , between them:
<AB> = A B cosLet A_{1}> and A_{2}> be vectors of length 1 ("unit vectors") such that <A_{1}A_{2}> = 0. (So the angle between these two unit vectors must be 90 degrees.) Then we can represent an arbitrary vector B> in terms of our unit vectors as follows:
B> = b_{1}A_{1}> + b_{2}A_{2}>For example, here is a graph which shows how B> can be represented as the sum of the two unit vectors A_{1}> and A_{2}>:
Figure 2: Representing B> by Vector Addition of Unit Vectors
Now the definition of the inner product <AB> has to be modified to apply to complex spaces. Let c* be the complex conjugate of c. (When c is a complex number of the form a ± bi, then the complex conjugate c* of c is defined as follows:
[a + bi]* = a biSo, for all complex numbers c, [c*]* = c, but c* = c just in case c is real.) Now definition of the inner product of A> and B> for complex spaces can be given in terms of the conjugates of complex coefficients as follows. Where A_{1}> and A_{2}> are the unit vectors described earlier, A> = a_{1}A_{1}> + a_{2}A_{2}> and B> = b_{1}A_{1}> + b_{2}A_{2}>, then
[a bi]* = a + bi
<AB> = (a_{1}*)(b_{1}) + (a_{2}*)(b_{2})
The most general and abstract notion of an inner product, of which we've now defined two special cases, is as follows. <AB> is an inner product on a vector space V just in case
(i) <AA> = A^{2}, and <AA>=0 if and only if A=0It follows from this that(ii) <BA> = <AB>*
(iii) <BA+C> = <BA> + <BC>.
(i) the length of A> is the square root of inner product of A> with itself, i.e.,A vector space is a set of vectors closed under addition, and multiplication by constants, an inner product space is a vector space on which the operation of vector multiplication has been defined, and the dimension of such a space is the maximum number of nonzero, mutually orthogonal vectors it contains.A = <AA>,and(ii) A> and B> are mutually perpendicular, or orthogonal, if, and only if, <AB> = 0.
Any collection of N mutually orthogonal vectors of length 1 in an Ndimensional vector space constitutes an orthonormal basis for that space. Let A_{1}>, ... , A_{N}> be such a collection of unit vectors. Then every vector in the space can be expressed as a sum of the form:
B> = b_{1}A_{1}> + b_{2}A_{2}> + ... + b_{N}A_{N}>,where b_{i} = <BA_{i}>. The b_{i}'s here are known as B's expansion coefficients in the Abasis.^{[5]}
Notice that:
(i) for all vectors A, B, and C in a given space,There is another way of writing vectors, namely by writing their expansion coefficients (relative to a given basis) in a column, like so:<AB+C> = <AB> + <AC>(ii) for any vectors M and Q, expressed in terms of the Abasis,
M> + Q> = (m_{1} + q_{1})A_{1}> + (m_{2} + q_{2})A_{2}> + ... + (m_{N} + q_{N})A_{N}>,and
<MQ> = m_{1}q_{1} + m_{2}q_{2} + ... + m_{n}q_{n}
where q_{i} = <QA_{i}> and the A_{i} are the chosen basis vectors.
Q> = q_{1}
q_{2}
When we are dealing with vector spaces of infinite dimension, since we can't write the whole column of expansion coefficients needed to pick out a vector since it would have to be infinitely long, so instead we write down the function (called the ‘wave function’ for Q, usually represented (i)) which has those coefficients as values. We write down, that is, the function:
(i) = q_{i} = <QA_{i}>Given any vector in, and any basis for, a vector space, we can obtain the wavefunction of the vector in that basis; and given a wavefunction for a vector, in a particular basis, we can construct the vector whose wavefunction it is. Since it turns out that most of the important operations on vectors correspond to simple algebraic operations on their wavefunctions, this is the usual way to represent statevectors.
When a pair of physical systems interact, they form a composite system, and, in quantum mechanics as in classical mechanics, there is a rule for constructing the statespace of a composite system from those of its components, a rule that tells us how to obtain, from the statespaces, H_{A} and H_{B} for A and B, respectively, the statespace  called the ‘tensor product’ of H_{A} and H_{B}, and written H_{A}H_{B}  of the pair. There are two important things about the rule; first, so long as H_{A} and H_{B} are Hilbert spaces, H_{A}H_{B} will be as well, and second, there are some facts about the way H_{A}H_{B} relates to H_{A} and H_{B}, that have surprising consequences for the relations between the complex system and its parts. In particular, it turns out that the state of a composite system is not uniquely defined by those of its components. What this means, or at least what it appears to mean, is that there are, according to quantum mechanics, facts about composite systems (and not just facts about their spatial configuration) that don't supervene on facts about their components; it means that there are facts about systems as wholes that don't supervene on facts about their parts and the way those parts are arranged in space. The significance of this feature of the theory cannot be overplayed; it is, in one way or another, implicated in most of its most difficult problems.
In a little more detail: if {v_{i}^{A}} is an orthonormal basis for H_{A} and {u_{j}^{B}} is an orthonormal basis for H_{B}, then the set of pairs (v_{i}^{A}, u_{j}^{B}) is taken to form an orthonormal basis for the tensor product space H_{A}H_{B}. The notation v_{i}^{A}u_{j}^{B} is used for the pair (v_{i}^{A},u_{j}^{B}), and inner product on H_{A}H_{B} is defined as:^{[6]}
<v_{i}^{A}u_{m}^{B}  v_{j}^{A}u_{n}^{B}> = <v_{i}^{A}  v_{j}^{A}> <u_{m}^{B}  u_{n}^{B}>It is a result of this construction that although every vector in H_{A}H_{B} is a linear sum of vectors expressible in the form v^{A}u^{B}, not every vector in the space is itself expressible in that form, and it turns out that
(i) any composite state defines uniquely the states of its components.(ii) if the states of A and B are pure (i.e., representable by vectors v^{A} and u^{B}, respectively), then the state of (A+B) is pure and represented by v^{A}u^{B}, and
(iii) if the state of (A+B) is pure and expressible in the form v^{A}u^{B}, then the states of A and B are pure, but
(iv) if the states of A and B are not pure, i.e., if they are mixed states (these are defined below), they do not uniquely define the state of (A+B); in particular, it may be a pure state not expressible in the form v^{A}u^{B}.
(i) O(A> + B>) = OA> + OB>, andJust as any vector in an Ndimensional space can be represented by a column of N numbers, relative to a choice of basis for the space, any linear operator on the space can be represented in a column notation by N^{2} numbers:(ii) O(cA>) = c(OA>).
where O_{ij} = < A_{i}  OA_{j}> and the A_{N}> are the basis vectors of the space. The effect of the linear operator O on the vector B is, then, given by
O = O_{11}
O_{21}O_{12}
O_{22}
OB> =Two more definitions before we can say what Hilbert spaces are, and then we can turn to quantum mechanics. B> is an eigenvector of O with eigenvalue a if, and only if, OB> = aB>. Different operators can have different eigenvectors, but the eigenvector/operator relation depends only on the operator and vectors in question, and not on the particular basis in which they are expressed; the eigenvector/operator relation is, that is to say, invariant under change of basis. Hermitean operators are linear operators, which have only real eigenvalues.
= O_{11}
O_{21}O_{12}
O_{22}× b_{1}
b_{2}
= (O_{11}b_{1} + O_{12}b_{2})
(O_{21}b_{1} + O_{22}b_{2})
= (O_{11}b_{1} + O_{12}b_{2})A_{1}> + (O_{21}b_{1} + O_{22}b_{2})A_{2}>
= B>
A Hilbert space, finally, is a vector space on which an inner product is defined, and which is complete, i.e., which is such that any Cauchy sequence of vectors in the space converges to a vector in the space. All finitedimensional inner product spaces are complete, and I will restrict myself to these. The infinite case involves some complications that are not fruitfully entered into at this stage.
b. Contexts of type 2 ("Measurement Contexts"):^{[9]} Carrying out a "measurement" of an observable B on a system in a state A> has the effect of collapsing the system into a Beigenstate corresponding to the eigenvalue observed. This is known as the Collapse Postulate. Which particular Beigenstate it collapses into is a matter of probability, and the probabilities are given by a rule known as Born's Rule:
prob(b_{i}) = <AB=b_{i}>^{2}.
There are two important points to note about these two kinds of contexts:
All of the physically consequential features of the behaviors of quantum mechanical systems are consequences of mathematical properties of those relations, and the most important of them are easily summarized:
(P1) Any way of adding vectors in a Hilbert space or multiplying them by scalars will yield a vector that is also in the space. 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 is the sum of a pair of eigenvectors of an observable B with distinct eigenvalues, it will not itself be an eigenvector of B, but will be associated, from (3.4b), with a set of probabilities for showing one or another result in Bmeasurements.If we make a couple of additional interpretive assumptions, we can say more. Assume, for instance, that(P2) For any Hermitian operator on a Hilbert space, there are others, on the same space, with which it doesn't share a full set of eigenvectors; indeed, it is easy to show that there are other such operators with which it has no eigenvectors in common.
(4.1) Every Hermitian operator on the Hilbert space associated with a system represents a distinct observable, and (hence) every normalized vector, a distinct state, andIt follows from (P2), by (3.1), that no quantum mechanical state is an eigenstate of all observables (and indeed that there are observables which have no eigenstates in common), and so, by (3.2), that no quantum mechanical system ever has simultaneous values for all of the quantities pertaining to it (and indeed that there are pairs of quantities to which no state assigns simultaneous values).(4.2) A system has a value for observable A if, and only if, the vector representing its state is an eigenstate of the Aoperator. The value it has, in such a case, is just the eigenvalue associated with that eigenstate.^{[11]}
There are Hermitian operators on the tensor product H_{1}H_{2} of a pair of Hilbert spaces H_{1} and H_{2} ... In the event that H_{1} and H_{2} are the state spaces of systems S1 and S2, H_{1}H_{2} is the statespace of the complex system (S1+S2). It follows from this by (4.1) that there are observables pertaining to (S1+S2) whose values are not determined by the values of observables pertaining to the two individually.
These are all straightforward consequences of taking vectors and operators in Hilbert space to represent, respectively, states and observables, and applying Born's Rule (and later (4.1) and (4.2)), to give empirical meaning to state assignments. That much is perfectly well understood; the real difficulty in understanding quantum mechanics lies in coming to grips with their implications  physical, metaphysical, and epistemological.
There is one remaining fact about the mathematical structure of the theory that anyone trying to come to an understanding about what it says about the world has to grapple with. It is not a property of Hilbert spaces, this time, but of the dynamics, the rules that describe the trajectories that systems follow through the space. From a physical point of view, it is far more worrisome than anything that has preceded. For, it does much more than present difficulties to someone trying to provide an interpretation of the theory, it seems to point either to a logical inconsistency in the theory's foundations.
Suppose that we have a system S and a device S* which measures an observable A on S with values {a_{1}, a_{2}, a_{3}...}. Then there is some state of S* (the ‘ground state’), and some observable B with values {b_{1}, b_{2}, b_{3}...} pertaining to S* (its ‘pointer observable’, so called because it is whatever plays the role of the pointer on a dial on the front of a schematic measuring instrument in registering the result of the experiment), which are such that, if S* is started in its ground state and interacts in an appropriate way with S, and if the value of A immediately before the interaction is a_{1}, then B's value immediately thereafter is b_{1}. If, however, A's value immediately before the interaction is a_{2}, then B's value afterwards is b_{2}; if the value of A immediately before the interaction is a_{3}, then B's value immediately after is b_{3}, and so on. That is just what it means to say that S* measures A. So, if we represent the joint, partial state of S and S* (just the part of it which specifies the value of [A on S & B on S*], the observable whose values correspond to joint assignments of values to the measured observable on S and the pointer observable on S*) by the vector A=a_{i}>_{s}B=b_{i}>_{s*}, and let "" stand in for the dynamical description of the interaction between the two, to say that S* is a measuring instrument for A is to say that the dynamical laws entail that,
A=a_{1}>_{s}B=ground state>_{s*} A=a_{1}>_{s}B=b_{1}>_{s*}Intuitively, S* is a measuring instrument for an observable A just in case there is some observable feature of S* (it doesn't matter what, just something whose values can be ascertained by looking at the device), which is correlated with the Avalues of systems fed into it in such a way that we can read those values off of S*'s observable state after the interaction. In philosophical parlance, S* is a measuring instrument for A just in case there is some observable feature of S* which tracks or indicates the Avalues of systems with which it interacts in an appropriate way.A=a_{2}>_{s}B=ground state>_{s*} A=a_{2}>_{s}B=b_{2}>_{s*}
A=a_{3}>_{s}B=ground state>_{s*} A=a_{3}>_{s}B=b_{3}>_{s*}
and so on.^{[12]}
Now, it follows from (3.1), above, that there are states of S (too many to count) which are not eigenstates of A, and if we consider what Schrödinger's equation tells us about the joint evolution of S and S* when S is started out in one of these, we find that the state of the pair after interaction is a superposition of eigenstates of [A on S & B on S*]. It doesn't matter what observable on S is being measured, and it doesn't matter what particular superposition S starts out in; when it is fed into a measuring instrument for that observable, if the interaction is correctly described by Schrödinger's equation, it follows just from the linearity of the U in that equation, the operator that effects the transformation from the earlier to the later state of the pair, that the joint state of S and the apparatus after the interaction is a superposition of eigenstates of this observable on the joint system.
Suppose, for example, that we start S* in its ground state, and S in the state
1/2A=a_{1}>_{s} + 1/2A=a_{2}>_{s}It is a consequence of the rules for obtaining the statespace of the composite system that the combined state of the pair is
1/2A=a_{1}>_{s}B=ground state>_{s*} + 1/2A=a_{2}>_{s}B=ground state>_{s*}and it follows from the fact that S* is a measuring instrument for A, and the linearity of U that their combined state after interaction, is
1/2A=a_{1}>_{s}B=b_{1}>_{s*} + 1/2A=a_{2}>_{s}B=b_{2}>_{s*}This, however, is inconsistent with the dynamical rule for contexts of type 2, for the dynamical rule for contexts of type 2 (and if there are any such contexts, this is one) entails that the state of the pair after interaction is either
A=a_{1}>_{s}B=b_{1}>_{s*}Indeed, it entails that there is a precise probability of 1/2 that it will end up in the former, and a probability of 1/2 that it will end up in the latter.or
A=a_{2}>_{s}B=b_{2}>_{s*}
We can try to restore logical consistency by giving up the dynamical rule for contexts of type 2 (or, what amounts to the same thing, by denying that there are any such contexts), but then we have the problem of consistency with experience. For it was no mere blunder that that rule was included in the theory; we know what a system looks like when it is in an eigenstate of a given observable, and we know from looking that the measuring apparatus after measurement is in an eigenstate of the pointer observable. And so we know from the outset that if a theory tells us something else about the postmeasurement states of measuring apparatuses, whatever that something else is, it is wrong.
That, in a nutshell, is the Measurement Problem in quantum mechanics; any interpretation of the theory, any detailed story about what the world is like according to quantum mechanics, and in particular those bits of the world in which measurements are going on, has to grapple with it.
If we don't want to lose the distinction between pure and mixed states, we need a way of representing the weighted sum of a set of pure states (equivalently, of the probability functions associated with them) that is different from adding the (suitably weighted) vectors that represent them, and that means that we need either an alternative way of representing mixed states, or a uniform way of representing both pure and mixed states that preserves the distinction between them. There is a kind of operator in Hilbert spaces, called a density operator, that serves well in the latter capacity, and it turns out not to be hard to restate everything that has been said about state vectors in terms of density operators. So, even though it is common to speak as though pure states are represented by vectors, the official rule is that states – pure and mixed, alike  are represented in quantum mechanics by density operators.
Although mixed states can, as I said, be used to represent our ignorance of the states of systems that are actually in one or another pure state, and although this has seemed to many to be an adequate way of interpreting mixtures in classical contexts, there are serious obstacles to applying it generally to quantum mechanical mixtures. These are left for detailed discussion in the other entries on 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 a discrete set; the mathematical niceties that are needed to generalize it to the case of continuous observables are complicated, 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 to approach the philosophical discussion of quantum mechanics, but it is only a first step. The more one learns about the relationships 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 statevectors move through the space, the better will be one's appreciation of both the nature and the difficulty of the problems associated with the theory. The funny backwards thing about quantum mechanics, the thing that makes it endlessly absorbing to a philosopher, is that the more one learns, the harder the problems get.
First published: November 29, 2000
Content last modified: November 29, 2000