## 1. Terminology




Physical systems are divided into **types** according to
their unchanging (or 'state-independent') properties, and
the **state** of a system at a time consists of a complete
specification of those of its properties that change with time (its
'state-dependent' properties). To give a complete
description of a system, then, we need to say what type of system it is
and what its state is at each moment in its history. 




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 real-valued
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 **state-space**
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 state-space
can be obtained by simply taking these as
 coordinates.[3]
 So, for instance, the state-space of a classical mechanical system
composed of \(n\) particles, obtained by specifying the values of
\(6n\) real-valued quantities -- three components of position, and
three of momentum for each particle in the system -- is a
\(6n\)-dimensional 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 state-spaces 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 state-spaces 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.


## 2. Mathematics


### 2.1 Vectors and vector spaces




A **vector** \(A\), written '\(\ket{A}\)', is a
mathematical object characterized by a length, \(|A|\), and a direction. A
normalized vector is a vector of length 1; i.e., \(|A| = 1\). Vectors can
be added together, multiplied by constants (including complex
numbers), and multiplied together. Vector addition maps any pair of
vectors onto another vector, specifically, the one you get by moving
the second vector so that its tail coincides with the tip of the
first, without altering the length or direction of either, and then
joining the tail of the first to the tip of the second. This addition
rule is known as the parallelogram law. So, for example, adding
vectors \(\ket{A}\) and \(\ket{B}\) yields vector \(\ket{C} (= \ket{A} + \ket{B})\) as
in Figure 1: 



![vector addition](figure1.gif)
Figure 1.
Vector Addition





Multiplying a vector \(\ket{A}\) by \(n\), where \(n\) is a
constant, gives a vector which is the same direction as \(\ket{A}\) but
whose length is \(n\) times \(\ket{A}\)'s length. 




In a real vector space, the
 (inner or dot) product
 of a pair of vectors \(\ket{A}\) and \(\ket{B}\), written
'\(\braket{A}{B}\)' is a scalar equal to the product of their
lengths (or 'norms') times the cosine of the angle,
\(\theta\), between them:



\[
\braket{A}{B} = |A| |B| \cos \theta
\]



Let \(\ket{A\_1}\) and \(\ket{A\_2}\) be vectors of length 1
("unit vectors") such that \(\braket{A\_1}{A\_2} = 0\). (So
the angle between these two unit vectors must be 90 degrees.) Then we
can represent any two-dimensional vector \(\ket{B}\) in terms of our unit vectors
as follows: 



\[
\ket{B} = b\_1 \ket{A\_1} + b\_2 \ket{A\_2}
\]



For example, here is a graph which shows how \(\ket{B}\) can be represented
as the sum of the two unit vectors \(\ket{A\_1}\) and \(\ket{A\_2}\): 



![figure2](figure2.gif)
Figure 2.
Representing \(\ket{B}\) by Vector Addition of Unit Vectors





Now the definition of the inner product \(\braket{A}{B}\) 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 \(a \pm bi\), then the complex conjugate
\(c^\*\) of \(c\) is defined as follows:



\[
[a+bi]^\* = a-bi \\
[a-bi]^\* = a+bi
\]



So, for all complex numbers \(c\), \([c^\*]^\* = c\),
but \(c^\* = c\) just in case \(c\) is real.) Now
definition of the inner product of \(\ket{A}\) and \(\ket{B}\) for complex spaces
can be given in terms of the conjugates of complex coefficients as
follows. Where \(\ket{A\_1}\) and \(\ket{A\_2}\) are the unit
vectors described earlier, \(\ket{A} = a\_1 \ket{A\_1} + a\_2 \ket{A\_2}\) and \(\ket{B} = b\_1 \ket{A\_1} + b\_2 \ket{A\_2}\), then 



\[
\braket{A}{B} = (a\_1^\*)(b\_1) + (a\_2^\*)(b\_2)
\]



The most general and abstract notion of an inner product, of which
we've now defined two special cases, is as follows. \(\braket{A}{B}\) is an
inner product on a vector space \(V\) just in case


1. \(\braket{A}{A} = |A|^2\), and \(\braket{A}{A}=0\) if and only if \(A=0\)
2. \(\braket{B}{A} = \braket{A}{B}^\*\)
3. \(\braket{B}{A+C}=\braket{B}{A} + \braket{B}{C}\).




It follows from this that 


1. the length of \(\ket{A}\) is the square root of inner
product of \(\ket{A}\) with itself, i.e.,

\[
|A| = \sqrt{\braket{A}{A}}
\]




and 


2. \(\ket{A}\) and \(\ket{B}\) are mutually perpendicular, or
**orthogonal**, if, and only if, \(\braket{A}{B} = 0\).




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. 




Any collection of \(N\) mutually orthogonal vectors of length 1 in an
\(N\)-dimensional vector space constitutes an **orthonormal
basis** for that space. Let \(\ket{A\_1}, \ldots, \ket{A\_N}\) be such a collection of unit vectors. Then every
vector in the space can be expressed as a sum of the form:



\[
\ket{B} = b\_1 \ket{A\_1} + b\_2 \ket{A\_2} + \ldots + b\_N \ket{A\_N},
\]



where \(b\_i = \braket{B}{A\_i}\). The \(b\_i\)'s here are
known as \(B\)'s **expansion coefficients** in the \(A\)-basis.[5]




Notice that:


1. for all vectors \(A\), \(B\), and \(C\) in a given space,

\[
\braket{A}{B+C} = \braket{A}{B} + \braket{A}{C}
\]
2. for any vectors \(M\) and \(Q\), expressed in terms of the \(A\)-basis,

\[
\ket{M} + \ket{Q} = \sum\_{i=1}^N (m\_i + q\_i) \ket{A\_i},
\]

 
 and



 \[
\braket{M}{Q} = \sum\_{i=1}^N m\_i^\* q\_i
 \]




There is another way of writing vectors, namely by writing their
expansion coefficients (relative to a given basis) in a column, like
so: 



\[
\ket{Q} = \begin{bmatrix}
 q\_1 \\
 q\_2
\end{bmatrix}
\]




where \(q\_i = \braket{Q}{A\_i}\) and the \(A\_i\) are the
chosen basis vectors. 




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 \(\psi(i))\) which has those coefficients as values. We write down, that is, the
function:



\[
\psi(i) = q\_i = \braket{Q}{A\_i}
\]



Given any vector in, and any basis for, a vector space, we can obtain
the wave-function of the vector in that basis; and given a
wave-function for a vector, in a particular basis, we can construct the
vector whose wave-function it is. Since it turns out that most of the
important operations on vectors correspond to simple algebraic
operations on their wave-functions, this is the usual way to represent
state-vectors. 




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 state-space of a composite system from those
of its components, a rule that tells us how to obtain, from the
state-spaces, \(H\_A\) and \(H\_B\) for \(A\) and \(B\),
respectively, the state-space -- called the 'tensor
product' of \(H\_A\) and \(H\_B\), and written
\(H\_A \otimes H\_B\) -- of the pair. There are two important
things about the rule; first, so long as \(H\_A\) and
\(H\_B\) are Hilbert spaces, \(H\_A \otimes H\_B\) will
be as well, and second, there are some facts about the way
\(H\_A \otimes H\_B\) 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 \(H\_A \otimes H\_B\). The
notation \(v\_i^A \otimes u\_j^B\) is used
for the pair \((v\_{i}^A,u\_{j}^B)\), and inner product on \(H\_A \otimes H\_B\)
is defined as:[6]



\[
\braket{v\_i^A \otimes u\_m^B}{v\_j^A \otimes u\_n^B} = \braket{v\_i^A}{v\_j^A} \braket{u\_m^B}{u\_n^B}
\]



It is a result of this construction that although every vector in
\(H\_A \otimes H\_B\) is a linear sum of vectors expressible in the form
\(v^A \otimes u^B\), not every vector in the space is itself
expressible in that form, and it turns out that 


1. any composite state defines uniquely the states of its
components.
2. 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
\(v^A \otimes u^B\),
 and
3. if the state of \((A+B)\) is pure and expressible in the form
 \(v^A \otimes u^B\),
 then the states of \(A\) and \(B\) are pure, but
4. 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
 \(v^A \otimes u^B\).


### 2.2 Operators




An **operator** \(O\) is a mapping of a vector space onto
itself; it takes any vector \(\ket{B}\) in a space onto another
vector \(\ket{B'}\) also in
the space; \(O \ket{B} = \ket{B'}\). **Linear operators** are operators
that have the following properties: 


1. \(O(\ket{A} + \ket{B}) = O \ket{A} + O \ket{B}\), and
2. \(O(c \ket{A}) = c(O \ket{A})\).




Just as any vector in an \(N\)-dimensional 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: 



\[O = \begin{bmatrix}
 O\_{11} & O\_{12} \\
 O\_{21} & O\_{22}
\end{bmatrix}\]



where \(O\_{ij} = \braket{A\_i}{O \mid A\_j}\) 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 



\[\begin{align}
O \ket{B} &=
 \begin{bmatrix} O\_{11} & O\_{12} \\
 O\_{21} & O\_{22} \end{bmatrix}
 \times \begin{bmatrix} b\_1 \\ b\_2 \end{bmatrix} \\
&= \begin{bmatrix}
 (O\_{11}b\_1 + O\_{12}b\_2) \\ (O\_{21}b\_1 + O\_{22}b\_2) \end{bmatrix} \\
&= (O\_{11}b\_1 + O\_{12}b\_2)|A\_1\rangle + (O\_{21}b\_1 + O\_{22}b\_2|A\_2\rangle \\
&=|B'\rangle
\end{align}\]



Two more definitions before we can say what Hilbert spaces are, and
then we can turn to quantum mechanics. \(\ket{B}\) is an
 **eigenvector** of \(O\) with
eigenvalue \(a\) if, and only if, \(O \ket{B} = a \ket{B}\).
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. A **Hermitean operator**
is an operator which has the property that there is an orthonormal
basis consisting of its eigenvectors and those eigenvalues are all
real. 




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 finite-dimensional 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.


## 3. Quantum Mechanics




Four basic principles of quantum mechanics are: 



(3.1)
**Physical States.**
Every physical system is associated with a Hilbert Space, every unit
vector in the space corresponds to a possible pure state of the system,
and every possible pure state, to some vector in the
 space.[7]


(3.2)
**Physical Quantities.**
Hermitian operators in the Hilbert space associated with a system
represent physical quantities, and their eigenvalues represent the
possible results of measurements of those quantities. 


There is an operator, called the Hamiltonian, that plays a special
role in quantum theory because the dynamics of a system can be
conveniently formulated by tracking its evolution. The Hamiltonian
- written \(H\), or \(\hat{H}\) - stands for the total
energy of the system. Its eigenvalues are the possible results that
might be obtained in measurements of total energy. It is given by
summing over the kinetic and potential energies of the system's
components.


(3.3)
**Composition.**
The Hilbert space associated with a complex system is the tensor
product of those associated with the simple systems (in the standard,
non-relativistic, theory: the individual particles) of which it is
composed. 


(3.4)
**Dynamics.**

a.
*Contexts of type 1*: Given the state of a system at
\(t\) and the forces and constraints to which it is subject, there is
an equation, '**Schrodinger's
equation**', that gives the state at any other time \(U
\ket{v\_t} \rightarrow \ket{v\_{t'}}\).[8]
 The important properties of \(U\) for our
purposes are that it is **deterministic**, which is to
say that it takes the state of a system at one time into a unique
state at any other, it is **unitary**, which means that
it is an automorphism of the Hilbert space on which it acts (i.e., a
mapping of that space onto itself that preserves the linear space
structure and inner product), and it is **linear**, which
is to say that if it takes a state \(\ket{A}\) onto the state
\(\ket{A'}\), and it takes the state \(\ket{B}\) onto the state
\(\ket{B'}\), then it takes any state of the form \(\alpha \ket{A} +
\beta \ket{B}\) onto the state \(\alpha \ket{A'} + \beta
\ket{B'}\).


b.
*Contexts of type 2* ("Measurement
 Contexts"):[9]
Carrying out a "measurement" of an observable \(B\) on a
system in a state \(\ket{A}\) has the effect of collapsing the system
into a \(B\)-eigenstate corresponding to the eigenvalue observed. This
is known as the **Collapse
Postulate**. Which *particular* \(B\)-eigenstate it
collapses into is a matter of probability, and the probabilities are
given by a rule known as **Born's Rule**:



\[
\Pr(b\_i) = |\braket{A}{B=b\_i}|^2
\] 



There are two important points to note about these two kinds of
contexts:


* The distinction between contexts of type 1 and 2 remains to be made
out in quantum mechanical terms; nobody has managed to say in a
completely satisfactory way, in the terms provided by the theory, which
contexts are measurement contexts, and
* Even if the distinction is made out, it is an open interpretive
question whether there *are* contexts of type 2; i.e., it is an
open interpretive question whether there are any contexts in which
systems are governed by a dynamical rule *other* than
Schrodinger's equation.


## 4. Structures on Hilbert Space




I remarked above that in the same way that all the information we have
about the relations between locations in a city is embodied in the
spatial relations between the points on a map which represent them, all
of the information that we have about the internal relations among (and
between) states and quantities in quantum mechanics is embodied in the
mathematical relations among the vectors and operators which represent
 them.[10]
 From a mathematical point of view, what
really distinguishes quantum mechanics from its classical predecessors
is that states and quantities have a richer structure; they form
families with a more interesting network of relations among their
members. 




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 \(B\)-measurements.


(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.





If we make a couple of additional interpretive assumptions, we can say
more. Assume, for instance, that 



(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, and


(4.2)
A system has a value for observable \(A\) if, and only if, the
vector representing its state is an eigenstate of the \(A\)-operator. The
value it has, in such a case, is just the eigenvalue associated with
that
 eigenstate.[11]





It 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). 




There are Hermitian operators on the tensor product
\(H\_1 \otimes H\_2\) 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 \(S2\),
\(H\_1 \otimes H\_2\) is the state-space 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.



Anyone trying to come to an understanding about what quantum mechanics
says about the world has to grapple with one remaining fact. This
problem is not an issue with Hilbert spaces, 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 discussed to this point. It not only presents
difficulties to someone trying to provide an *interpretation*
of the theory, but also seems to point 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
\(\ket{A=a\_i}\_s \ket{B=b\_i}\_{s^\*}\), and let "\(\rightarrow\)"
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,



\[\begin{align}
\ket{A=a\_1}\_s \ket{B=\text{ground state}}\_{s^\*} &\rightarrow \ket{A=a\_1}\_s \ket{B=b\_1}\_{s^\*} \\
\ket{A=a\_2}\_s \ket{B=\text{ground state}}\_{s^\*} &\rightarrow \ket{A=a\_2}\_s \ket{B=b\_2}\_{s^\*} \\
\ket{A=a\_3}\_s \ket{B=\text{ground state}}\_{s^\*} &\rightarrow \ket{A=a\_3}\_s \ket{B=b\_3}\_{s^\*}
\end{align}\]



and so on.[12]




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 \(A\)-values 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 \(A\)-values
of systems with which it interacts in an appropriate way. 




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
Schrodinger'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 Schrodinger'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



\[
\frac{1}{\sqrt{2}} \ket{A = a\_1}\_s + \frac{1}{\sqrt{2}} \ket{A=a\_2}\_s
\]

It is a consequence of the rules for obtaining the state-space of the
composite system that the combined state of the pair is 



\[
\frac{1}{\sqrt{2}} \ket{A=a\_1}\_s \ket{B=\text{ground state}}\_{s^\*} + \frac{1}{\sqrt{2}} \ket{A = a\_2}\_s \ket{B = \text{ground state}}\_{s^\*}
\]

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 



\[
\frac{1}{\sqrt{2}} \ket{A=a\_1}\_s \ket{B=b\_1}\_{s^\*} + \frac{1}{\sqrt{2}} \ket{A = a\_2}\_s \ket{B = b\_2}\_{s^\*}
\]


 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* 



\[
\ket{A=a\_1}\_s \ket{B=b\_1}\_{s^\*}
\]



or



\[
\ket{A=a\_2}\_s \ket{B=b\_2}\_{s^\*}
\]



Indeed, it entails that there is a precise probability of \(\frac{1}{2}\) that it
will end up in the former, and a probability of \(\frac{1}{2}\) that it will end up
in the latter. 




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 post-measurement 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.


### Loose Ends


**Mixed states** are weighted sums of pure states, and
they can be used to represent the states of ensembles whose components
are in different pure states, or states of individual systems about
which we have only partial knowledge. In the first case, the weight
attached to a given pure state reflects the size of the component of
the ensemble which is in that state (and hence the objective
probability that an arbitrary member of the ensemble is); in the second
case, they reflect the epistemic probability that the system in
question to which the state is assigned is in that state.



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 state-vectors 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.