Quantum Mechanics
Quantum mechanics is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic particles -- or, at least, of the measuring instruments we use to explore those behaviors -- and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had. Mathematically, the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to quantum mechanics. Minimally interpreted, the theory describes a set of facts about the way the microscopic world impinges on the macroscopic one, how it effects our measuring instruments, described in everyday language or the language of classical mechanics. Disagreement centers on the question of what a microscopic world, which affects our apparatuses in the prescribed manner, is, or even could be, like intrinsically; or how those apparatuses could themselves be built out of microscopic parts of the sort the theory describes.[1]
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.
· 1. Terminology
· 2. Mathematics
· 3. Quantum Mechanics
· 4. Structures on Hilbert Space
· Bibliography
· Other Internet Resources
· Related Entries
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
Vectors and vector spaces
A vector A, written ‘|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 it's 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 |A> and |B> yields vector |C> (= |A> + |B>) as in Figure 1:
Figure 1: Vector Addition
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.
In a real vector space, the (inner or dot) product of a pair of vectors |A> and |B>, written ‘’ is a scalar equal to the product of their lengths (or ‘norms’) times the cosine of the angle, , between them:
= |A| |B| cos
Let |A1> and |A2> be vectors of length 1 ("unit vectors") such that and the Ai 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 (i)) which has those coefficients as values. We write down, that is, the function:
(i) = qi =
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, HA and HB for A and B, respectively, the state-space -- called the ‘tensor product’ of HA and HB, and written HA HB -- of the pair. There are two important things about the rule; first, so long as HA and HB are Hilbert spaces, HA HB will be as well, and second, there are some facts about the way HA HB relates to HA and HB, 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 {viA} is an orthonormal basis for HA and {ujB} is an orthonormal basis for HB, then the set of pairs (viA, ujB) is taken to form an orthonormal basis for the tensor product space HA HB. The notation viA ujB is used for the pair (viA,ujB), and inner product on HA HB is defined as:[6]