For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.
For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.
Complex Hilbert spaces play an important role in the description of quantum systems. In fact, with every quantum system there is associated an infinite dimensional or a finite dimensional separable complex Hilbert space ℍ that consists of the states of the quantum system. In physics terminology, the Hilbert space ℍ is usually referred to as the space of (pure) states. Throughout this monograph, the mathematical description of a quantum system shall be based on a certain complex (separable) Hilbert space ℍ, and therefore the quantum system will simply be denoted by ℍ. The quantum system ℍ is said to be a finite-dimensional system if ℍ is a finite-dimensional complex Hilbert space. Otherwise, the quantum system ℍ is said to be an infinite-dimensional system. The Hilbert space ℍ representing a composite quantum system that consists of n subsystems is a tensor product of the Hilbert spaces of n component systems described by ℍ 1 , … ℍ n , i.e., ℍ = ℍ 1 ⊗ … ⊗ ℍ n . In the theory of Hilbert spaces, a tensor product of Hilbert spaces is a way to create a new Hilbert space out of 2 (or more) Hilbert spaces. In the following, we illustrate the concept and construction of tensor product ℍ ⊗ 핂 of only 2 Hilbert spaces ℍ and 핂. The concepts and constructions can be easily extended to more than 2 Hilbert spaces such as ℍ 1 ⊗ ℍ 2 ⊗ … ⊗ ℍ n . Recall that the tensor product of ℍ with 핂 denoted by ℍ ⊗ 핂 is a new Hilbert space that consists of elements ϕ ⊗ φ ( ϕ ∈ ℍ and φ ∈ 핂) and is equipped with the Hilbertian inner product 〈·, ·〉 ℍ⊗핂 :(ℍ ⊗ 핂) × (ℍ ⊗ 핂) → C defined by 〈 ϕ 1 ⊗ φ 1 , ϕ 2 ⊗ φ 2 〉 ℍ⊗핂 = 〈 ϕ 1 , ϕ 2 〉 ℍ 〈 φ 1 , φ 2 〉 핂 , ϕ i ∈ ℍ and φ i ∈ 핂 for i = 1, 2.
The superposition principle is a founding principle of quantum mechanics; we have already made use of it in interpreting the Young's slit experiment. Quantum mechanics is a linear theory, and so it is natural that vector spaces play an important role in it. We shall see that a physical state is represented mathematically by a vector in a space whose characteristics we shall define; this is called the space of states . A second founding principle, which can also be deduced from the Young's slit experiment, is the existence of probability amplitudes. These probability amplitudes will be represented mathematically by scalar products defined on the space of states. In the theory of waves, the use of complex numbers is just a convenience, but in quantum mechanics the probability amplitudes are fundamentally complex numbers – the scalar product will a priori be a complex number. Physical properties like momentum, position, energy, and so on will be represented by operators acting in the space of states. In this chapter we shall introduce the essential properties of Hilbert spaces, that is, vector spaces on which a positive-definite scalar product is defined, and we shall limit ourselves to the case of finite dimension. This restriction will be lifted later on, because the space of states is in general of infinite dimension. The mathematical theory of Hilbert spaces of infinite dimension is much more complicated than that of spaces of finite dimension, and we shall put off studying them until Chapter 7. The reader familiar with vector spaces of finite dimension and operators in such spaces can proceed directly to Chapter 3 after reviewing the notation.
In Chapter 4 we saw that the canonical commutation relations force us to use a space of states of infinite dimension, in which rigor would require the use of advanced mathematical tools. Fortunately, physicists generally need only to carry the results for finite dimension over to infinite dimension with some simple modifications which we shall indicate here, without embarking on sophisticated mathematics. Nevertheless, it is useful to be aware of the lapses in rigor which are customarily made in physics in order to avoid possible unpleasant surprises. The objective of this chapter is, on the one hand, to present some concrete examples illustrating the new features which arise in infinite dimension and, on the other, to give the rules for practical calculations, in particular to write down the spectral decomposition of Hermitian and unitary operators. The mathematics we use is a bit more detailed than commonly found in most quantum mechanics textbooks. The reader interested purely in the practical aspects can proceed directly to Section 7.3, where the results essential for later on are summarized. Hilbert spaces Definitions The space of states of quantum mechanics is a Hilbert space ℌ, which in general is of infinite dimension. The axiomatic definition of a Hilbert space is the following. It is a vector space which, for the needs of quantum mechanics, is defined on complex numbers. The vectors of this space are denoted |ϕ〉. […]
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
The quantum analogue of a finite set X (in its role as a configuration space in classical mechanics) is the finite-dimensional Hilbert space ℓ2(X) , by which we mean the vector space of functions ψ:X→C , equipped with the inner product ⟨ψ,φ⟩=∑x∈Xψ(x)ϕ(x).
In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.