Hilbert Spaces 1 Introduction Hilbert spaces are the mathematical structures that underpinns modern quan- tum mechanics. Here we go through the essential concepts needed before learning how to apply this to the study of nature. Familarity with mathematical sym- bols, calculus and linear algebra is assumed. Some of the notation we will use are: ) leads to 8 for all 2 belongs to −! maps ⊆ subset of , equivalent ! goes to j with the condition 2 Linear spaces We want to consider the topic of linear spaces. In linear algebra, we have the following composition rules for vectors. Note that the latin letters u; v; w; ::: denote vectors while the greek letters λ, µ, ::: denote scalars. (L1) u + v = v + u (commutative law) u + (v + w) = (u + v) + w (associative law) u + 0 = u (0 is the null element) u + (−1)u = 0 (-u is the opposite element ) (L2) λ(µu) = (λµ)u 1 · u = u 0 · u = 0 λ · 0 = 0 (0 is the null element) (L3) (λ + µ)u = λu + µu (distributive law) λ(u + v) = λu + λv (distributive law) Remark: The same rules apply for other entities, such as n-tiples in Rn and m × n-matrices. For n-tiples, addition is defined as (x1; x2; ::: ; xn) + (y1; y2; ::: ; yn) = (x1 + y1; x2 + y2; ::: ; xn + yn): (1) and multiplication with a scalar is defined as λ(x1; x2; ::: ; xn) = (λx1; λx2; ::: ; λxn) (2) The rules (L1)-(L3) also apply to functions, given appropriate definitions of the operations ’addition’ and ’multiplication with scalar’. Such spaces are called function spaces. 1 We now define the term linear space, a space in which (L1)-(L3) are axioms and further properties are derived from these axioms. Definition: A linear space over R is a set H on which the following opera- tions are defined: addition: u 2 H, v 2 H =) u + v 2 H, multiplication with scalar: λ 2 R, u 2 H =) λu 2 H, such that (L1)-(L3) applies. The space can also be over C instead of R (we then talk about a linear space over C). The elements on H are called vectors. Instead of linear space, the term vector space may be used. Example: For the complex n-tiples n C = f(z1; z2; ::: ; zn)jzk 2 C; (k = 1; ::: ; n)g addition and multiplication with a scalar are defined as in (1) and (2). It can be shown that (L1)-(L3) applies to this set, so Cn is a linear space over C. Example: A subset Ω of R can be a limited or unlimited interval of R. Given end points a and b, some possible subsets Ω are: (a; b) = fxja < x < bg ; (a; b] = fxja < x ≤ bg ; [a; 1) = fxja ≤ xg Let Φ = Φ(Ω) = fall functions f :Ω −! Rg 1be the set of all real functions on Ω. For two functions f; g 2 Φ(Ω); λ 2 R, the functions f + g and λf are defined as f + g : x 7! f(x) + g(x); x 2 Ω; λf : x 7! λf(x); x 2 Ω; Using these operations, it can be shown that (L1)-(L3) applies to Φ and it is therefore a linear space over R. (Here, 0 represents the null function.) This is an example of a function space. We simply regard the functions as vectors in the linear space Φ(Ω). If one replaces R with C, it can be shown that the set of all complex functions on Ω is a linear space over C. Definition: U ⊆ H is a linear subspace of H if u 2 U; v 2 U =) u + v 2 U; and λ 2 R (or C); u 2 U =) λu 2 U: Because U is a subspace of H, (L1)-(L3) still apply. Example: If H is a set containing all geometrical vectors in 3-dimensional space and U is a set containing all vectors parallel to a given plane π, then U is a subspace of H. It can be shown that if u and v are two vectors in the plane π, both the vector u + v and all vectors λu, with λ 2 R are also in π. 1The notation should be read as: The set of all functions f that maps points from the interval Ω to R 2 Example: The set n n X U = f(z1; ::: ; zn) 2 C j zk = 0g k=1 is a subspace of Cn, because n n n 0 00 X 0 00 X 0k X 00 z 2 U; z 2 U =) (zk + zk ) = z + zk = 0g; k=1 k=1 k=1 n X λ 2 C; z 2 U =) λzk = 0 k=1 However, the set n n X U = f(z1; ::: ; zn) 2 C j zk = 1g k=1 is not a subspace of Cn. This is because the zero vector 0 = (0; :::; 0) 62 U. All subspaces must contain the zero vector, because 0 · z = 0 for all z. Example: The set 2 U = f(x1; x2) 2 R j x1x2 = 0g is not a subspace of R2. We can easily prove this with an example: x0 = (0; 1) 2 U; x00 = (1; 0) 2 U =) x0 + x00 = (1; 1) 62 U Example: The set Πn(R) containing all possible polynomials n X k akx k=0 with a degree ≤ n and real coefficients ak can be shown to be a subspace of Φ(R). If p and q are polynomials and λ 2 R, both p + q and λp are polynomials as well. This also holds true for polynomials with complex coefficients, only in this case Πn(C) is a subspace of Φ(C). Definition: Consider a linear space H. The vectors u1; u2; :::; un 2 H are said to be linearly independent if n X λjuj = 0 =) λj = 0; 8j j=1 If any vector v 2 H can be expressed as a linear combination of the vectors uj, the set u1; ::: ; un is said to be a basis in H. Remark: This is certainly true in a finite space. To show this for an infinite space, more care is required. All bases in a linear space H have the same number of elements. The number of elements in a basis of H is the dimension of H. It is not possible to have a finite basis if the space is infinite-dimensional. Example: The dimension of n-dimensional real space is n dim R = n: 3 The dimension of n-dimensional complex space is n dim C = n The dimension of the set containing all n-degree polynomials is dim Πn = n + 1; 2 n because the basis of Πn is [1; x; x ; :::; x ]. The set Πn is a subspace of all poly- nomials (any degree), which is denoted Π. We will now define the important function spaces C(Ω) and Ck(Ω). Ω is a con- nected domain in Rn, that may be open (the boundary points are not part of the domain) or closed (all boundary points are part of the domain). • C(Ω): Functions that are continuous in Ω. This is a linear space and a subspace of Φ(Ω). • Ck(Ω): Functions whose derivatives of order ≤ k are continuous in Ω. This is a linear space and a subspace of Φ(Ω). 3 Scalar product and norm Given two elements in a linear space u; v we denote their scalar product by (ujv). In the ordinary Rn case, the scalar product can be written n X (ujv) = ukvk: (3) k=1 The length of a vector can be expressed with the help of a scalar product: n 1=2 X 2 1=2 kuk = (uju) = ( uk) (4) k=1 The length of a vector is usually called the norm. The norm kuk ≥ 0 for all u 2 H. In the C case, we must modify the definition of the scalar product. The norm should still be a real, positive number and be defined by the scalar product. Equation (4) will usually not give a real number if the numbers uk are complex. If the scalar product is defined as n X ∗ (ujv) = ukvk; (5) k=1 then the norm can be written n 1=2 X 2 1=2 kuk = (uju) = ( jukj ) : k=1 This is a real number ≥ 0. The following rules hold true for the scalar product (ujv), as defined by (3) and (5): (S1) (ujλ1v1 + λ2v2) = λ1(ujv1) + λ2(ujv2) (S2) (ujv) = (vju)∗ (S3) (uju) ≥ 0 (equality for u = 0) ∗ ∗ (S4) (λ1u1 + λ2u2jv) = λ1(u1jv) + λ2(u2jv)) 4 Note that (S4) follows from (S1) and (S2). Definition: A scalar product on a linear space H is a rule which associates two elements u; v 2 H to a scalar, (ujv), so that the rules (S1)-(S4) apply. A linear space with a scalar product is called a pre-Hilbert space. Example: In the continuous case C(Ω) the scalar product is defined by Z (ujv) = u∗(x)v(x)dx: Ω It is easily verified that rules (S1)-(S4) apply. Example: Now for the general case. Given a positive function w > 0; w 2 C(Ω), Z ∗ (ujv)w = u (x)v(x)w(x)dx (6) Ω is a scalar product on C(Ω). When Ω is a finite domain in R this also holds for piecewise continuous functions with finite-value discontinuities. The func- tion w is called a weight function. Different weight functions define different pre-Hilbert spaces. Definition: For a pre-Hilbert space H, i) u; v are orthogonal if (ujv) = 0.