wang
(Wang)
#1
• The norm onHprovides with a natural notion of distance, which induced the distance
topology onH, and allows us to investigate the convergence of sequences inH. The definition
of Hilbert space includes the requirement thatHbe acomplete space.
By definition, a spaceHis complete if every Cauchy sequence{|un〉}{n}⊂Hconverges.
(Recall that a sequence{|un〉}{n}⊂ H is a Cauchy sequence provided||um−un|| →0 if
m,n → ∞implies that there exists a|u〉 ∈ Hsuch that the sequence converges to|u〉,
namely||un−u||→0 asn→ ∞.) This property is automatic when the dimension ofHis
finite, but is an extra requirement when the dimension is infinite.
• The Hilbert spaces in quantum mechanics are required to beseparable, which means
that they admit a countable orthonormal basis. If the number of orthormal basis vectors of
a separable Hilbert spaceHisN <∞, thenHis isomorphic toCN. On the other hand, all
separable Hilbert spaces of infinite dimension are isomorphic to one another.
3.1.1 Triangle and Schwarz Inequalities
The distance defined by the norm satisfies the triangle inequality,
||u+v||≤||u||+||v|| (3.5)
for all|u〉,|v〉∈H, which in turn implies the Schwarz inequality,
|〈u|v〉|≤||u||·||v|| (3.6)
Both may be shown using the following arguments. If|v〉= 0, both inequalities hold trivially.
Henceforth, we shall assume that|v〉6= 0. Positivity of the norm implies that for any complex
numberλ, we have
0 ≤||u+λv||^2 =||u||^2 +|λ|^2 ||v||^2 +λ〈u|v〉+λ∗〈u|v〉∗ (3.7)
We now choose the phase ofλto be such thatλ〈u|v〉is real and negative; as a result,λ〈u|v〉=
−|λ||〈u|v〉|. Using the fact that|v〉 6= 0, we may choose|λ|=||u||/||v||. Substituting the
corresponding value forλinto (3.7) immediately yields (3.6). Using nowλ= 1 in (3.7) and
bounding|〈u|v〉|using the Schwarz inequality readily gives (3.5).
3.1.2 The construction of an orthonormal basis
Hilbert spaces of finite dimension and separable Hilbert spaces of infinite dimension share the
property that one can construct an orthonormal basis using theGramm-Schmidt procedure.
Let{|un〉}nbe a basis ofH, whereneither runs over the finite set{ 1 , 2 ,···,N}or over all
