proposition 12.106 The plane wave is a generalized momentum eigenvector

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2860 · p. 444

Rests on

depends_on definition 12.105 Generalized eigenvector ¶
- depends_on definition 12.103 Gelfand triple ¶
  - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
    - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.46 Functional ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
      - depends_on theorem 12.18 Projection theorem ¶
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-25 ¶
  - depends_on corollary 12.19 Double complement; the density criterion ¶
    - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on definition 12.16 Orthogonal complement ¶
      - proves proof ch:10-hilbert-spaces@proof-8 ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-10 ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
- depends_on definition 12.50 Point, continuous and residual spectrum ¶
  - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
  - depends_on definition 12.49 Resolvent set; spectrum ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
    - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶
depends_on equation 12.54 eq:hilbert-momentum-operator ¶
depends_on example 12.104 The Schwartz triple ¶
- depends_on definition 12.103 Gelfand triple ¶ ↺
- depends_on example 12.11 The function space $L^{2}$ ¶
  - depends_on definition 12.2 Hilbert space ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
  - depends_on definition 5.18 Inner product ¶ ↺
- depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
  - depends_on example 12.11 The function space $L^{2}$ ¶ ↺
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-1 ¶
  - proves proof ch:10-hilbert-spaces@proof-48 ¶
proves proof ch:10-hilbert-spaces@proof-49 ¶

Supports

depends_on example A.283 Momentum on the line ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Generalized eigenvector	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2867
`depends_on`	→	eq:hilbert-momentum-operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2867
`depends_on`	→	The Schwartz triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2867
`depends_on`	←	Momentum on the line	declared	appendices/A-long-proofs.tex:14226
`proves`	←	ch:10-hilbert-spaces@proof-49	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2871