theorem 9.72 Connection formulae

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2543 · p. 302

Rests on

depends_on corollary 9.62 The semiclassical wave ¶
- depends_on equation 9.62 eq:slt-wkb-ansatz ¶
- depends_on equation 9.61 eq:slt-wkb-kappa ¶
- depends_on proposition 9.61 Eikonal and transport equations ¶
  - depends_on equation 9.64 eq:slt-wkb-series ¶
  - depends_on proposition 9.60 Exact Riccati form ¶
    - depends_on equation 9.62 eq:slt-wkb-ansatz ¶ ↺
    - depends_on equation 9.60 eq:slt-wkb-normal-form ¶
    - proves proof ch:07-odes-sturm-liouville@proof-30 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-31 ¶
- proves proof ch:07-odes-sturm-liouville@proof-32 ¶
depends_on equation 9.79 eq:slt-wkb-airy-ai ¶
depends_on equation 9.80 eq:slt-wkb-airy-bi ¶
depends_on equation 9.76 eq:slt-wkb-airy-equation ¶
depends_on equation 9.74 eq:slt-wkb-airy-scale ¶
proves proof ch:07-odes-sturm-liouville@proof-37 ¶

Supports

depends_on postulate A.795 Matching, quoted ¶
- depends_on corollary A.799 The impressed pressure gradient ¶
depends_on proposition 23.57 The wave resolution at a turning surface ¶
- depends_on remark 23.58 The caustics that are not turning surfaces ¶
- depends_on remark 23.59 What is observed ¶
depends_on proposition 23.56 The phase jump at a fold ¶
depends_on proposition 9.74 Connection at a turning point of odd order ¶
depends_on remark 23.51 Where the expansion breaks, and what is owed ¶
- depends_on proposition 23.57 The wave resolution at a turning surface ¶ ↺
depends_on theorem 9.78 Semiclassical quantisation of a well ¶
- depends_on example 9.80 The harmonic oscillator ¶
- depends_on proposition 23.38 Bohr–Sommerfeld and Einstein–Brillouin–Keller quantization ¶
  - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
- depends_on proposition 52.24 The p-mode cavity ¶
  - depends_on phenomenon 52.25 The Sun rings, and its interior can be inverted ¶
    - depends_on phenomenon 52.22 The solar-neutrino deficit, and its resolution ¶
depends_on theorem 9.76 Semiclassical transmission through a barrier ¶
- assumes experiment Cockcroft and Walton: lithium split by accelerated protons ¶
  - cites (data) derived bib 3261 Disintegration of Lithium by Swift Protons ¶
  - cites (evidence) derived bib 3261 Disintegration of Lithium by Swift Protons ¶ ↺
  - cites (method) derived bib 830 Experiments with high velocity positive ions.---(I) Further developments in the method of obtaining high velocity positive ions ¶
  - cites (data) derived bib 831 Experiments with high velocity positive ions. II. -The disintegration of elements by high velocity protons ¶
  - cites (data) derived bib 831 Experiments with high velocity positive ions. II. -The disintegration of elements by high velocity protons ¶ ↺
  - cites (evidence) derived bib 831 Experiments with high velocity positive ions. II. -The disintegration of elements by high velocity protons ¶ ↺
  - cites (context) derived bib 3-4 Zur Quantentheorie des Atomkernes ¶
  - cites (context) derived bib 3-4 Zur Quantentheorie des Atomkernes ¶ ↺
  - cites (context) derived bib 3073 Wave Mechanics and Radioactive Disintegration ¶
  - cites (context) derived bib 3073 Wave Mechanics and Radioactive Disintegration ¶ ↺
  - cites (context) derived bib 2 Quantum Mechanics and Radioactive Disintegration ¶
  - cites (context) derived bib 2 Quantum Mechanics and Radioactive Disintegration ¶ ↺
  - … 1 more
- depends_on example 9.81 Gamow factor for an alpha particle, in SI ¶
  - depends_on example 9.82 Checking the validity criterion for the barrier above ¶
- depends_on phenomenon 117.5 The reaction proceeds far below the Coulomb barrier ¶
  - tests experiment Cockcroft and Walton: lithium split by accelerated protons ¶ ↺
- depends_on theorem 107.63 Alpha decay rate ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The semiclassical wave	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2566
`depends_on`	→	eq:slt-wkb-airy-ai	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2566
`depends_on`	→	eq:slt-wkb-airy-bi	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2566
`depends_on`	→	eq:slt-wkb-airy-equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2566
`depends_on`	→	eq:slt-wkb-airy-scale	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2566
`depends_on`	←	Matching, quoted	declared	appendices/A-long-proofs.tex:38939
`depends_on`	←	The wave resolution at a turning surface	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1945
`depends_on`	←	The phase jump at a fold	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1910
`depends_on`	←	Connection at a turning point of odd order	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2659
`depends_on`	←	Where the expansion breaks, and what is owed	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1762
`depends_on`	←	Semiclassical quantisation of a well	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2904
`depends_on`	←	Semiclassical transmission through a barrier	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2783
`proves`	←	ch:07-odes-sturm-liouville@proof-37	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2569