proposition 9.61 Eikonal and transport equations

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2121 · p. 297

Rests on

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 ¶

Supports

depends_on corollary 9.62 The semiclassical wave ¶
- depends_on proposition 9.63 The transport equation is flux conservation ¶
  - 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 ¶
- depends_on theorem 9.72 Connection formulae ¶
  - 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 theorem 9.76 Semiclassical transmission through a barrier ¶ ↺
- depends_on theorem 9.78 Semiclassical quantisation of a well ¶ ↺
- depends_on theorem 9.76 Semiclassical transmission through a barrier ¶ ↺
depends_on proposition 23.49 The Hamilton–Jacobi equation is the leading order ¶
- depends_on remark 23.50 The dictionary completed ¶
depends_on proposition 9.65 Validity criterion ¶
- depends_on example 9.82 Checking the validity criterion for the barrier above ¶ ↺
- depends_on remark 23.51 Where the expansion breaks, and what is owed ¶ ↺
- depends_on remark 23.50 The dictionary completed ¶ ↺

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`	→	eq:slt-wkb-series	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2139
`depends_on`	→	Exact Riccati form	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2139
`depends_on`	←	The semiclassical wave	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2183
`depends_on`	←	The Hamilton–Jacobi equation is the leading order	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1674
`depends_on`	←	Validity criterion	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2268
`proves`	←	ch:07-odes-sturm-liouville@proof-31	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2142