proposition 9.60 Exact Riccati form

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2075 · p. 297

Rests on

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 ¶

Supports

depends_on proposition 9.61 Eikonal and transport equations ¶
- 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 ¶
      - depends_on example 9.81 Gamow factor for an alpha particle, in SI ¶
      - depends_on phenomenon 117.5 The reaction proceeds far below the Coulomb barrier ¶
      - 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 proposition 52.24 The p-mode cavity ¶
    - 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-ansatz	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2082
`depends_on`	→	eq:slt-wkb-normal-form	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2082
`depends_on`	←	Eikonal and transport equations	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2139
`proves`	←	ch:07-odes-sturm-liouville@proof-30	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:2085