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Bohr radius

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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
ϵ = E / ( Z 1 2 Z 2 2 m c 2 α 2 / 2 ) .
For Z 1 Z 2 = 1 and m = m e , the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, a 0 = / ( m e c α ) , and to a multiple of the Rydberg constant, …
2: 5.5 Functional Relations
§5.5(iv) Bohr–Mollerup Theorem
3: 5.18 q -Gamma and q -Beta Functions
Also, ln Γ q ( x ) is convex for x > 0 , and the analog of the Bohr–Mollerup theorem (§5.5(iv)) holds. …
4: 5.19 Mathematical Applications
The volume V and surface area S of the n -dimensional sphere of radius r are given by …
5: 2.11 Remainder Terms; Stokes Phenomenon
2.11.8 n = ρ p + α ,
2.11.14 a 2 ( θ , α ) = 1 12 ( 6 α 2 6 α + 1 ) α 1 + e i θ + 1 ( 1 + e i θ ) 2 .
2.11.16 c ( θ ) = 2 ( 1 + e i θ + i ( θ π ) ) ,
2.11.18 h 0 ( θ , α ) = e i α ( π θ ) 1 + e i θ i c ( θ ) .
6: 1.5 Calculus of Two or More Variables
With 0 r < , 0 ϕ 2 π , … With 0 r < , 0 ϕ 2 π , < z < , … With 0 ρ < , 0 ϕ 2 π , 0 θ π , …
1.5.39 ( x , y ) ( r , ϕ ) = r (polar coordinates) .
1.5.41 ( x , y , z ) ( ρ , θ , ϕ ) = ρ 2 sin θ (spherical coordinates) .
7: 22.18 Mathematical Applications
In polar coordinates, x = r cos ϕ , y = r sin ϕ , the lemniscate is given by r 2 = cos ( 2 ϕ ) , 0 ϕ 2 π . …
22.18.4 l ( r ) = ( 1 / 2 ) arccn ( r , 1 / 2 ) .
22.18.5 r = cn ( 2 l , 1 / 2 ) ,
8: 1.9 Calculus of a Complex Variable
where
1.9.4 r = ( x 2 + y 2 ) 1 / 2 ,
If h ( w ) is continuous on | w | = R , then with z = r e i θ
§1.9(vi) Power Series
The circle | z z 0 | = R is called the circle of convergence of the series, and R is the radius of convergence. …
9: 18.39 Applications in the Physical Sciences
b) The Bohr Quantum Number
where n is now the Bohr Principle Quantum Number. Physical scientists use the n of Bohr as, to 0 th and 1 st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first. … The solution, (18.39.29), of the spherical radial equation (18.39.28), now expressed in terms of the Bohr quantum number n , is … which is Bohr’s quantization for the Coulomb bound state energies (18.39.31). …
10: 3.4 Differentiation
Taking C to be a circle of radius r centered at x 0 , we obtain
3.4.18 1 k ! f ( k ) ( x 0 ) = 1 2 π r k 0 2 π f ( x 0 + r e i θ ) e i k θ d θ .
3.4.19 1 k ! = 1 2 π r k 0 2 π e r cos θ cos ( r sin θ k θ ) d θ .