© Kamal K. Barley, Andreas Ruffing, and Sergei K. Suslov
“Old Quantum Mechanics by Bohr and Sommerfeld from Modern Perspective”

© К. Барли, А. Руффинг, С.К. Суслов
“Старая квантовая механика Бора и Зоммерфельда с современной точки зрения”

Успехи физических наук, Январь 2026 г., Том 196, № 1, с. 83-103

DOI: https://doi.org/10.3367/UFNr.2025.08.040014

(Last modified/executed on December 4, 2026;  12:10 PM Arizona time.)

Appendix E/Д: Mathematica Derivation of Equations (3.18)-(3.21)

Introduce function s(θ)=1/r from (3.9):

s[θ_]:=(1+ϵ*Cos[ω*θ])/(a*(1-(ϵ^2)))

s[θ]

and evaluate its derivative:

D[s[θ],θ]

The r.h.s. of (3.4), evaluate:

(D[s[θ],θ])^2+(s[θ])^2

and substitute:

%/.((Sin[θ ω])^2)->1-(((Cos[θ ω]))^2)

Introduce S and eliminate cosine:

Solve[S==((1+ϵ Cos[θ ω])/(a (1-(ϵ^2)))),Cos[θ ω]]

FullSimplify[%%/.Cos[θ ω]->((-1+a S-a S (ϵ^2))/ϵ)]

As a result, in terms of S, the above expressions becomes:

Collect[%,S]

Equation (3.4), define the difference between r.h.s. and l.h.s.:

P=((((En/(m (c^2)))+((Z (e^2) S )/(m (c^2)))))^2)-1-(((((ℏ (Subscript[n, θ]))/(m c))))^2)*(((ω^2)/((a^2) (-1+(ϵ^2))))-((2 S (ω^2))/(a (-1+(ϵ^2))))-(S^2) (-1+(ω^2)))

and simplify

%/.(e^2)->α*ℏ*c ;

Collect[%,S]

To satisfy (3.4), all coefficients of this quadratic polynomials in S must vanish:

{(Subscript[C, 0])=(((Z^2) (α^2) (ℏ^2))/((c^2) (m^2)))+(((-1+(ω^2)) (ℏ^2) (Subscript[n, θ]^2))/((c^2) (m^2))),(Subscript[C, 1])=((2 En Z α ℏ)/((c^3) (m^2)))+((2 (ω^2) (ℏ^2) (Subscript[n, θ]^2))/(a (c^2) (m^2) (-1+(ϵ^2)))),(Subscript[C, 2])=-1+((En^2)/((c^4) (m^2)))-(((ω^2) (ℏ^2) (Subscript[n, θ]^2))/((a^2) (c^2) (m^2) (-1+(ϵ^2))))}

From the vanishing leading term, we choose the positive solution as in (3.19):

Solve[(Subscript[C, 0])==0,ω]

From the linear term, find the energy in terms of a as follows:

Solve[(Subscript[C, 1])==0,En]

Alternatively,  from (3.17):

Solve[((En Z α)/((Subscript[n, θ]^2)*c*ℏ*(ω^2)))==(1/(a (1-(ϵ^2)))),En]

We verify that the linear term is vanished:

(Subscript[C, 1])/.En->-((c (ω^2) ℏ (Subscript[n, θ]^2))/(a Z α (-1+(ϵ^2))))

The constant term can be simplified in terms of (3.16):

(Subscript[C, 2])/.(1/((-1+(ϵ^2))))->-((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)]))^2)/((ω^2) (Subscript[n, θ]^2)))

The same for the linear term, in order to eliminate epsilon from the expression for energy:

(Subscript[C, 1])/.(1/((-1+(ϵ^2))))->-((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)]))^2)/((ω^2) (Subscript[n, θ]^2)))

Solve[%==0,En]

Exclude energy from the constant term:

-1+((En^2)/((c^4) (m^2)))+(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/((a^2) (c^2) (m^2)))/.En->((c ℏ ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/(a Z α))

Now solve the modified equation for A=a^2:

Solve[(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/((c^2) (m^2)))+(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^4))/((c^2) (m^2) (Z^2) (α^2)))==A,A]

Transform the result as follows:

Factor[(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/((c^2) (m^2)))+(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^4))/((c^2) (m^2) (Z^2) (α^2)))]

An equivalent expression:

(((ℏ^2) ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2)*(((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2)+(Z^2) (α^2)))/((c^2) (m^2) (Z^2) (α^2)))

Verification:

Simplify[%-%%]

Finally, substitute the parameter a into the previously found expression for energy En:

((c ℏ ((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/(a Z α))/.a->((ℏ ((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])) (Sqrt[(Z^2) (α^2)+((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2)]))/(c m Z α))

Compare the square of this result with the reciprocal square of equation (3.18):

Simplify[((Z^2) (α^2)+((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2))/((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2)-(1+(((Z^2) (α^2))/((((Subscript[n, r])+(Sqrt[-(Z^2) (α^2)+(Subscript[n, θ]^2)])))^2)))]

Thus we derived equations (3.18)-(3.21) with the help of Mathematica computer algebra system (see also our other complementary Mathematica notebook, BohrAtomMathematica.nb, for more details).