The Phase-Integral Method, The Bohr-Sommerfeld Condition and The Restricted Soap Bubble: with a proposition concerning the associated Legendre equation

2011-06-30

The Phase-Integral Method, The Bohr-Sommerfeld Condition and The Restricted Soap Bubble

with a proposition concerning the associated Legendre equation

Hazhar Ghaderi

Supervisor: Staffan Yngve

Kandidatprogrammet i Fysik

Department of Physics and Astronomy

CONDITION AND THE RESTRICTED SOAP BUBBLE WITH A PROPOSITION CONCERNING THE ASSOCIATED LEGENDRE

EQUATION.

Bachelor Thesis for The Bachelor of Science Programme in Physics, 15hp.

Examensjobb för Kandidatprogrammet i fysik, 15hp.

Supervisor: Staffan Yngve

HAZHAR GHADERI

Abstract. After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman & Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we de- rive an expression for the angular frequencies of a restricted (0 ≤ ϕ ≤ β) soap bubble and also give a proposition concerning the parameters # and m of the associated Legendre differential equation.

SAMMANFATTNING. Vi använder Fröman & Fröman’s Fas-Integral Metod till- sammans med Yngve & Thidé’s plattformfunktion för att härleda kvantiser- ingsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade ap- sidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sek- tion fyra tittar vi på en såpbubbla begränsad till området 0 ≤ ϕ ≤ β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna # och m tillhörande den associerade Legendreekvationen.

Date: June, 2011.

i

Acknowledgement

I would like to thank my supervisor Staffan Yngve.

Contents

1. Background 1

2. The Phase-Integral Method 3

2.1. Phase-Integrals generated from an unspecified base function 3

2.2. The Connection Formulæ 7

3. The Bohr-Sommerfeld Quantization Condition. 8

3.1. The Bohr-Sommerfeld Condition in terms of apsidal angle. 8 3.2. The hydrogen atom and the isotropic 3-D harmonic oscillator. 11 3.3. Higher order quantization condition for the radial problem. 13

4. The vibrating soap bubble and a proposition. 15

4.1. The angular frequency of a restricted soap film and a proposition. 15

5. Conclusion 19

References 20

Appendix A. The limits of the integral!(t^!!)

(t^!) q(z) dz. 21

Appendix B. Quantization Condition: Radial single-well potential 22 Appendix C. δ for a hydrogen potential of the form: V (r) = −µa^!20r−r^C3 24 Appendix D. Alternative computation of the hydrogen energy eigenvalues. 25

1. Background

The so-called WKB approximation (see [1], [2], [3], [5], [6] for instance) has some serious deficiencies (in higher order terms) due to it’s unsatisfactory form (between the preexponential factor and the integrand in the exponent) and it’s lack of flexibil- ity [11]. A well known example where one of these deficiencies appear is the radial Schrödinger equation with a Coulomb singularity in the potential at the origin and with " "= 0. In this case, the WKB half-integer quantization condition does not, in general, yield the correct energy levels. What is more for the radial Schrödinger equation with a Coulomb or a Centrifugal potential, the first-order WKB wave- function does not have the correct r^!+1 behavior at the origin.

These issues are remedied if one makes the substitution "("+1) −→ ("+1/2)²empiri- cally motivated and suggested by Kramers [16]. Langer [8] sought to justify Kramers substitution and found by making the transformation¹ r = e^x, ψ(r) = e^x/2u(x), that the centrifugal potential (in the first-order WKB hydrogen radial wavefunction) transformed to h²(" + 1/2)²/2µr² thus justifying Kramers substitution. But the Kramer-Langer substitution does not always im- prove the results and can even worsen the approximations in some cases as was shown (only a decade after Kramers publication) by L. Yost, J. A. Wheeler and G.

Breit [10]. All of the above led Fröman and Fröman to develop and generalize and make rigorous the WKB method [5] into what is now called The Phase-Integral Method [11], [15].

The Phase-Integral Method (PIM) is superior to the usual WKB method in that it does not suffer from the first of the deficiencies mentioned above, i.e. the lack of a simple relation between the preexponential factor and the integrand [11], which will be seen below. The second deficiency (WKB’s lack of flexibility) is also absent here since the PIM contains an unspecified function, the so-called base function Q(z), which one can choose with some degree of freedom and which can (if one chooses the square of the base function Q²(z)wisely) simplify expressions and yield exact results (e.g. for phase shifts or eigenvalues).

The PIM is also in many (not all) cases superior and preferable to regular per- turbation methods as shown by Yngve & Linnaeus ( [12], [13] and [14] ) in that it is more accurate and valid in a wider range, for instance for phase shift (produced by complex potentials) calculations [14]. The PIM can also be used in regions were perturbation theory completely breaks down cf. fig2. of [12].

1Actually it has been shown that the Langer exponential transformation r = e^x is completely irrelevant and that only ψ(r) = e^x/2u(x) = r^1/2u(x)is needed, see [17] and the references given there.

Another important part of the PIM is the platform function which in the case of Fröman and Fröman is denoted by $0. Since in applying the Phase-Integral method, one often ends up doing a lot of closed contour integrals, it is of utmost interest to find total derivatives in each order of approximation, because then calculating the contributions from these terms becomes a trivial task. In the case of Fröman and Fröman, total derivatives are not found until one reaches the fifth order (cf. eq.

2.2.10c in [15]), but by using the platform function Ps(z) of Yngve and Thidé one finds a total derivative already in the third order [18]. Moreover Ps(z)depends on the parameter s which depending on what value one assigns to it, defines the base function Q(z) and hence can simplify calculations for specific problems.

In this paper, we will write the Bohr-Sommerfeld quantization condition (1^st and 3rd order), in terms of the apsidal angle as defined in section 3 (cf. eq. 8.31 of [19]

which is 2× our definition of apsidal angle). Since the apsidal angle for different classical motions in a central potential is known, the Bohr-Sommerfeld quantization condition (in first order approximation) thus reduces to a simple integral as will be seen and we will obtain the exact energy eigenvalues for the hydrogen atom and the 3-dimensional harmonic oscillator. We will also find the angular frequency of vibration for the restricted soap bubble and give an interesting proposition concern- ing the relation between " and m, the degree and order of the associated Legendre equation respectively.

2. The Phase-Integral Method

The Phase-Integral Method can be applied to equations of the form d²u

dz² + R(z)u = 0, (2.1)

where R(z) is a meromorphic² function of the complex variable z. We call these Schrödinger-type³ equations. Any linear second-order homogeneous differential equation of the general form

y^!!+ A(z)y^!+ B(z)y = 0 (2.2)

can be transformed into a Schrödinger type equation (2.1) by the transformation:

ln y = ln u−1 2

"

A(z)dz. (2.3)

Ris then given by

R(z) = B(z)−1

2A^!(z)−1

4A²(z) (2.4)

For reference we also give R for an equation given in Sturm-Liouville form d

dz

# p(z)dy

dz

$ +%

λ ˜˜w(z)− ˜q(z)&

y = 0, (2.5)

where ˜λ (the eigenvalue) is a constant ˜q(z) a function and ˜w(z)is a known (weight) function. In this case, applying the transformation (2.3) or (2.4) one finds

R(z) = λ ˜˜w(z)− ˜q(z) p(z) −1

2 d dz

' 1 p(z)

dp dz

(

−1 4

' 1 p(z)

dp dz

(2

. (2.6)

We will be using this in section (4) where we will among other things tackle the problem of finding the eigenfrequencies of a restricted (in angular coordinate ϕ) soap bubble with the phase-integral method.

2.1. Phase-Integrals generated from an unspecified base function.

In trying to find two linearly independent solutions of (2.1), we make the ansatz

u_±(z) = A(z)e^±iw(z), (2.7)

where A(z) and w(z) are two admissible complex functions. Utilizing the Wronskian ([20],[21]) of the two functions (2.7), we find that two linearly independent solutions of (2.1) can be written in the form

u₊(z) = q⁻¹²(z)e^+iw(z) (2.8a) u₋(z) = q⁻¹²(z)e^−iw(z) (2.8b) where

w(z) =

" z

q(z^#)dz^# (2.9)

2A function R(z) is said to be meromorphic in a domain D if at every point of D it is either analytic or has a pole.

3The 1-D Schrödinger equation (−!²/2m)(d²/dz²)ψ + V (z)ψ = Eψ, can be written as (d²/dz²)ψ +!

2m[E− V (z)]/!²"

ψ = 0⇒ R(z) = 2m[E − V (z)]/!², where V (z) may be the ac- tual or an effective potential.

is called the phase integral and q(z) the phase integrand. Inserting (2.8) and (2.9) into (2.1) we find (cf. chapter 3 of [5] or any of these [11], [15])

q⁻³² d²

dz²q⁻¹² + R(z)q⁻²− 1 = 0 (2.10) or equivalently

q⁺¹² d²

dz²q⁻¹² − q²+ R(z) = 0. (2.11) Thus, for any solution satisfying the q-equation (2.11), the functions (2.8) and (2.9) are linearly independent solutions of (2.1). Suppose that we some- how have found a function Q(z) which we will call the base function, such that Q(z) is an approximate solution of the q-equation (2.11). That is, substituting the approximate solution Q(z) into (2.10) we require that the quantity

Q⁻³² d²

dz²Q⁻¹² + R(z)Q⁻²− 1 = $0, (2.12) be small compared to unity⁴. To take this smallness into account Fröman and Fröman [11], [15], introduce a bookkeeping parameter λ to be set to unity in the end and study the auxiliary equation

d²u dz² +

#Q²

λ² + R(z)− Q²

$

u = 0. (2.13)

Inserting (2.8) and (2.9) into (2.13) we get q^+1/2 d²

dz²q^−1/2− q²+

#Q²

λ² + R(z)− Q²

$

= 0, (2.14)

or (cf. ([18])) alternatively, 'qλ

Q (2

− λ² 1

Q² [qλ]^1/2 d²

dz²[qλ]^−1/2− λ²R(z)− Q²

Q² = 1, (2.15) which upon defining

ζ :=

" _z

z0

Q(z^#)dz^# (2.16)

(where typically Q²(z₀) = 0) becomes 'qλ

Q (2

− λ² 'qλ

Q (1/2

d² dζ²

'qλ Q

(1/2

− λ²R(z)− Q²

Q² − λ² 1 Q^3/2

d² dz²

1 Q^1/2 = 1.

(2.17)

4In the global problem, the condition |%0(z)| % 1 might not always be sufficient (p.21 [15]).

Using (2.16) and the identity 1

Q^3/2 d² dz²

1 Q^1/2 ≡ 1

Q d dz

'1 2z^s d

dz 1 z^sQ

(

− '1

2z^s d dz

1 z^sQ

(2

+s(s− 2) 4z²Q² .

(2.18)

equation (2.17) can be rewritten as 'qλ

Q (2

− λ² 'qλ

Q

(1/2 d² dζ²

'qλ Q

(_−1/2

+ λ²

)Q²− R(z) − ^s(s_4z⁻²⁾²

Q² −dPs(z)

dζ + P_s²(z)

*

= 1,

(2.19)

where

P_s(z) := 1 2z^s d

dz 1

z^sQ (2.20)

is the platform function of Yngve & Thidé, [18]. This can be compared with

$₀of (2.12) which is the platform function of Fröman & Fröman [15] by using eq. (12) of [18], that is

R(z)− Q²

Q² +dP0(z)

dζ − P0²(z)≡ $0. (2.21) Now, to obtain a formal solution of (2.19), we insert the following ansatz⁵

qλ Q =

+∞ n=0

Y2nλ²ⁿ (2.22)

into (2.19) and by recognizing powers of λ we find (cf. [18] and [26])

Y0 = 1. (2.23)

Y₂=−1

2P_s²+1 2

dP_s

dζ +R(z)− Q²+^s(s_4z⁻²⁾2

2Q² . (2.24)

5The reason for the absence of odd terms in the series (2.22) is well explained in pp. 12-15 [11]

Y₄=−1

8P_s⁴− 1 8

'dP_s dζ

(2

+1 4

d dζ

'1

3P_s³− Ps

dP_s dζ +1

2 d²P_s

dζ² (

−1 4

R(z)− Q²+^s(s_4z⁻²⁾2

Q²

'dP_s dζ − Ps²

(

−1 8

,

R(z)− Q²+^s(s−2)_4z2

-2

Q⁴

−1 8

d² dζ²

R(z)− Q²+^s(s_4z⁻²⁾2

Q² .

(2.25)

Truncating the series in our ansatz (2.22) at n = N and putting λ = 1 we obtain

q^{(2N +1)}(z) = Q(z) +N n=0

Y_2n. (2.26)

Inserting (2.26) into (2.8) and (2.9), we obtain the phase-integral approxi- mation of the order⁶ (2N + 1), generated from the base function Q(z), as the approximate solution of the original differential equation (2.1). Thus the phase-integral (2.9) of order 2N + 1 is given by

w^{(2N +1)}(z) =

" _z

(t)

q^{(2N +1)}(z^#) dz , (2.27)

where the notation !z

(t) is explained in appendix A. Thus for instance, the 3rd order (N = 1 in (2.26)) approximate solution of (2.1) is given by

u_±= [Q(Y₀+ Y₂)]^−1/2× e^{±i#zQ(Y0+Y2)dz!}, which, using the identity,

1

2QdP_s(z) dζ ≡ 1

2

dP_s(z)

dz . (2.28)

6See p.14 of [11] for a discussion on why it should be order 2N + 1.

can be written as

u_±= )1

2 dP_s

dz + '

1−1 2P_s²

(

Q + R(z)− Q²+^s(s_4z⁻²⁾2

2Q

*_−1/2

× exp .

±i /1

2P_s+

" _z) (1−1

2P_s²)Q +R(z^#)− Q²+ ^s(s−2)_4z!2 2Q

* dz^#

01 (2.29)

from which it is obvious that one should choose the base function Q(z) such that

Q²(z) = R(z) +s(s− 2)

4z² . (2.30)

The choice of the parameter s in the platform function (2.20) basically de- termines (the square of) the base function Q(z). Hence for s = 0, 1 or −2"

we get for the above (2.30) choice of base function:

Q²(z) =







R(z), for s = 0

R(z)− 1/4z² for s = 1 R(z) + "(" + 1)/z² for s = −2"

, (2.31)

which corresponds to; the unmodified base function, the Kramer-Langer modification discussed in section 1 (or pp. 59-61 [15] and pp. 7-8 in [27]) or the omission of the centrifugal barrier respectively.

2.2. The Connection Formulæ. Having found two linearly independent asymptotic solutions q^−1/2e^±iw(z) of (2.1) valid sufficiently far away from any transition points, we have solved the local problem. However, the linear combination of two asymptotic solutions of ψ in a certain region of the complex plane might not represent ψ even approximately in another region of the complex plane [3], [7] hence a different linear combination has to be used. This is due to the Stokes phenomenon and the problem of determining the linear combinations of asymptotic solutions that represent ψ in different regions is called the connection problem or the global problem. This can be done quite generally for transition points lying anywhere in the complex plane and for complex valued functions R(x) and Q²(x), x ∈ R, by utilizing the properties of the so-called F -matrix introduced by Fröman and Fröman [5]. Here, we consider the more restricted case where the functions R(x) and Q²(x)are real on the real z-axis and the transition zero is a turning point on this axis. Thus to make the connection across the turning point (or points⁷) we use the following connection formulæ (eqs. (3.10.10^#) and (3.12.9) resp.

7The turning points need to be sufficiently far from each other in order for the approximations to be valid.

in [15]):

|q^−1/2(x)|e^−|w(x)| −→ 2|q^−1/2(x)| cos(|w(x)| − π/4), (2.32a)

sin(δ)|q^−1/2(x)|e^+|w(x)|←− |q^−1/2(x)| cos(|w(x)| + δ − π/4), (2.32b) (for δ "≈ kπ, k ∈ Z,)

where the left and right hand-side of the expressions above denotes the clas- sically forbidden and allowed regions respectively. Notice that when connect- ing solutions from the classically allowed region to the classically forbidden region, (2.32b) is valid only when δ is not too close to an integer multiple of π. Also one should notice the one-directionality of the arrows, there is an example (pp. 133-7 in [15]) where the connection is made in the wrong way with approximately correct end result (though wrong intermediate results), other than that one should always respect the one-directionality of the con- nection formulæ (pp. 74-83 in [15]). Intuitively, this should be obvious in (2.32a); since a small change of the wave function on the right hand-side might give rise to an exponentially increasing term in the function on the left (p. 170 [9]). Finally these connection formulæ can be used to obtain the quantization condition for a system, this is done in appendix B and we’ll use the results derived there in the next section.

3. The Bohr-Sommerfeld Quantization Condition.

The Bohr-Sommerfeld quantization condition (BSQC) has not only an important historical value but is also of pedagogical interest, it is linked to the WKB method which gives the connection between classical mechanics and quantum mechanics. In the present section, which is an adaption of a paper by Yngve and Thidé, we shall express the BSQC (of order 1 and higher) in terms of the apsidal angle (to be defined later) of a classical mo- tion in a central potential. To show the user-friendliness of this new version of the BSQC we shall apply it to the hydrogen atom and the isotropic 3-D harmonic oscillator and show how one can with relative ease find the exact energy eigenvalues in the mentioned cases.

3.1. The Bohr-Sommerfeld Condition in terms of apsidal angle. The half-integral Bohr-Sommerfeld quantization condition (see for instance p.

171 [9]) reads:

6

p_qdq = (n_q+ 1/2)h, n_q= 0, 1, 2, ... (3.1) where; q is a generalized coordinate describing a degree of freedom of a quan- tal system in a semiclassical description, pq is the corresponding generalized

momentum, h = 2π! is Planck’s constant , and the integration is to be car- ried over one period of the classical motion of the particle. For an atomic electron moving in a central potential V (r) with the polar coordinates as generalized coordinates, the Lagrangian expressed in terms of the (reduced) mass µ is

L = 1

2µ ˙r²+1

2µr²ϕ˙²− V (r). (3.2) From the Lagrangian we find

p_r= ∂L

∂ ˙r = µ ˙r, (3.3a)

p_ϕ= ∂L

∂ ˙ϕ = µr²ϕ,˙ (3.3b)

where pr is the classical radial momentum and pϕ the angular momentum.

Since ϕ is a cyclic coordinate, pϕ is conserved and from (3.1) it follows that 7p_ϕdϕ =!_2π

0 p_ϕdϕ = p_ϕ!_2π

0 dϕ = p_ϕ2π≡ (" + 1/2)2π!, whence we obtain the semiclassical angular momentum

p_ϕ = (" + 1/2)!, (3.4)

where " is the angular momentum quantum number. For the radial mo- mentum integral 7

p_rdr = 7

µ ˙r dr, we observe that a complete cycle of r pertains going from r1 to r2, and then back to r1, where r1 is the position closest to the centre of force and r2 is the position farthest away. Hence 7p_rdr = !r2

r1p_rdr + !r1

r2−prdr = 2!r2

r1p_rdr, since ˙r and therefore pr is negative (r decreases) on the way back. It follows that (compare with eq.

(B.15)) " r2

r1

p_rdr = (k + 1/2)π!, k = 0, 1, 2, ... (3.5) where k is the radial quantum number. Another conserved quantity is the energy of the atom E = ¹₂µ ˙r²+¹₂µr²ϕ˙²+V (r) = _2µ^p2r+_2µr^p2ϕ₂+V (r), which upon substituting for pϕ the expression for the semiclassical angular momentum (3.4) becomes,

E = p²_r

2µ+ (" + 1/2)²!²

2µr² + V (r). (3.6)

Solving (3.6) for pr ≥ 0 we get p_r =

8

2µ[E− V (r)] −(" + 1/2)²!²

r² (3.7)

and we write (3.5) as

" r2

r1

8

2µ[E− V (r)] − (" + 1/2)²!²

r² dr = (k + 1/2)π!, k = 0, 1, 2, . . . .

(3.8)

In principle we are done, solving (3.8) for E = Ekfor a given potential V (r) yields the energy eigenvalues of the system. To simplify calculations we shall write the expression above (3.8) in terms of a notion well known from the classical theory of central motion (cf. Ch. 8 in [19]), namely that of the apsidal angle.

Definition 3.1. The apsidal angle is the angular displacement from r1 to r2 in a central field of force where r1 is the position closest to the centre of force and r2 the position farthest away. We denote it by

∆ϕ_a≡ π + δπ = π(1 + δ) (3.9)

where δ ∈ R is some real number.

Using ^ϕ˙_˙r = ^dϕ_dr, we obtain from (3.3), (3.4) and (3.7) for pr≥ 0, dϕ

dr = pϕ

r²p_r = (" + 1/2)! r²

9

2µ[E− V (r)] −^(#+1/2)_r2 ^2!2

. (3.10)

As r varies from r1 to r2 the variation in ϕ is (see p. 5 and eq.(39) in [22]

respectively)

∆ϕ_a=

.π, for V (r) ∝ 1/r

π/2, for V (r) ∝ r² , (3.11) which corresponds to (using (3.9))

δ =

.0, for V (r) ∝ 1/r

−1/2, for V (r) ∝ r² . (3.12) If the hydrogen atom potential has an additional ‘small’ 1/r³-term,

V (r) =− !² µa₀r − C

r³ (3.13)

we have approximately (see eq. (C.7) of appendix C)

δ = 3Cµ

(" + 1/2)⁴!²a₀, (3.14) where a0 is the Bohr-radius. Integrating (3.10) over r from r1 to r2 and over ϕfrom 0 to ∆ϕa≡ π(1 + δ) we obtain

π(1 + δ) =

" r2

r1

(" + 1/2)! dr r²

9

2µ[E− V (r)] − ^(#+1/2)_r^{2 2!2}

(3.15)

and dividing (3.8) by ! we rewrite the radial Bohr-Sommerfeld condition as (k + 1/2)π = 1

!

" _r2

r1

2µ[E− V (r)] − ^(#+1/2)_r2 ^2!2

9

2µ[E− V (r)] −^(#+1/2)_r^{2 2!2} dr

=

" r2

r1

2µ

!²[E− V (r)]

92µ

!²[E− V (r)] −^(#+1/2)_r2 ²

dr

− (" + 1/2)

" _r2

r1

(" + 1/2)! r²

9

2µ[E− V (r)] −^(#+1/2)_r2 ^2!2

dr,

(3.16)

where we recognize the last integral as π(1 + δ), (3.15). Thus, we write the radial Bohr-Sommerfeld quantization condition in the following form

" r2

r1

2µ!²[E− V (r)]

92µ

!²[E− V (r)] − ^(#+1/2)_r^{2 2}

dr = (k + " + 1)π + (" + 1/2)δπ, k, " = 0, 1, 2, . . . .

(3.17)

Equation (3.17) is what we set out to accomplish: a different version of the radial Bohr-Sommerfeld condition expressed implicitly in terms of the apsidal angle which simplifies calculations for different potentials and utilizes ones knowledge of classical motion as we will show next.

3.2. The hydrogen atom and the isotropic 3-D harmonic oscillator.

3.2.1. The Hydrogen Energy Eigenvalues. In the case of the hydrogen⁸atom with the potential V (r) = −!²/µa₀r ∝ 1/r eq. (3.9) =⇒ δ = 0 and since the energy is negative we let E −→ −|E|. Thus in this case (3.17) becomes

(k + " + 1)π =

" r2

r1

2µ

!²

,−|E| +_µa^!2_0r- 8

2µ!²

,−|E| +_µa^!2_0r-

− ^(#+1/2)_r2 ²

dr

=

" r2

r1

−^2µ_!^|E|2 r +_a² 8 0

−^2µ_!^|E|2 ,

r²−_µa^!₀²_|E|-

− (" + 1/2)² dr

=

" _r2

r1

−^2µ_!^|E|2 r +_a² : 0

+^2µ_!^|E|2

#

−,

r−_2µa^!₀²_|E|-2

+,

!² 2µa0|E|

-2

−^(#+1/2)_2µ|E|^2!2

$ dr

(3.18)

8See appendix D for an alternative way of calculating the eigenvalues.

and after making the substitution R := r −_2µa^!₀²_|E| it is easy to see from the last integral in (3.18) that the values of R at the turning points are ±R2, where

R₂=

:' !² 2µa₀|E|

(2

−(" + 1/2)²!²

2µ|E| . (3.19)

Substituting R, the limits, the differential and keeping only even terms (3.18) becomes

(k + " + 1)π =

: !² 2µa²₀|E|

" +R2

−R2

; dR

R²₂− R², (3.20) where the integral!+R2

−R2

√ dR

R²₂−R² = πand thus |E| = _2µa^{2 !2}

0(k+#+1)² or since E is negative |E| = −E hence

E_k,#=− !²

2µa²₀(k + " + 1)² k = 0, 1, 2, . . . ,

" = 0, 1, 2, . . . .

(3.21)

We have thus obtained the well-known result that the Bohr-Sommerfeld con- dition reproduces the exact eigenvalues for the hydrogen atom.

3.2.2. The Isotropic 3-D Harmonic Oscillator. In the classical orbit of the isotropic three-dimensional harmonic oscillator, there is an angle (eq. (3.9))

∆ϕ_a = π/2 between the shortest and the longest radius vector from the centre of force to the particle. Thus we use δ = −1/2 together with the potential V (r) = ¹₂µω²r² in (3.17) i.e.

(k + "/2 + 3/4)π =

" _r2

r1

2µE

!² −^µ2ω_!2^2r2

92µE

!² −^µ2ω_!2^2r2 −^(#+1/2)_r2 ²

dr

=

" r2

r1

2µE

!² −^µ2ω_!^22r2 8

−µ²ω²

!²r²

%

r⁴−_µω^2E2r²+ ^(#+1/2)_µ2ω^22!2

& dr,

(3.22)

from which we see that r²_1,2= E

µω²



1 ∓ :

1− (" + 1/2)²'!ω E

(2

 . (3.23)

Substituting









A = r²− _µω^E2

A₂ = r₂²− _µω^E2

dr = ^dA

2$ A+_µω2^E

(3.24)

(3.22) becomes

(k + "/2 + 3/4) =

" _A2

−A2

dA 29

A +_µω^E₂

µE

!² −^µ2_!^ω2²A 8@_µω

!

A2 A²₂−A² A+_µω2^E

(3.25)

and keeping only even terms we arrive at

(k + "/2 + 3/4)π = E 2!ω

" A2

−A2

; dA

A²₂− A², (3.26) noting that the integral is !A2

−A2

√ dA

A²₂−A² = π, we obtain

E_k,#= (2k + " + 3/2)!ω. (3.27) Thus the Bohr-Sommerfeld condition yields the exact eigenvalues for the isotropic three-dimensional harmonic oscillator, which is a well-known fact.

3.3. Higher order quantization condition for the radial problem.

The quantization condition (B.15) derived in appendix B can using (2.22) with λ := 1 be written

(k + 1/2)π =

" (t^!!) (t^!)

q(r, E_k) dr =

" (t^!!) (t^!)

Q(r) +∞ n=0

Y_2ndr

=

" (t^!!) (t^!)

Q(r)× [Y0+ Y₂+higher order terms] dr

≡ 1 2

6

Λ

Q(r)× [Y0+ Y₂+higher order terms] dr.

(3.28)

It is among other situations here that we can rip the benefit from the platform function of Yngve and Thidé as compared to that of Fröman and Fröman [11], [15]. For in the case of Fröman and Fröman: no total derivatives are found until in the fifth order contribution (cf. pp. 16-17 in [15]) i.e. in Y4

(as a reminder: Y2n is said to be the (2n + 1)th order contribution); while for the platform function defined by (2.20) we already see a total derivative in the 3rd order contribution using the identity (2.28) and as will be shown

now it simplifies calculations. Thus starting from (3.28) we have that

" _(t^!!₎

(t^!)

Q(r)× [Y0+ Y₂+higher order terms] dr =

" (t^!!) (t^!)

Q(r)× /

1−1

2P_s²+1 2

dPs

dζ +R(r)− Q²+ ^s(s_4r⁻²⁾2

2Q² +h.o.t 0

dr =

" _(t^!!₎

(t^!)

Q(r) dr +

" _(t^!!₎

(t^!)

)−Q

2 P_s²+1

2Q(r)dP_s

dζ +R(r)− Q²+^s(s−2)_4r2

2Q

* dr +higher order terms.

(3.29) Following (2.31) we choose s = 1 and the (square of the) base function

Q²(r) = R(r) +s(s− 2)

4r² = R(r)− 1

4r², for s = 1 (3.30) (where R(r) is given by (B.2)) and keeping only terms up to 3rd order the above (3.29) quantization condition becomes

(k + 1/2)π =

" (t^!!) (t^!)

Q(r) dr +

" (t^!!) (t^!)

'−Q

2 P₁²+1

2Q(r)dP₁ dζ

(

dr. (3.31) The first integral in the above expression (3.31) is our old friend (3.8) divided by ! and can be rewritten as

" r2

r1

2µ!²[E− V (r)]

92µ

!²[E− V (r)] −^(#+1/2)_r2 ²

dr− (" + 1/2)δπ, (3.32) with δ according to definition 3.1. Using the identity (2.28), the second term of the second integral can be written as a total derivative

1

2Q(r)dP_s(r) dζ = 1

2

dP_s(r)

dr (3.33)

and hence won’t contribute⁹ to the integral when we use the Fröman &

Fröman definition of the limits!(t^!!)

(t^!) as explained in appendix A. Using the above (3.32), (3.33) and the definition of the integral !(t^!!)

(t^!) , (A.4) we write the 3rd order quantization condition as

" r2

r1

2µ

!²[E− V (r)]

92µ

!²[E− V (r)] − ^(#+1/2)_r2 ²

dr−1 4

6

Γt(r)

QP₁²dr

=(k + " + 1)π + (" + 1/2)δπ.

(3.34)

9Except if we’re dealing with branch cuts, but then the contribution is trivially calculated.

4. The vibrating soap bubble and a proposition.

The soap film is an invaluable¹⁰tool for realizing concepts of nature which might be hard to grasp or get a sense for [25], such as the degeneracy of a system and how one can affect it. For instance, the degeneracy of the nor- mal modes of a soap film suspended in a perfectly square rigid rim of sides a and b, can be broken (see p. 184 of [24]) if one breaks the symmetry of the square by making it a rectangle (a −→ c "= ^m_nb,for m, n ∈ N). The soap film can also be used to show a variety of other features of a system, see [25]

for some interesting applications. One of the major deficits of the semiclas- sical description considered in section 3 is that it hides the degeneracy due to the fact that the energy levels do not depend on the azimuthal quantum number m, we will look into that in this section. We will also obtain the eigenfrequencies of a restricted soap film and give a proposition concerning the parameters " and m of the associated Legendre differential equation.

4.1. The angular frequency of a restricted soap film and a propo- sition. The governing equation for the vibrations of a soap membrane of constant radius a is

∇²U = 1 c²

∂²U

∂t² , (4.1)

where U is the deviation from the equilibrium position of an element of the soap membrane at position θ, ϕ and at time t and c is the speed of the transverse waves on the membrane. Here ∇² is the Laplacian in spherical coordinates with r = a, i.e. (4.1) can be written as

1 a²

1 sin θ

∂

∂θ '

sin θ∂U

∂θ (

+ 1

a²sin²(θ)

∂²U

∂ϕ² = 1 c²

∂²U

∂t² . (4.2) The time-independent wave equation can be obtained by making the substi- tution _∂t^∂22 −→ −ω², for a specific angular frequency ω and we end up with Helmholtz’s equation ∇²U =−@_ω

c

A2

U. Thus 1

sin θ

∂

∂θ '

sin θ ∂

∂θU (θ, ϕ) (

+ 1

sin²θ

∂²

∂ϕ²U (θ, ϕ) + ω²a²

c² U (θ, ϕ) = 0, (4.3) is the time-independent wave equation for a vibrating soap bubble of radius a. Let us now consider a vibrating soap bubble restricted to the region

0≤ ϕ ≤ β < 2π,

0≤ θ ≤ π, (4.4)

10Super cheap, widely available.

with the boundary conditions

U (θ, 0) = U (θ, β) = 0,

|U(θ, ϕ)|_θ=0,π <∞, BB

BB ∂

∂θU (θ, ϕ) BB BB

θ=0,π

<∞.

(4.5)

Seperating variables

U (θ, ϕ) = Θ(θ)Φ(ϕ) (4.6)

gives

d²Φ

dϕ² + m²Φ(ϕ) = 0, for 0 ≤ ϕ ≤ β (4.7) and

1 sin θ

d dθ

'

sin θdΘ dθ

( +

'ω²a²

c² − m² sin²θ

(

Θ(θ) = 0. (4.8) Equation (4.7) has a solution consisting of a linear combination of sin(mϕ) and cos (mϕ) but the latter one is ruled out due to the boundary condition Φ(0) = 0. The other boundary condition implies that for non-trivial solutions sin(mβ) = 0⇔ mβ = nπ, and since this is an eigenvalue problem we expect n ≥ 0. Now for 0 ≤ ϕ ≤ 2π and periodic boundary conditions n would take the values n = 0, 1, 2 . . . (see p. 69 in [4]), but in our case we have a restricted soap bubble with the boundary conditions given thus for β "= 0 we get

m = nπ

β , for n = 1, 2, 3, . . . . (4.9) We recognize the θ-dependent equation (4.8) as the associated Legendre equation with "(" + 1) = ^ω_c²2^a2 in usual notation, which is the equation of the θ-dependent part of the hydrogen atom wavefunction, but here " is not necessarily an integer. Thus, our problem of finding the angular frequencies of standing waves in a restricted soap bubble is equivalent to solving the associated Legendre equation

1 sin θ

d dθ

'

sin θdΘ dθ

( +

'

"(" + 1)− m² sin²θ

(

Θ(θ) = 0 (4.10) with

"(" + 1) = ω²a²

c² (4.11)

for a given m and with the natural boundary conditions that U and its derivative be bounded for θ = 0 and θ = π, (4.5). Notice that we don’t have any restrictions on ", it doesn’t have to be an integer. Making the

substitution z = cos θ which is one-to-one on −1 ≤ z ≤ 1 for 0 ≤ θ ≤ π, equation (4.10) becomes:

(1− z²)d²Θ

dz² − 2zdΘ dz +

'

"(" + 1)− m² 1− z²

(

Θ =0 . (4.12) Now, we want to be able to apply the phase-integral method to this problem hence we write (4.12) in the form of (2.2)

d²Θ

dz² + −2z 1− z²

dΘ dz +

'"(" + 1)

1− z² − m² (1− z²)²

(

Θ = 0 (4.13) and apply the transformation (2.3)

ln Θ = ln u−1 2

"

−2z

1− z² dz (4.14)

or

u(z) = (1− z²)^1/2Θ (4.15) to obtain the differential equation in the form of (2.1)

d²u

dz² + R(z)u = 0, for − 1 ≤ z ≤ 1,

with u(−1) = u(1) = 0, (4.16) where R(z) is given by (2.4), that is

R(z) = "(" + 1)

1− z² − m²− 1

(1− z²)². (4.17) To obtain the function Q(z) to be used in the Bohr-Sommerfeld condition we now perform a double Langer modification (cf. sec. (1)) by letting

"(" + 1)−→ (" + 1/2)², m²− 1 = 4

'm 2 − 1

2 ( 'm

2 −1 2 + 1

(

−→ 4,m 2

-2

= m². (4.18) hence

Q²(z) = (" + 1/2)²

1− z² − m²

(1− z²)². (4.19) Thus the Bohr-Sommerfeld condition for obtaining the proper values of " is

" _t

−t

Q(z) dz = (k + 1/2)π, for k = 0, 1, 2 . . . (4.20) where

Q(z) = :

(" + 1/2)²

1− z² − m²

(1− z²)² (4.21)

and ±t are the zeros of Q(z) i.e.

(" + 1/2)²(1− t²) = m²⇒ t = :

1− m²

(" + 1/2)². (4.22)

The integral (4.21) can be written

" t

−t

Q(z^#) dz^# = '" z

−t

+

" +t z

(

Q(z^#) dz^# = '" z

−t−

" z +t

(

Q(z^#) dz^#, (4.23)

where the integrals in the last parenthesis have been computed in Appendix C of [26] with the result

'" z

−t−

" z +t

(

Q(z^#) dz^#= π

2 [−m + (" + 1/2) − m + (" + 1/2)] . (4.24) From (4.24) and the Bohr-Sommerfeld condition (4.20) it follows that

"− m = k = 0, 1, 2 . . . , (4.25) thus we have shown that

Proposition 4.1. the difference between the parameters " and m in the associated Legendre equation (4.10) is a non-negative integer irrespective of the value of m provided that the region for θ is 0 ≤ θ ≤ π with natural boundary conditions at θ = 0 and θ = π.

Finally from "(" + 1) = ^ω2_c2^a2 with " = m + k = k +^nπ_β we obtain: the an- gular frequencies of standing waves for a restricted soap bubble (0 ≤ ϕ ≤ β) of radius a and with c being the speed of transverse waves on the membrane;

ω_nk = c a

:'

k + nπ β

( ' k + nπ

β + 1 (

,

k = 0, 1, 2, . . . , n = 1, 2, 3, . . . , 0 < β < 2π.

(4.26)

These frequencies can be studied by using a mechanical vibrator to vibrate the wire (or whatever the boundaries are) that delimits the soap film. Al- ternatively one can excite the film by letting a speaker in conjunction with a tone generator transmit the frequencies through the air [25].