Isoperiodic Potentials via Series Expansion

Display

Morse / Pöschl–Teller V(x)

isochronous V(x)

big equation for T(α)

invert parameters?

No

Yes

c

1

⟷

c

2

1

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3

⟷

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1

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5

⟷

c

6

1

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7

⟷

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1

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9

⟷

c

10

1

T(α) /

T

0

=

1

+α (

15

2

c

1

16

-

3

c

2

4

)

+

2

α

(

3465

4

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1

1024

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945

128

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+

105

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32

+

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64

-

15

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4

16

)

+

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α

(

255255

6

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225225

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4096

+

15015

512

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+

45045

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256

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+

315

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1155

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64

-

35

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32

)

+

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α

(

334639305

8

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1

4194304

-

101846745

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6

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1

262144

+

14549535

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3

5

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1

65536

+

72747675

2

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131072

-

3828825

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32768

-

3828825

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225225

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)

+

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α

(

29113619535

10

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67108864

-

45176306175

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468495027

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1405485081

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61108047

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65536

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305540235

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65536

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14549535

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3

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16384

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14549535

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135135

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2909907

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135135

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8192

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135135

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4096

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135135

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)

In a one-dimensional oscillation obeying conservation of energy, the potential function determines the period of motion

T

as a function of dimensionless energy

α

. However, the period function

T(α)

only uniquely determines the potential

V(x)

in the case of parity symmetry, where

V(x)=V(-x)

. In all other cases, it is possible to construct an uncountable infinity of potentials with the same period

T(α)

. Regardless of symmetry, we call any two potentials isoperiodic if they have the same period function

T(α)

[1]. Careful examination of inverse functions leads to a precise definition of isoperiodic potentials using power series expansion[2]. As in[3], we use a phase-space technique to write

T(α)

as a function of the potential expansion coefficients around a stable minima. The general form of

T(α)

leads to a set of linear constraints between the expansion coefficients of isoperiodic potentials (see Details).

The examples here explore energy-dependent potentials between Morse and Pöschl–Teller, and energy-independent potentials including the familiar quadratic

2

x

. The validity and convergence of intermediate potentials can be examined again by numerical time evolution along the potential surface.

Details

For details of published calculations, see[1, 2]. Our approach follows[3]. Starting with a Hamiltonian

2H=α=

2

p

+

2

q

+

∞

∑

n=1

c

n

n+2

q

,

we transform to the polar coordinates of phase space

α=

2

Ψ

+

∞

∑

n=1

n+2

c

n

(ΨQ)

,

where

Q=cosϕ

. The preceding algorithm approximately solves this implicit equation by series inversion, producing a truncated sum

Ψ(α,Q)≈

11

∑

n=1

ψ

n

(Q)

n/2

α

.

The corresponding approximate period is then calculated using

T(α)=

T

0

1+

5

∑

n=1

t

n

(c)

n

α

+(

6

α

)=

d

dH

-2π

∫

0

1

2

Ψ

dϕ=

d

dα

-2π

∫

0

2

Ψ

dϕ

.

Analyzing the general period function

T(α)

first reported in[4], we prove that every power of the Hamiltonian energy

n

α

attaches to a function of the potential expansion coefficients

t

n

(c)

with a pair

(

c

2n

,

c

2n-1

)

that does not occur in the coefficient of any

m

α

with

m<n

. This fact allows order-by-order construction of isoperiodic potentials as series expansions around a stable minima.

The isoperiodic constraint between two distinct potentials with expansion coefficients

c

and

c'

is

t

n

(c)=

t

n

(c')∀n

.

Fixing the

c

values and applying the isoperiodic constraint to the

c'

yet leaves one continuous degree of freedom in every coefficient

t

n

(c')

. These continuous degrees of freedom are controlled by sliders in this Demonstration, which directly enables you to calculate a range of isoperiodic potentials. Methods used here have also contributed to award-winning posts on Wolfram Community[5, 6].

By direct evaluation of the period function for the expansion coefficients of the Morse potential and the Pöschl–Teller potential, it is possible to prove approximate isoperiodicity order-by-order, as in the commented code at the end of the initialization section. Comparing coefficients, we expect direct evaluation of both period integrals to yield

T(α)=

T

0

1

1-α

.

References

[1] M. Asorey, J. F. Cariñena, G. Marmo and A. Perelomov, "Isoperiodic Classical Systems and Their Quantum Counterparts," Annals of Physics, 322(6), 2007 pp. 1444–1465. doi:10.1016/j.aop.2006.07.003.

[2] E. T. Osypowski and M. G. Olsson, "Isynchronous Motion in Classical Mechanics," American Journal of Physics, 55(8), 1987 pp. 720–725. doi:10.1119/1.15063.

[3] B. Klee, "Plane Pendulum and Beyond by Phase Space Geometry." arxiv.org/abs/1605.09102.

[4] The On-Line Encyclopedia of Integer Sequences. (Apr 4, 2017) oeis.org/A276816.

[5] B. Klee, "A Period Function for Anharmonic Oscillations" from Wolfram Community—A Wolfram Web Resource. (Apr 4, 2017) community.wolfram.com/groups/-/m/t/984488.

[6] B. Klee, "Plotting the Contours of Deformed Hyperspheres" from Wolfram Community—A Wolfram Web Resource. (Apr 4, 2017) community.wolfram.com/groups/-/m/t/1023763.

Permanent Citation

Brad Klee

"Isoperiodic Potentials via Series Expansion"
http://demonstrations.wolfram.com/IsoperiodicPotentialsViaSeriesExpansion/
Wolfram Demonstrations Project
Published: April 21, 2017