Space fractional Schrödinger equation for a quadrupolar triple Dirac - potential

$Page 1: Space fractional Schrödinger equation for a quadrupolar triple Dirac - potential$
Space fractional Schrödinger equation for a quadrupolar triple

Dirac- potential

Jeffrey D. Tare* and Jose Perico H. EsguerraNational Institute of Physics, University of the Philippines, Diliman, Quezon City 1101

*Presenter

Presented at the 7th Jagna International WorkshopResearch Center for Theoretical Physics, Jagna, Bohol

6–9 January 2014

2

Objective

Obtain the solutions to the time-independent space fractional Schrödinger equation for a quadrupolar triple Dirac- potential for all energies E

3

I. Time-independent space fractional Schrödinger equation A. Position representation B. Momentum representation

II. Quadrupolar triple Dirac- potentialIII. MethodologyIV. Solutions to the time-independent space fractional

Schrödinger equationA. (bound state) B. (scattering state)

V. Evaluation of some integrals in terms of Fox’s H-functionVI. Summary

Contents

0E

0E

4

I. Time-independent space fractional Schrödinger equation

A. Position representation

(1)

Riesz fractional derivative

(2)

Fourier-transform pair

(3)

When , , with m being the mass of the particle.

22 , 1 2D x V x x E x

22 1

2ipx ipxx dpe p e x dx

1,2

ipx ipxp x e dx x p e dp

2 2 1 2D m

5

B. Momentum representation

(4)

Fourier convolution integral

(5)

(6)

2

V pD p p E p

V p V p p p dp

ipxV p V x e dx

I. Time-independent space fractional Schrödinger equation

6

In the framework of standard quantum mechanics the quadrupolar triple Dirac-delta (QTD-delta) potential has been considered by Patil1 and has the form

(7)

Figure 1. QTD-delta potential

1S. H. Patil, Eur. J. Phys. 30, 629 (2009)

II. Quadrupolar triple Dirac- potential

0 02 , 0V x V x a x x a V

7

III. Methodology

TISFSE with QTD-delta potential

Fourier transform to obtain the corresponding momentum representation

Consider the cases E < 0 for bound state and E ≥ 0 for scattering state

For E < 0 obtain energy equations for V0 > 0 and V0 < 0 and solve graphically to

determine the number of bound states

For E ≥ 0 obtain an expression for the wave function

Obtain the wave function and normalize it

Inverse Fourier transform the wave functions to obtain the corresponding

position representation

Express the wave functions in terms of Fox’s H-function

8

After Fourier transforming the TISFE with a QTD-delta potential the following expression for the wave function in the momentum representation is obtained:

(8)

where

(9)

IV. Solutions to the time-independent space fractional Schrödinger equation

00 1 22 0 ,

2iap iap Vp e C a C e C a

Dp E D

0 2 1, 0iapC a C a e p dp C p dp

9

For this case we define

(10)

Then

(11)


A. (bound state)

0E

, 0ED

0 1 22 0iap iapp e C a C e C ap

10

A. (bound state)

(12)

with (13)

(14)

0E


0 0

1 1

2 2

0 2 22 0 0 0

2 2 0

T T a T a C a C aT a T T a C CT a T a T C a C a

1 1

0

cos2 ,

1yq

T y dq q pq

11


A. (bound state) To obtain nontrivial solutions we impose the condition

(15)

where

(16)

0E

0 T I

0 2 2 1 0 02 0 , 0 1 0

2 2 0 0 0 1

T T a T aT a T T aT a T a T

T I

12

A. (bound state)

Energy equation

(17)

where (18)

(19)

(20)

3 11 3R A R B


0E

10, 2a R D a V

2 2 23 0 4 2A T T T

2 2 2 22 0 0 2 2 2 2B T T T T T T

13

A. (bound state)

Figure 2. Plots of the functions and for some values of α and R = 2. Red dots mark the intersection points.


0E

1,f R A 3 13,g R B

14

A. (bound state)

The wave function can be obtained by inverse Fourier transforming

(21)

that is,

(22)with

(23)


0E

x

0 1 22 0iap iapp e C a C e C ap

1

1 0x C W T x a T x Z T x a

0 2

1 1 0

cos, , 2

2 0 2 0 1C a C a yq

W Z T y dq q pC C q

15

A. (bound state)

To normalize the technique of de Oliveira et al.2 is adapted.

Parseval’s theorem

(24)

2E. C. de Oliveira, F. S. Costa, and J. Vaz, Jr., J. Math. Phys. 51, 062102 (2010)


0E

x

12

x x dx p p dp

16

A. (bound state)

For the wave function be normalized to unity,

(25)

with

(26)

(27)

2 1

1 0 , ,4

C F W Z


0E

x

1 2

2 2 2, , 1 1 csc 4 2 2F W Z W Z W Z I a WZ I a

2

0cos 1 ,I y yq q dq q p

17


A. (bound state)

Normalized wave function

(28)

where (29)

(30)

0E

1 1 ,1 , 1 2, 22,12,3

0, , 1 1 ,1 , 1 2, 2y H y

x N W x a x Z x a

, ,N F W Z

18


A. (bound state)

Figure 3. Plots of the wave function as a function of for some values of α and W = Z = 2.

0E

N x x

19


A. (bound state)

Consider the case when the strength of the QTD-delta potential .

Let . Then the QTD-delta potential becomes

(31)

0E

2 , 0V x g x a x x a g

0 0V

0 0V g g

Figure 4. QTD-delta potential when V0 < 0

20


A. (bound state)

Equations for Cn (n = 0,1,2)

(32)

(33)

0E

0 0

1 1

2 2

0 2 22 0 0 0

2 2 0

T T a T a C a C aT a T T a C CT a T a T C a C a

1 1g

21


A. (bound state)

Energy equations

(34)

(35)where

0E

1(2 ) 0T T Q

1 2 2 20 2 2 9 0 6 2 0 2 16T T Q T T T T T

1, 2a Q D a g

0

cos2 ,

1yq

T y dq q pq

22


A. (bound state)

Figure 4. Plots of the energy equation (34) for Q = 2 and some values of α. Red dots mark the intersection points.

0E

23


A. (bound state)

Figure 5. Plots of the energy equation (35) for Q = 2 and some values of α. Red dots mark the intersection points.

0E

24


A. (bound state)

Normalized wave function

(36)

where (37)

(38)

0E

1 1 ,1 , 1 2, 22,12,3

0, , 1 1 ,1 , 1 2, 2y H y

g x N W x a x Z x a

, ,N F W Z

25


A. (bound state)

Figure 6. Plots of the wave function as a function of for W = Z = 2 and some values of α.

0E

g N x x

26


B. (scattering state)

For this case define (39)

and use the property of the delta-function to write

(40)

0E

, 0ED

0f x x f x

1 2 0 1 22 0iap iapp A p A p e C a C e C ap

27



Equations for

(41)where

42)

(43)

0E

0,1,2nC n

0 0 01

1 1 1

2 2 2

0 2 20 0 2 0 0

2 2 0

C a M a S S a S a C aC M S a S S a CC a M a S a S a S C a

0 2 1 2 1 1 2, 0ia iaM a M a Ae A e M A A

1

02 cos 1S y yq q dq q p

28



Equations for

(44)

(45)

(46)

0E

0,1,2nC n

2

1 2 2 10 0 1 2

2 2

2 3 1 22 0

1 1

l l l lUC a M a M M a

l l

1 0 2 1 20 1 0UC M a l M M a

2 1 12 0 1 2

2 2

2 2 2 10

1 1l l l

UC a M a M M al l

29



Notation

0E

1 11 1 11 0 , 1 2 0 ,S S S a

11 2, 4 1jl S ja U l l

30



The wave function after inverse Fourier transforming

now reads

(47)

where

0E

x

1 2 0 1 22 0iap iapp A p A p e C a C e C ap

1

1 2 0 12 02

i x i xx A e A e C a S x a C S x

2C a S x a

1

02 1,2 , 2 cos 1j jA A j S y yq q dq q p

31



In terms of Fox’s H-function

(48)

with

(49)

0E

1

1 2 0 1 22 02

i x i xx A e A e C a x a C x C a x a

1 1 ,1 , 1 2 2 , 2 22,12,3

0, , 1 1 ,1 , 1 2 2 , 2 2y H y

1 1 ,1 , 1 2 2 , 2 22,12,3

0, , 1 1 ,1 , 1 2 2 , 2 2H y

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V. Evaluation of some integrals in terms of Fox’s H-function

In terms of Fox’s H-function evaluate integral of the form

(50)

Mellin-transform pair

(51)

(52)

0

cos2

1qy

T y dqq

1

0

zf z y f y dy

12

c i z

c if y y f z dz

i

33

Mellin transform of

(53)

Definitions/properties

(54)

(55)

(56)

T y

0

,2

1I q z dq

T zq

1

0, coszI q z y qy dy

1

0, Re , , , z iqy z zI q z J q z J q z y e dy i q z

, cos sin

2 2z z zJ q z q z i


34

Mellin transform of

(57)

Useful formulas

(58)

(59)

T y

0

12 cos 2 cos csc

2 1 2

z zz q zT z z dq zq

sin

1w

w w

1sin cos

2 2z z


35

Mellin transform of

(60)

T y

1

2 1 1 1 11 12 2

z z z zT z z


36

Fox’s H-function3

Definition

(61)

(62)

3A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications (Springer, New York, 2009)

1 1

1 1

( , ) ( , ), ,( , ), , ,, , ,( , ) ( , ), ,( , )

12

p p p p

q q q q

a A a A a Am n m n m n sp q p q p qb B b B b B L

H z H z H z s z dsi

1 1

1 1

1

1

m n

j j j jj jq p

j j j jj m j n

b B s a A ss

b B s a A s


37

Fox’s H-function

Important property (change of independent variable)

(63)

( , )( , ), ,, ,( , ) ( , )

, 0p pp p

q q q q

a kAa Am n m n kp q p qb B b kB

H z kH z k


38

Recall the Mellin transform of T(y)

(64)

Inverse transform

(65)

1

2 1 1 1 11 12 2

z z z zT z z

1

2 1 1 1 1 11 12 2 2

c i

z

c i

z z z zT y z y dzi


39

Identification of indices and parameters

2, 1, 2, 3m n p q

1 2 1 21 1 , 1 2; 1 , 1 2a a A A

1 2 3 1 2 30, 1 1 , 1 2; 1, 1 , 1 2b b b B B B


40

Comparison with the definition of Fox’s H-function

(66)

(67)

1 1 ,1 , 1 2,1 22,1

2,3 0,1 , 1 1 ,1 , 1 2,1 2

2T y H y

1 1 ,1 , 1 2, 22,12,3

0, , 1 1 ,1 , 1 2, 22T y H y


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VI. Summary

The time-independent space fractional Schrödinger equation for a QTD-delta potential is solved for all energies.

For the case E < 0 equations satisfied by the bound-state energy are derived. Graphical solutions show that, for V0 > 0, there is only one bound-state energy

for each fractional order α considered; and, for V0 < 0, there are two bound-state energies for each fractional order α considered. All the eigenenergies shift to higher values as α decreases.

Symmetric bound-state wave function is observed when W = Z; valley-like structures that become steeper with respect to the symmetry axis as α decreases are observed.

For E ≥ 0 an expression for the wave function is obtained as a precursor to analyzing scattering by the QTD-delta potential.

Wave functions are expressed in terms of Fox’s H-function.

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References

• N. Laskin, Phys. Lett. A268, 298 (2000).• N. Laskin, Phys. Rev. E 63, 3135 (2000).• N. Laskin, Phys. Rev. E 66, 056108 (2000).• J. P. Dong and M. Y. Xu, J. Math. Phys. 49, 052105 (2008).• X. Y. Guo and M. Y. Xu, J. Math. Phys. 47, 082104 (2006).• J. P. Dong and M. Y. Xu, J. Math. Phys. 48, 072105 (2007).• E. C. de Oliveira, S. F. Costa, and J. Vaz, Jr., J. Math. Phys. 51, 123517 (2010).• M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Scharwz, J. Math. Phys. 51, 062102

(2010).• S. H. Patil, Eur. J. Phys. 30, 629 (2009).• A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and

Applications (Springer, New York, 2009).

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Thank you!

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Space fractional Schrödinger equation for a quadrupolar triple Dirac - potential