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description
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
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Objective
Obtain the solutions to the time-independent space fractional Schrödinger equation for a quadrupolar triple Dirac- potential for all energies E
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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
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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
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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
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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
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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
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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
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For this case we define
(10)
Then
(11)
IV. Solutions to the time-independent space fractional Schrödinger equation
A. (bound state)
0E
, 0ED
0 1 22 0iap iapp e C a C e C ap
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A. (bound state)
(12)
with (13)
(14)
0E
IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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A. (bound state)
Energy equation
(17)
where (18)
(19)
(20)
3 11 3R A R B
IV. Solutions to the time-independent space fractional Schrödinger equation
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
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A. (bound state)
Figure 2. Plots of the functions and for some values of α and R = 2. Red dots mark the intersection points.
IV. Solutions to the time-independent space fractional Schrödinger equation
0E
1,f R A 3 13,g R B
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A. (bound state)
The wave function can be obtained by inverse Fourier transforming
(21)
that is,
(22)with
(23)
IV. Solutions to the time-independent space fractional Schrödinger equation
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
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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)
IV. Solutions to the time-independent space fractional Schrödinger equation
0E
x
12
x x dx p p dp
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A. (bound state)
For the wave function be normalized to unity,
(25)
with
(26)
(27)
2 1
1 0 , ,4
C F W Z
IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
B. (scattering state)
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
B. (scattering state)
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
B. (scattering state)
Notation
0E
1 11 1 11 0 , 1 2 0 ,S S S a
11 2, 4 1jl S ja U l l
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IV. Solutions to the time-independent space fractional Schrödinger equation
B. (scattering state)
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
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IV. Solutions to the time-independent space fractional Schrödinger equation
B. (scattering state)
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
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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Mellin transform of
(60)
T y
1
2 1 1 1 11 12 2
z z z zT z z
V. Evaluation of some integrals in terms of Fox’s H-function
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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
V. Evaluation of some integrals in terms of Fox’s H-function
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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
V. Evaluation of some integrals in terms of Fox’s H-function
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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!