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UNIVERSIDAD DE
GUADALAJARA
CENTRO UNIVERSITARIO DE CIENCIAS
EXACTAS E INGENIERÍAS
DEPARTAMENTO DE INGENIERÍA DE PROYECTOS
Particle in a Finite Dual Well Potential System:
A Simple Quantum Model for Well Super-
Localization and Ultra-Dispersion Duality Effect
An Alternative Model for the Hydrogen Atom
Emmanuel Saucedo-Flores
Rubén Ruelas-Lepe
Víctor M. Rangel
-
Cobián
Research Report No. CA378/13
5 de marzo de 2014
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 2
Particle in a Finite Dual Well Potential System: A Simple Quantum
Model for Well Super-Localization and Ultra-Dispersion Duality Effect.
An Alternative Model for the Hydrogen Atom.
Emmanuel Saucedo Flores, Rubén Ruelas Lepe and Víctor Manuel Rangel Cobián
DIP-CUCEI Universidad de Guadalajara
José Guadalupe Zuno No. 48, 45101 Zapopan, Jalisco, México
emmmauel.saucedo@cucei.udg.mx, ruben.ruelas@cucei.udg.mx, victor.rangel@cucei.udg.mx
Resumen—Se estudia un sistema de una partícula en dos pozos finitos de energía potencial.
Se encuentran niveles de energía que crean estados degenerados duales con tuneleo normal o
prohibido para la partícula. Si este tipo de nivel de energía es cercano a la altura de los pozos,
la partícula muestra la dualidad súper-localización/ultra-dispersión (entrelazamiento). La noción
de sistemas cuánticos ‘gemelos’ se usa para modelar las líneas espectrales del hidrógeno.
Palabras clave—tuneleo prohibido, entrelazamiento, computación cuántica, sistemas gemelos
cuánticos, modelo del átomo de hidrógeno.
Abstract—The study of particle in a finite dual well potential energy system is presented. It is
found that some particle energy levels create degenerated states with an allowed/forbidden
tunneling duality. If this type of energy level is near to the well height, the particle shows a
super-localization/ultra-dispersion duality (entanglement). The concept of ‘twin’ quantum
systems is used to model the hydrogen atom spectral lines.
Key Words—forbidden tunneling, entanglement, quantum computing, quantum twin system,
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 3
System Definition
On this work we solve the time independent Schrödinger equation to determine the
wave functions describing the quantum behavior of a system consisting of a particle
interacting with a one-dimensional potential function U(x) conformed by two finite wells
which give rise to five regions defined by
U(x) = 0 a ≤ |x| ≤ L
= U
0
|x| < a
= U
0
|x| > L
(1)
This function is depicted in Figure 1.
Figure 1. Potential energy function definition.
The widths of the barrier on region III and of the wells on regions II and IV are 2a and
b, respectively; the barrier and wells height is U
0
. On what follows, we’ll consider the
particle to be an electron with charge q and rest mass m
0
; the factors m
1
, m
2
and m
3
are
introduced to define the effective particle mass for regions I and V, II and IV and III,
respectively.
Solution Derivation
Schrödinger equation set
We’ll consider first the case when the particle energy E is such that
0 < E < U
0
(2)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 4
Then, the time-independent Schrödinger equations for each of the five regions in
Figure 1 are given by
L , x(x)α
dx
(x)d
I
I
−≤=− 0
2
1
2
2
ϕ
ϕ
(3)
axL , (x)α
dx
(x)d
II
II
−<<−=+ 0
2
2
2
2
ϕ
ϕ
(4)
axa , (x)α
dx
(x)d
III
III
≤≤−=− 0
2
3
2
2
ϕ
ϕ
(5)
Lx, ax
dx
xd
IV
IV
<<=+ 0)(
)(
2
2
2
2
ϕα
ϕ
(6)
L, xx
dx
xd
V
V
≥=− 0)(
)(
2
1
2
2
ϕα
ϕ
(7)
where φ
I-V
are the wave functions corresponding to each region and the following
definitions were introduced
/m)(EUFEF EUF 1 , ,
03322011
−≡≡−≡
ααα
(8)
)//1( /2 , /2 , /2
2/12
033
2
022
2
011
eVmqmmFqmmFqmmF hhh ≡≡≡
(9)
here ħ
2
stands for Planck’s constant h divided by 2π .
Even wave function solution
The general solution for the above equation system is
L), xm/ (eeA(x)
xα
xα
I,ev
B
−≤=
−
+
1
11
11
ϕ
(10)
axL), m/x) ((αBx) (αA(x)
II,ev
−<<−+= 1cossin
2222
ϕ
(11)
axa, m eBeA(x)
x
x
III,ev
≤≤−+=
−
)/1(
33
33
α
α
ϕ
(12)
Lx), am/x) ((αBx) (αA(x)
IV,ev
<<+= 1cossin
2424
ϕ
(13)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 5
L, x)m/ (
xα
eeA(x)
B
xα
V,ev
≥=
−
+ 1
11
55
ϕ
(14)
where all A’s and B’s are integration constants to be determined next.
We first address an even wave function solution. Then, given that for x = +/- ∞ (10)
and (14) must vanish, it follows that
0,
5
1
=
AB
(15)
and from the given symmetry, it is obtained
51
BA =
(16)
Similarly, from (11) and (13)
eBeA eBeAx)((x)
xxxx
III,evIII,ev 3333
3333
α
α
α
α
ϕϕ
+=+=−=
−
−
then
33
BA
=
(17)
From (12), the symmetry involved allows to write
)cos()sin()cos()sin(
22222222
xBxAxBxAx)((x)
IV,evII,ev
ααααϕϕ
−+−=+=−=
and
B, BAA
4224
0===
(18)
With (15) to (18), (10) to (14) become
L), xm/ (eA(x)
xα
I,ev
−≤= 1
1
1
ϕ
(19)
axL), m/x) ((αB(x)
II,ev
−<<−= 1cos
22
ϕ
(20)
axa, m xA(x)
III,ev
≤≤−= )/1( )cosh(2
33
αϕ
(21)
Lx), am/x) ((αB(x)
IV,ev
<<= 1cos
22
ϕ
(22)
L), xm/ (
xα
eA(x)
V,ev ≥=
−
1
1
1
ϕ
(23)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 6
Let’s now consider the continuity implications of the wave functions and their
derivatives at the well walls. For x = - L, from (19) and (20), it follows that
)cos(
221
1
LB eAL)(L)(
L
II,evI,ev
αϕϕ
α
−==−=−
−
and
eLBA
L
1
)cos(
221
α
α
=
(24)
For the derivatives at the same site we get the expression
LBeA
L
)sin(
22211
1
ααα
α
−−=
−
so then
eLBA
L
1
)sin(
2
1
2
21
α
α
α
α
=
(25)
From (24) and (25), we derive
122
)tan(
α
α
α
=
L
(26)
For x = - a, (20) and (21) provide
)cosh(2)cos(
2322
aA aBa)(a)(
III,evII,ev
α
α
ϕ
ϕ
−
=
−
=
−
=
−
so that
a
a
BA )cosh( )cos(
2
3
2
23
α
α
=
(27)
For the derivatives evaluated at the same site we obtain
aAaB )sinh(2)sin(
333222
α
α
α
α
−
=
−
−
a
a
BA )sinh( )sin(
2
3
2
3
2
23
α
α
α
α
−=
(28)
Then, from (27) and (28),
)tanh()tan(
3322
a a
αααα
−=
(29)
We’ll return to (26) and (29), later on. For the moment, considering (25) and (27), (19)
to (23) are transformed into
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 7
L), xm/ (L)e(αB(x)
Lxα
I,ev
−≤=
+
1cos
)(
1
22
ϕ
(30)
axL )m/x) ((αB(x)
II,ev
−<<−= ,1cos
22
ϕ
(31)
axa, mx) (α
a)(α
a)(α
B(x)
III,ev
≤≤−= )/1( cosh
cosh
cos
3
3
2
2
ϕ
(32)
Lx), am/x) ((αB(x)
evIV
<<= 1cos
22,
ϕ
(33)
L, x)m/ (L)e(αB(x)
Lxα
V,ev
≥=
−
−
1cos
)(
1
22
ϕ
(34)
To determine B
2
lets make use of one of the quantum mechanics postulate, namely
dx(x)xdx(x)x
evevevev
∫∫
∞−
∞
∞−
==
0**
)(2)(1
ϕϕϕϕ
(35)
where φ
ev
(x) is the system’s global wave function and φ*
ev
(x) is its complex conjugate,
their product φ*
ev
(x) φ
ev
(x) is the particle probability density. The integral on the right
consists of three terms, the first of which we consider for evaluation is
dx(x)xI
L
evIevIev
∫
−
∞−
=
,
*,,1
)(
ϕϕ
and with (30), this becomes
L
Lx
LLx
ev
e
LB
dxeLBI
−
∞−
+
−
∞−
+
== ∫
)(2
1
2
22
2
)(2
2
22
2,1
11
2)(cos
)(cos
αα
αα
α
evaluating this expression provides
2
2,1
1
2
22
2
,1
2)(cos BT
LB
I
evev
==
αα
(36)
where the following definition was introduced
(m)
L
T
ev
1
2
,1
2
)(cos
2
α
α
≡
(37)
The second integral term involves the wave function for region II, so using (31)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 8
2
2,2
2
2
2
22
22
2,2
)
4)2sin(
2
()(cos BT
xx
BdxxBI
ev
a
L
a
L
ev
=+==
−
−
−
−
∫
α
α
α
(38)
where
(m)
a
L
b
T
ev
2
22
,2
4
)
2
sin(
)
2
sin(
2
α
α
α
−
+≡
(39)
The third integral term involves (32) and gives
2
2,3
0
3
3
3
22
22
2
0
3
2
3
22
22
2
,2
)
4)2sinh(
2
(
)(cosh )(cos
)(cosh
)(cosh )(cos BT
xx
aaB
dxx
aaB
I
ev
a
a
ev
=+==
−
−
∫
α
α
α
α
α
α
α
(40)
with
) (m)
α
a)α(
a
(
a)(α
a)(α
T
,ev
3
3
3
22
2
3
4
2sinh
2cosh
cos +≡
(41)
Adding up (36), (38) and (40) in (35), B
2
is found to be
)m/ (
TTT
B
,ev,ev,ev
1
)(2
1
321
2
++
=
(42)
so, wave functions (30) to (34) are now fully determined.
Allowed energy levels for even wave functions
Let’s return to (26) and (29). Elimination of α
2
and using (8) and (9) allows to derive
the following transcendental equation
)E(LF)E(aF
)EU(aFρ
)E(LF)E(aF
EvEv
Ev
EvEv 22
3
22
sincos
tanh
cossin
0
−=
−
(43)
from which it is possible to obtain, graphically or numerically, the particle allowed energy
levels for the even wave function case; above, ρ is defined as
1
3
m
m
=
ρ
(44)
Using a couple of trigonometric identities, equation (43) can be transformed into
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 9
)])sin(()[sin())sin(()sin( 22122 EvEvEvEv EFaLEbFTEFaLEbF +−=++
ρ
(45)
where T
1
stands for
)tanh(
0
31
Ev
EUaFT −≡
(46)
An analytical expression for selected E
Ev
values will be derived from (45) further
down.
Odd wave function solution
We now consider the odd wave function solution for equation set (3) to (7). A similar
derivation as above provides,
L), xm/ (L)e(αA(x)
Lxα
I,odd
−≤−=
+
1sin
)(
1
22
ϕ
(47)
axL), m/x) ((αA(x)
II,odd
−<<−= 1sin
2
2
ϕ
(48)
axa, mx) (α
a)(α
a)(α
A(x)
III,odd
≤≤−= )/1( sinh
sinh
sin
3
3
2
2
ϕ
(49)
Lx), am/x) ((αA(x)
oddIV
<<= 1sin
22
,
ϕ
(50)
L, x)m/ (L)e(αA(x)
Lxα
V,odd
≥=
−−
1sin
)(
1
22
ϕ
(51)
where A
2
is a constant given by
), m/ (
TTT
A
oddoddodd
1
)(2
1
,3,2,1
2
++
=
(52)
and where the following definitions were introduced
(m)
α
L)(α
T
,odd
1
2
2
1
2
sin
≡
(53)
(m)
aL
b
T
odd
2
22
4
)2sin()2sin(
2
,2
α
α
α
−
−≡
(54)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 10
) (m)
a
α
a)α(
(
a)(α
a)(α
T,odd 24
2sinh
sinh
sin
3
3
3
22
2
3−≡
(55)
Allowed energy levels for odd wave functions
Again, from boundary continuity considerations expected to be met by the wave
functions (47) to (51) and their derivatives, it is possible to establish the following
transcendental equation
)E(LF)E(aF
)EU(aFρ
)E(LF)E(aF
OddOdd
Odd
OddOdd 22
3
22
sincos
tanh
cossin
0
1
−=
−
−
(56)
from which it is possible to obtain, graphically or numerically, the particle allowed energy
levels for the odd wave function case. ρ is the same as in (44).
As for the even case, (56) can be converted to
)])sin(()[sin())sin(()sin( 22222
1
OddOddOddOdd
EFaLEbFTEFaLEbF +−−=+−−
−
ρ
(57)
where a negative sign has been introduced on both sides of the equation for plotting
clarity when compared to (45) and T
2
is defined as
)tanh(
0
32 Odd
EUaFT −≡
(58)
As for E
Ev
values in (45), an analytical expression for selected E
Odd
values will be
derived from (57) further down.
Solution for E = 0
Schrödinger equation set
The time-independent Schrödinger equation set for this particle energy case is given
by
L , x(x)α
dx
(x)d
I
I
−≤=− 0
0,1
2
0,
2
2
ϕ
ϕ
(59)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 11
axL ,
dx
(x)d
II
−<<−= 0
20,
2
ϕ
(60)
axa , (x)α
dx
(x)d
III
III
≤≤−=− 0
0,3
20,
2
2
ϕ
ϕ
(61)
Lx, a
dx
xd
IV
<<= 0
)(
2
0,
2
ϕ
(62)
L, xx
dx
xd
V
V
≥=− 0)(
)(
0,
2
0,
2
2
,1
ϕα
ϕ
(63)
where φ
I-V,0
are the wave functions corresponding to each region in Figure 1. α
1,3
are
calculated with (8) and E = 0.
Using a similar derivation procedure as above we can find again two solutions for this
case, one for even and one for odd wave functions. The even solution case is given by
L), xm/ ( eB(x)
Lxα
evI,
−≤=
+
1
)(
1
0,2,0
ϕ
(64)
axL)m/( B(x)
evII,
−<<−= ,1
0,2,0
ϕ
(65)
axa, mx) (α
a)(α
B
(x)
evIII,
≤≤−= )/1( cosh
cosh
3
3
0,2
,0
ϕ
(66)
Lx), am/( B(x)
evIV
<<= 1
0,2,0,
ϕ
(67)
L, x)m/( eB(x)
Lxα
evV,
≥=
−−
1
)(
1
0,2,0
ϕ
(68)
where
)m/ (
TTT
B
ev,ev,ev,
1
)(2
1
,03,02,01
0,2
++
=
(69)
and where the following definitions were introduced
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 12
(m)T
ev
1
21
,0,1
α
≡
(70)
b (m)T
ev
≡
,0,2
(71)
) (m)
α
a)α(
a
(
a)(α
T
ev, 3
3
3
2
,03
4
2sinh
2
cosh
1+≡
(72)
The odd case solution is given by
L), xm/ ( LeA(x)
Lxα
oddI,
−≤−=
+
1
)(
1
0,2
,0
ϕ
(73)
axL)m/(x A(x)
oddII,
−<<−= ,1
0,2
,0
ϕ
(74)
axa, mx) (α
a)(α
aA
(x)
oddIII,
≤≤−= )/1( sinh
sinh
3
3
0,2
,0
ϕ
(75)
Lx), am/(x A(x)
oddIV
<<= 1
0,2
,0,
ϕ
(76)
L, x)m/( LeA(x)
Lxα
oddV,
≥=
−−
1
)(
1
0,2
,0
ϕ
(77)
where
)m/m (
TTT
A
odd,odd,odd,
/1
)(2
1
,03,02,01
0,2
++
=
(78)
with
) (m
α
L
T
,odd, 3
1
01
2
2
≡ (79)
) (m
aL
T
odd
3
33
3
,0,2
−
≡
(80)
)) (m
a
α
a)α(
(
a)(α
a
T
,odd,
3
2
24
2sinh
sinh
3
3
3
2
03
−≡
(81)
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 13
Special Energy Levels
Particle forbidden tunneling or well super-localization (WSL). Even-odd degeneracy
By inspecting the right term of expression (43) it is found that when the particle
energy is such that
0coscos
22
== a)()E(aF
Ev
α
(82)
then the left term must meet
0coscos
22
== L)()E(LF
Ev
α
(83)
Then, we can conclude that the even wave functions (30), (32) and (34) would
vanish so that the particle will be fully confined inside the wells, that is, a forbidden
tunneling effect is present for all those energy levels satisfying (82) and (83). Another
way to conclude this is by observing that the integral terms (37) and (41) will evaluate to
zero.
Now, given that the above expressions also appear in (56), they will be met with E
Odd
values equal to E
Ev
, so that these special allowed energy values will form degenerated
states having an even-odd duality. However, the odd wave functions will not show the
forbidden tunneling effect because none of the (47) to (51) expressions would be null.
We now follow a similar derivation using the sine factor on the right term of (56); if it is
such that
0sinsin
22
== L)()E(LF
Odd
α
(84)
then, its left term must give
0sinsin
22
== a)()E(aF
Odd
α
(85)
This time, we can establish that the odd wave functions (47), (49) and (51) would
vanish so that the particle will be fully confined inside the wells, that is, a forbidden
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tunneling effect is present for all odd wave functions for those particle energy levels
satisfying (84) and (85). Also, note that the integral terms in (53) and (55) will become
null.
As for the previous case, the above sine terms also appear in (43), so the
corresponding E
Ev
levels will match the E
Odd
ones and the two-fold degeneracy is again
obtained but the even wave functions (30) to (34) will not show the forbidden tunneling
condition. So, the obtained energy states will also have the even-odd duality but with
the opposite tunneling behavior as compared to the previous case.
When the effective mass factors m
1
and m
3
are equal, ρ in (44) is equal to one and
given that T
1
in (46) and T
2
in (58) are positive numbers, we can derive from (45) and
(57) that the energy values providing degenerate states with or without the WSL
behavior must comply with
,...,,nπn , Ea)F(L n321
2
==+
(86)
while the energy values giving rise to degenerate states with one wave function
symmetry having the particle WSL must additionally satisfy
,...
,,sπs , Ea)F(L
s
321
2
==−
(87)
from here, given that the energy value must be the same, it follows that
ssk s
b
a
s
b
ab
s
a
L
aL
n>+=+=
+
=
−
+
=)1()
2
1(
2
(88)
where k, the barrier width to well width ratio, is defined as
b
a
k2
≡ (89)
Then, from (86) and using (9), the general expression for the system even-odd
degenerate energy levels is given by
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,...3,2,1 )(
8)2/(2)2(
2
02
2
02
=
+
=
+
=n, eV )
qmmba
πn
( )
qmmba
πn
(
n
Ehh
(90)
while from (87) and (9), the expression for the energy levels with one of the wave
function symmetries showing the WSL condition, alternatively with the other symmetry,
is given by
,...3,2,1 )(
2
2
02
== s, eV )
qmmb
πs
(E
S
h
(91)
with s complying with expression (88).
It is worth to note the inverse dependence on system parameters for the energy
values in (90) and (91). Noticeable cases are when the particle is quasi-neutral or/and it
is almost mass free, that is q ≈ 0 or/and m
2
m
0
≈ 0 , which can be additionally
compounded with a narrow system dimension, that is L + a ≈ 0. For such conditions, the
first excited allowed energy level, if any at all, would become too highly separated from
the system’s energy ground level E ≈ 0 which might end up as being the only particle
allowed level unless the system’s potential well depth U
0
happens to be high enough.
On this writing, we will not go into further detail for these extreme cases typical of
nuclear particles, dimensions and processes; we will focus on modeling an electron in
systems of atomic dimensions.
On the other hand, the opposite effect might be observed, that is, the number of
allowed excited states might become too close to each other, particularly near to E = 0,
when the system dimensions and or the particle effective mass is increased.
Calculations examples given below will neatly illustrate this behavior.
Also, from (90) it is foreseen that it is possible to obtain the same excited allowed
energy levels with different combinations of systems dimensions a and b while keeping
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constant the term a + b/2, that is, the center point of the well; other systems parameters
might stay constant or be chosen to vary proportionally in the opposite sense as the
geometrical data. Despite being different, these type of twin systems will not lend
themselves to be differentiated by their energy transitions among their levels unless the
wave function symmetry nature and/or their WSL behavior have an impact on the
particle transition probabilities. In a forthcoming writing, it will be demonstrated that ±(a
+ b/2) are modal values for the particle x-coordinate for all WSL wave functions.
Emitted photon wavelengths for transitions among energy levels
When the system is such that its energy levels are given by (90), it is possible to
derive an expression for the emitted photon wavelength when the particle transits from
an initial higher energy level ni down to a final lower energy level nf
. By introducing the
hydrogen atomic radius r
H
in the numerator and denominator in (90) we get for the
initial energy level
321
2)(
2
2
2
02
, . . .,, ni(eV),
aL nir
m
K
qmmaLr
nir
ni
E
He
H
H
=
+
=
+
=
π
h
(92)
where the constant K
e
has been defined as
)( 28163.134
22
2
0
eV )
qmr
h
(K
H
e
=≡
(93)
Similarly, the final energy level expression is given by
ni . . . ,,, nf (eV)
aL nfr
m
K
nf
E
H
e
<=
+
=3210
2
2
(94)
Now, subtracting (94) from (92) and using the Planck relation of a photon energy, E,
versus its wavelength λ, given by
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(eV)
λ
hc
E=
where c is the speed of light, we obtain
,...,ni,,nf...,, (m), ni
nfni
raL
K
hcm
H
e
ni,nf,DWPS
λ
1210321
1
22
2
2
−=>=
−
+
=
(95)
For the sake of comparison, the corresponding general expression for the hydrogen
Bohr model emission lines for the energy transitions is given by the Rydberg equation
as follows
... ,,nf...,ni (m),
nfni nfni
R
Hnfni
λ
,321,4,32
1
22
22
,,
=>∞=
−
= (96)
where R, the Rydberg constant, has a value of 1.097373x10
7
(1/m). The above
expression gives rise to six well known spectral series named according to the n
f
value,
that is, Lyman, Balmer, Paschen, Bracket, Pfund and Humphreys for n
f
= 1, 2, 3, 4, 5
and 6, respectively.
Particle well super-localization /ultra-dispersion duality
For the particular case when the system parameters are such that in (91) E
S
≈ U
0
, it
is observed that the degenerated super-localized wave function is present for one
symmetry case of the wave functions, that is, the particle is confined inside the wells
while for the other symmetry the wave function becomes strongly dispersed as is
expected for a nearly free particle; which wave function symmetry goes super-localized
will depend on which expression of (82) or (84) goes to nearly zero while the ultra-
dispersed case becomes present due to a significant reduction - over one order of
magnitude - of either B
2
or A
2
in (42) or (52), respectively, which in turn makes the
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probability density function integral for |x| < L to vary from practically 1.0 for the
forbidden tunneling or well super-localized case down to less than 0.01 for the ultra-
dispersed case.
This short-long range distribution (in-out) behavior is an example of a degenerated
quantum state for which the particle is expected to exist in any of two completely
different space distributions without requiring energy nor time to transit among them, a
phenomenon known as quantum entanglement or as Einstein referred to a “spooky
action at a distance” effect; this weird denomination is mainly due to its implication of
teleportation of quantum entities which nowadays has found important practical use in
quantum computing (quantum bit teleportation) (Qubit, 2014).
Calculation examples
System cases I and II. Examples of the WSL/UD duality
In this section we study two cases of the dual well potential system to illustrate the
well super-localization/ultra-dispersion duality. System case I parameters are provided
in Figure 2 which also shows the plots of equations (45) and (47) along with their energy
values solutions, that is, the allowed energy levels for the system. As can be seen, all
five energy levels are two fold degenerated with the even-odd duality nature; the E-
values where all plots cross at zero correspond to energy levels presenting the particle
WSL behavior for one of the two possible wave functions. For the case of system in
hand, these values can also be analytically derived using equation (90) or (91).
Figure 3a) provides examples of the obtained wave functions using equations sets (30)-
(34), (47)-(51), (64)-(68) and (73)-(77); particle probability densities for positive x are
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depicted in Figure 3b) while Figure 3c) gives the cumulative probability plots for the n-
levels 2 and 4; note that the WSL wave function for n = 2 has the even symmetry.
Figure 2. Left, parameters used for system case I having a total of 5 allowed energy levels.
Right, plots of equations (45), that is, left term LT
Ev
and right term RT
Ev
, and (57), that is, left
term LT
Odd
and right term RT
Odd
. The four excited energy levels can be either extracted from the
plot or calculated using expression (90).
For n = 4 (s = 2), that is, the case when E
4
≈ U
0
, the odd wave function is WSL while
the even one is distributed through the whole system, so the particle has the UD nature.
Note that the plot for this case in Figure 3c) has a small slope so the wave function
φ
4
,ev(x) trend to zero for x → ∞ is met as requested by quantum mechanics postulates.
The system case I potential energy function U
I
(x), the particle allowed energy levels
E
n
and wavelengths λ for all possible transitions among the energy levels are shown in
Figure 4. Note that some of the obtained wavelengths correspond to the hydrogen
emission spectral lines known as Lyman series (UV), Balmer series (visible) and
Bracket series (IR). λ-values in bold correspond to limit values on the mentioned
spectral series.
System case II parameter definition is given in Figure 5; compared to system case I,
the case II has wider barrier and well dimensions. As can be seen in Figure 5 b), c) and
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d), this system has six excited energy levels, all of them degenerated and with n = 2, 4,
and 6 having the WSL nature for one of their wave functions. This time, for n = 6 (s = 3),
that is, when E
6
≈ U
0
, the even wave function is WSL while the odd one has U
D
behavior
with the proper trend for x → ∞.
Figure 3. Examples of calculated wave functions, a), probability densities, b) and cumulative
probabilities, c) for system case I with a total of 5 allowed energy levels. Observe that all four n
= 0 and n = 1 wave functions show a normal barrier tunneling effect (the same as for case n = 3,
not provided here for the sake of clarity). For n = 2 (s = 1), the even wave function case is fully
confined inside the wells while the odd counterpart shows a normal tunneling behavior.
Figure 4. Left, potential energy function definition, U
I
(x) and allowed particle energy levels,
E
n
, for a system case with 5 allowed energy levels. The table on the right provides the emitted
photon wavelengths corresponding to all possible particle energy steps between the two energy
levels indicated in the transitions.
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Figure 5. a) System case II parameter definition, b) plots of equations (45) and (57), c)
examples of system’s even wave functions, d) examples of system’s odd wave functions and e)
example of cumulative probabilities for the negative x-axis.
The case II potential energy function U
II
(x), the particle allowed energy levels En and
wavelengths λ for all possible transitions among energy levels are shown in Figure 6.
Figure 6. Left, potential energy function definition, U
II
(x) and allowed particle energy levels,
E
n
, for the 7 energy level system. Right, calculated emitted photon wavelengths corresponding
to all possible particle energy steps between the two energy levels indicated in the transitions.
This time, some additional typical spectral lines appear.
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This time some additional emission spectral lines of the Lyman and Balmer series
appear along with some lines on the Paschen and Humphreys series. λ-values in bold
correspond to limit values on the mentioned spectral series
Different systems with the same energy levels. Twin systems for system case II
In this section we study three different potential functions U
2,3,4
(x) whose dimension
parameters for all of them are such that a + 0.5b = 0.5r
H
, that is, the center point of the
wells are equal as for system case II above; other non geometrical system parameters
used in case II are kept the same for this calculation example. The new potential
functions are shown in Figure 7 a) along with their allowed energy levels which happen
to be the same as for system case II above. Figures 7 b) and c) give the plots of
equations (45) and (57) for potential functions U
2
(x) and U
4
(x), respectively; a couple of
wave functions corresponding to n = 3 and n = 6 for each of these potentials are
presented in Figures 7 d) and e). Please compare these plots with those in Figure 5.
Figure 7. a) Potential energy functions U
2,3,4
(x) and allowed particle energy levels. Plots of
equations (45) and (57) for U
2
(x), b), and U
4
(x), c). Examples of wave functions for U
2
(x), d), and
for U
4
(x), e).
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Given that the allowed energy levels for all these systems are equal, also their
corresponding absorption or emission wavelengths sets will be equal to the one given in
Figure 6. So, even the narrower and wider wells potentials U
2
(x) and U
4
(x) systems will
be hard to differentiate by energy transition spectra unless the presence of the WSL
level n,s = 3,1 on system U
4
(x) might have an impact on the particle transition
probability involving this type of levels as compared to the non-localized type of levels.
An alternative model for the hydrogen atom. System case III.
On the previous cases it has been observed the appearance of some λ-values
associated to well-known spectral line series of the hydrogen atom. On this section we
study the case of a dual well potential system (DWPS) with 24 excited energy levels on
which we find many more λ-values corresponding to all documented H absorption-
emission spectral lines. As commented on previous section, the energy level set given
rise to λ-values similar as above can be achieved with more than one system
configurations; for the case in hand, we have chosen first the case IIIA defined in Figure
8 which also shows the plots of the U
III
A(x) potential function, the allowed energy levels
E
n
and some associated wave functions. A twin system, case IIIB is given in Figure 9 for
which a 4X higher effective mass factor has been introduced along with a 0.5X narrower
well; the ratio L/a is kept the same for both cases so the particle allowed energy levels
and their corresponding wave function natures are equal.
The λ-values, calculated using expression (95), for all 300 transitions involving the
allowed energy levels on system case III are given in Table I. The zigzagged bold cell
line gives a reference for all transitions implying an emission energy lower or equal than
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3.4 eV, or with no more than 12 energy level difference involved in the transition; note
that the 12 n-level delta from n = 12 to n = 0 corresponds to an energy transition of 3.4
eV, that is, a λ-value of 364.7 nm which is an end value of the Balmer series which also
appears other three times along the zigzagged line.
Figure 8. a) case IIIA parameter definition for a 25 energy level DWPS; b) potential energy
function U
IIIA
(x) and allowed particle energy levels; c) example of wave functions for a few n
values.
Figure 9. a) twin system case IIIB parameter definition; b) potential energy function U
IIIB
(x)
and allowed particle energy levels; c) example of wave functions for a few n and s values.
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Table I. Emitted photon wavelengths for all possible energy level transitions among the 25
energy levels of the DWPS cases IIIA and IIIB.
A plot of the above λ-values is given in Figure 10 along with the photon wavelengths
calculated with expression (96) for all typical hydrogen spectral series.
Figure 10. Photon wavelengths calculated with expression (96) for typical hydrogen spectral
series, bottom, and for the 25 energy level Dual Well Potential System (DWPS) case IIIA using
expression (95), top. The lines on Balmer series identified as Hα, Hβ, Hγ and Hδ correspond to
656.4, 486, 434 and 410 nm, respectively.
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Figure 11 provides a closer look to the Balmer series data, that is, the near UV and
visible regions of the electromagnetic spectrum and how it compares to the Fraunhofer
sun absorption lines; the corresponding case III data are also plotted indicating the five
possible transition types, that is, transitions from any excited n-level, odd or even, down
to n = 0, transitions among even n-levels or among odd n-levels, or from even to odd n-
levels or vice versa.
Figure 11. Balmer series, bottom, compared to Fraunhofer sun absorption lines, top. The 25
energy level DWPS case III calculated wavelengths are given at the middle for the five possible
different type transitions according to the type of the two n-values involved. Note that the Hα
line, at 656.4 nm, occurs for three different transition types. The Fraunhofer picture was taken
from (Fraunhofer, 2014).
Let’s now make a comparison of the two λ-lines sets obtained with expressions (95)
and (96) with respect to the absorption spectrum processes based on the Bohr model of
the hydrogen atom. First, let’s consider the absorption steps needed for the generation
of the spectral lines on Balmer series. As shown in Figure 12, in order to have an
absorption process involving n = 2 as the Balmer series requires, it is necessary to first
send the electron to this level by a previous absorption of a photon with an energy of
10.2 eV. Once this is the case, a second photon absorption with any of the energies on
the right side of Figure 12 will contribute to a spectrum line on the Balmer series related
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to each of the next four particle allowed energy levels; these lines are H
α
= λ
2,3
, H
β
=
λ
2,4
, H
γ
= λ
2,5
and H
δ
= λ
2,6
.
It is interesting to note that most re-emitted photons will have energies higher than or
equal to 10.2 eV, that is, λ-lines on the Lyman series with transitions down to n = 1. It is
worth mentioning that for the Bohr model, in all absorption or emission events, the
electron is considered to transit instantly among precise circular n-orbits each having a
predicted electron velocity and angular momentum.
Figure 12. Energy level transitions required to produce the absorption spectral lines on
Balmer series based on the hydrogen Bohr model. Left, a high energy photon is first absorbed
so that the electron is transferred from bottom level n = 1 to first excited level n = 2. Right, a
second photon is absorbed so that the electron goes to an energy level between n = 3 to 6
according to the photon energy involved; this creates the absorption spectrum.
For the dual well potential system case III we need to consider its first 12 energy
levels as shown in Figure 13, where the even numbered levels are identified with a
hexagon and the odd ones with a rhombus. Again, we consider a two-step process, on
the left side of Figure 13 the electron is first excited with a low energy photon absorption
from level n = 0 to any level with either n = 1, 4, 6 or 8; strictly speaking, two or more
energy steps can also be considered to reach any of the last three mentioned levels.
Next, as shown on the right side of Figure 13, the electron absorbs another photon with
any one of the indicated energies and it is transferred to either level n = 9 or n = 12, this
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creates three of the series lines, a dual H
α
= λ
1,9
= λ
8,12
, H
β
= λ
6,12
and H
δ
= λ
4,12
. The
direct transition from n = 0 to n = 11 corresponds to the 434 nm absorption line, that is,
H
γ
= λ
0,11
is shown on the left side of Figure 13.
Figure 13. Energy level transitions required to produce the absorption spectral lines on
Balmer series based on the 25 energy level system case III. Left, a low energy photon is
absorbed, the electron jumps from level n = 0 to n = 1, 4, 6 or 8. The direct, or one step, n = 0 to
n = 11 level transition corresponds to the 434 nm absorption line (Hγ). Right, a second energy
photon is absorbed, the electron jumps from levels n = 1, 4, 6 or 8 to either level n = 9 or level n
= 12 to create the indicted absorption λ-lines, one of them being a dual 656.4 nm case (Hα).
From this figure we can observe that three of these transitions occur among two even
n-values, one occurs among two odd n-values and one from an odd n-value to n = 0.
There is little experimental documentation on regards to transition likelihood with
respect to energy levels involved on absorption spectra; one useful source of
information is site (Köppen, 2007a) which provides interesting absorption count data for
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many elements; the reported numbers for H lines (Köppen, 2007b) are given in Figure
14 along with a plot of their relative absorption count.
Figure 14. Relative intensity of main absorption lines on Balmer series. The dual Hα line at ≈
656.3 nm is over two times stronger than all the other four lines shown in the chart combined.
Raw data table on the left was obtained on site (Köppen, 2007a).
For this system, unless a further photon absorption occurs at n = 12 level, the energy of
the re-emitted photons will not exceed 3.4 eV, that is, λ-lines on the Lyman series are
excluded. Additionally, as can be seen in Table I and Figure 11, the DWPS case III
predicts the presence of a few green and yellow lines, that is, lines on the wavelength
range from 500 to 600 nm, which the Bohr model fail to predict and are not present on
the Balmer series. For the DWPS the concept of different precise orbits for the particle
is not present, instead the particle changes its oscillation pattern from one energy state
to another keeping two fixed modal x-position values, namely, the center points of the
wells at x = ± (a + 0.5 b).
Conclusions
On this work, the quantum behavior of a particle in a dual well potential system
(DWPS) has been established by solving the time independent Schrödinger equation
and deriving various sets of system wave functions. The studied particle has been an
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electron with charge q and rest mass m
0
. It has been demonstrated that for certain
system parameter combinations, there can be a set of particle allowed energy levels
giving rise to two-fold degenerated states for which the wave functions show an odd-
even symmetry duality; additionally, for a subset of these states, an allowed-forbidden
tunneling duality is found to be present, that is, for these states the particle can have a
normal tunneling capability through the well walls for one of the wave function
symmetries or show a well super-localization (WSL) effect for the other symmetry case;
the wave function symmetry nature giving rise to this duality is interchanged for
consecutive quantum state numbers. One particular case of such states can be found at
a near-well height particle energy level E ≈ U
0
whose degeneracy is such that the
particle is either WSL for one wave function symmetry or shows an ultra-dispersed (UD)
behavior for the other symmetry.
This short-long range distribution (in-out) behavior is an example of a degenerated
quantum state for which the particle is expected to exist in any of two completely
different space distributions without requiring energy nor time to transit among them, a
phenomenon known as quantum entanglement or as Einstein referred to a “spooky
action at a distance” effect; this weird denomination is mainly due to its implication of
teleportation of quantum entities which nowadays has found important practical use in
quantum computing. All the above provided calculation examples of dual well potential
systems illustrate the WSL/UD duality.
The DWPS is demonstrated to be such that a given set of particle allowed energy
levels can be obtained with different configurations of system parameters given rise to a
series of hard to distinguish twin systems. Calculation examples are given for systems
Particle in a Finite Dual Well Potential System: A simple Quantum Model for Universidad de Guadalajara
Well Super-Localization and Ultra-Dispersion Duality Effect. DIP-CUCEI
An Alternative Model for the Hydrogen Atom. Research Report No. CA378/13
Emmanuel Saucedo Flores, Rubén Ruelas Lepe, Víctor M. Rangel Cobián 31
which provide allowed energy levels able to produce a good fitting to the main spectral
line series reported for the hydrogen atom (Lyman, Balmer, Paschen, Bracket, Pfund
and Humphreys) and to the Fraunhofer sun absorption lines. A comparison of a
possible DWPS energy set and the one provided by the Bohr model is made in order to
describe the required processes to produce the hydrogen absorption spectral lines. As
can be seen, although further refining might be needed, the DWPS can provide a
relatively good model for the emission/absorption processes on the hydrogen atom.
References
1. Qubit. (2014, February 28). In Wikipedia, The Free Encyclopedia. Retrieved 00:21,
March 8, 2014, from:
http://en.wikipedia.org/w/index.php?title=Qubit&oldid=597468251
2. Fraunhofer lines. (2014, February 20). In Wikipedia, The Free Encyclopedia.
Retrieved 00:22, March 8, 2014, from:
http://en.wikipedia.org/w/index.php?title=Fraunhofer_lines&oldid=596325883.
3. Köppen, J. (2007, June 21). Spectra of Gas Discharges, Strasbourg/Illkirch/Kiel,
Retrieved March 8, 2014, from: http://astro.u-strasbg.fr/~koppen/discharge/
4. Köppen, J. (2007, June 21). Spectra of Gas Discharges, Strasbourg/Illkirch/Kiel,
Retrieved March 8, 2014, from:
http://astro.u-strasbg.fr/~koppen/discharge/hydrogen.txt
