Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics
Using the position as an independent variable, and time as the dependent variable, we derive the function P ( ± ) = ± 2 m ( H - V ( q ) ) , which generates the space evolution under the potential V ( q ) and Hamiltonian H . No parametrization is used. Canonically conjugated variables are the time an...
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| description | Using the position as an independent variable, and time as the dependent variable, we derive the function
P
(
±
)
=
±
2
m
(
H
-
V
(
q
)
)
, which generates the space evolution under the potential
V
(
q
)
and Hamiltonian
H
. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (
-
H
). While the classical dynamics do not change, the corresponding Quantum operator
P
^
(
±
)
naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of
P
^
(
±
)
and the kinetic energy
K
0
(separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are
±
ħ
ω
/
2
, and the corresponding pair of states are coupled for
K
0
≠
0
. No time evolution is present for
K
0
=
0
, and the ground state with energy
ħ
ω
/
2
is stable. For
K
0
>
0
, the ground state becomes coupled to the state with energy
-
ħ
ω
/
2
, and
this coupling
allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of
K
0
=
k
ħ
ω
(
k
=
1
,
2
,
…
). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case
K
0
=
0
leads to plane-waves-like solutions at the threshold. Above the threshold (
K
0
>
0
), we obtain a plane-wave-like solution, and for the bounded states (
K
0
<
0
), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus. |
| doi_str_mv | 10.1007/s10701-024-00787-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3085103257</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3085103257</sourcerecordid><originalsourceid>FETCH-LOGICAL-c200t-61a8005ffd30112719d9939df8959f806491f90225d3f97d6da5f407b1910cb23</originalsourceid><addsrcrecordid>eNp9kEFLAzEQhYMoWKt_wFPAc-xMttlsjlJbLVRUqF5Dupu0Kd1sTbYF_73bVvDmZYZ5vPcYPkJuEe4RQA4SggRkwIesOwvJ8Iz0UEjOlMD8nPQAUDAFWFySq5TWAKBkPuyR7VuTfOubQE2iJtBpqOzWdiO09NNEbxYb2-kVbVeWjmsblzaUljbuKOCAs7mvLZ1EUx5azIY-2uj3pvV7S32g7zsT2l1NX2y5MsGX6ZpcOLNJ9uZ398nHZDwfPbPZ69N09DBjJQdoWY6mABDOVRkgcomqUipTlSuUUK6AfKjQKeBcVJlTssorI9wQ5AIVQrngWZ_cnXq3sfna2dTqdbOL3YNJZ1AIhIwL2bn4yVXGJqVond5GX5v4rRH0gaw-kdUdWX0kq7ELZadQ6sxhaeNf9T-pH_IfehA</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3085103257</pqid></control><display><type>article</type><title>Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics</title><source>SpringerLink Journals</source><creator>Beims, Marcus W. ; de Lara, Arlans J. S.</creator><creatorcontrib>Beims, Marcus W. ; de Lara, Arlans J. S.</creatorcontrib><description>Using the position as an independent variable, and time as the dependent variable, we derive the function
P
(
±
)
=
±
2
m
(
H
-
V
(
q
)
)
, which generates the space evolution under the potential
V
(
q
)
and Hamiltonian
H
. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (
-
H
). While the classical dynamics do not change, the corresponding Quantum operator
P
^
(
±
)
naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of
P
^
(
±
)
and the kinetic energy
K
0
(separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are
±
ħ
ω
/
2
, and the corresponding pair of states are coupled for
K
0
≠
0
. No time evolution is present for
K
0
=
0
, and the ground state with energy
ħ
ω
/
2
is stable. For
K
0
>
0
, the ground state becomes coupled to the state with energy
-
ħ
ω
/
2
, and
this coupling
allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of
K
0
=
k
ħ
ω
(
k
=
1
,
2
,
…
). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case
K
0
=
0
leads to plane-waves-like solutions at the threshold. Above the threshold (
K
0
>
0
), we obtain a plane-wave-like solution, and for the bounded states (
K
0
<
0
), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.</description><identifier>ISSN: 0015-9018</identifier><identifier>EISSN: 1572-9516</identifier><identifier>DOI: 10.1007/s10701-024-00787-1</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Classical and Quantum Gravitation ; Classical Mechanics ; Coupling ; Dependent variables ; Derivatives ; Dirac equation ; Evolution ; Exact solutions ; Ground state ; Hamiltonian functions ; Harmonic oscillators ; History and Philosophical Foundations of Physics ; Hydrogen atoms ; Independent variables ; Kinetic energy ; Operators (mathematics) ; Parameterization ; Philosophy of Science ; Physics ; Physics and Astronomy ; Plane waves ; Quantum mechanics ; Quantum Physics ; Relativity Theory ; Separation ; Statistical Physics and Dynamical Systems ; Time dependence</subject><ispartof>Foundations of physics, 2024-08, Vol.54 (4), Article 55</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-61a8005ffd30112719d9939df8959f806491f90225d3f97d6da5f407b1910cb23</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10701-024-00787-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10701-024-00787-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Beims, Marcus W.</creatorcontrib><creatorcontrib>de Lara, Arlans J. S.</creatorcontrib><title>Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics</title><title>Foundations of physics</title><addtitle>Found Phys</addtitle><description>Using the position as an independent variable, and time as the dependent variable, we derive the function
P
(
±
)
=
±
2
m
(
H
-
V
(
q
)
)
, which generates the space evolution under the potential
V
(
q
)
and Hamiltonian
H
. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (
-
H
). While the classical dynamics do not change, the corresponding Quantum operator
P
^
(
±
)
naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of
P
^
(
±
)
and the kinetic energy
K
0
(separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are
±
ħ
ω
/
2
, and the corresponding pair of states are coupled for
K
0
≠
0
. No time evolution is present for
K
0
=
0
, and the ground state with energy
ħ
ω
/
2
is stable. For
K
0
>
0
, the ground state becomes coupled to the state with energy
-
ħ
ω
/
2
, and
this coupling
allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of
K
0
=
k
ħ
ω
(
k
=
1
,
2
,
…
). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case
K
0
=
0
leads to plane-waves-like solutions at the threshold. Above the threshold (
K
0
>
0
), we obtain a plane-wave-like solution, and for the bounded states (
K
0
<
0
), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.</description><subject>Classical and Quantum Gravitation</subject><subject>Classical Mechanics</subject><subject>Coupling</subject><subject>Dependent variables</subject><subject>Derivatives</subject><subject>Dirac equation</subject><subject>Evolution</subject><subject>Exact solutions</subject><subject>Ground state</subject><subject>Hamiltonian functions</subject><subject>Harmonic oscillators</subject><subject>History and Philosophical Foundations of Physics</subject><subject>Hydrogen atoms</subject><subject>Independent variables</subject><subject>Kinetic energy</subject><subject>Operators (mathematics)</subject><subject>Parameterization</subject><subject>Philosophy of Science</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Plane waves</subject><subject>Quantum mechanics</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Separation</subject><subject>Statistical Physics and Dynamical Systems</subject><subject>Time dependence</subject><issn>0015-9018</issn><issn>1572-9516</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKt_wFPAc-xMttlsjlJbLVRUqF5Dupu0Kd1sTbYF_73bVvDmZYZ5vPcYPkJuEe4RQA4SggRkwIesOwvJ8Iz0UEjOlMD8nPQAUDAFWFySq5TWAKBkPuyR7VuTfOubQE2iJtBpqOzWdiO09NNEbxYb2-kVbVeWjmsblzaUljbuKOCAs7mvLZ1EUx5azIY-2uj3pvV7S32g7zsT2l1NX2y5MsGX6ZpcOLNJ9uZ398nHZDwfPbPZ69N09DBjJQdoWY6mABDOVRkgcomqUipTlSuUUK6AfKjQKeBcVJlTssorI9wQ5AIVQrngWZ_cnXq3sfna2dTqdbOL3YNJZ1AIhIwL2bn4yVXGJqVond5GX5v4rRH0gaw-kdUdWX0kq7ELZadQ6sxhaeNf9T-pH_IfehA</recordid><startdate>20240801</startdate><enddate>20240801</enddate><creator>Beims, Marcus W.</creator><creator>de Lara, Arlans J. S.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20240801</creationdate><title>Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics</title><author>Beims, Marcus W. ; de Lara, Arlans J. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-61a8005ffd30112719d9939df8959f806491f90225d3f97d6da5f407b1910cb23</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Classical and Quantum Gravitation</topic><topic>Classical Mechanics</topic><topic>Coupling</topic><topic>Dependent variables</topic><topic>Derivatives</topic><topic>Dirac equation</topic><topic>Evolution</topic><topic>Exact solutions</topic><topic>Ground state</topic><topic>Hamiltonian functions</topic><topic>Harmonic oscillators</topic><topic>History and Philosophical Foundations of Physics</topic><topic>Hydrogen atoms</topic><topic>Independent variables</topic><topic>Kinetic energy</topic><topic>Operators (mathematics)</topic><topic>Parameterization</topic><topic>Philosophy of Science</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Plane waves</topic><topic>Quantum mechanics</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Separation</topic><topic>Statistical Physics and Dynamical Systems</topic><topic>Time dependence</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Beims, Marcus W.</creatorcontrib><creatorcontrib>de Lara, Arlans J. S.</creatorcontrib><collection>CrossRef</collection><jtitle>Foundations of physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Beims, Marcus W.</au><au>de Lara, Arlans J. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics</atitle><jtitle>Foundations of physics</jtitle><stitle>Found Phys</stitle><date>2024-08-01</date><risdate>2024</risdate><volume>54</volume><issue>4</issue><artnum>55</artnum><issn>0015-9018</issn><eissn>1572-9516</eissn><abstract>Using the position as an independent variable, and time as the dependent variable, we derive the function
P
(
±
)
=
±
2
m
(
H
-
V
(
q
)
)
, which generates the space evolution under the potential
V
(
q
)
and Hamiltonian
H
. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian (
-
H
). While the classical dynamics do not change, the corresponding Quantum operator
P
^
(
±
)
naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of
P
^
(
±
)
and the kinetic energy
K
0
(separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are
±
ħ
ω
/
2
, and the corresponding pair of states are coupled for
K
0
≠
0
. No time evolution is present for
K
0
=
0
, and the ground state with energy
ħ
ω
/
2
is stable. For
K
0
>
0
, the ground state becomes coupled to the state with energy
-
ħ
ω
/
2
, and
this coupling
allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of
K
0
=
k
ħ
ω
(
k
=
1
,
2
,
…
). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case
K
0
=
0
leads to plane-waves-like solutions at the threshold. Above the threshold (
K
0
>
0
), we obtain a plane-wave-like solution, and for the bounded states (
K
0
<
0
), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10701-024-00787-1</doi></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0015-9018 |
| ispartof | Foundations of physics, 2024-08, Vol.54 (4), Article 55 |
| issn | 0015-9018 1572-9516 |
| language | eng |
| recordid | cdi_proquest_journals_3085103257 |
| source | SpringerLink Journals |
| subjects | Classical and Quantum Gravitation Classical Mechanics Coupling Dependent variables Derivatives Dirac equation Evolution Exact solutions Ground state Hamiltonian functions Harmonic oscillators History and Philosophical Foundations of Physics Hydrogen atoms Independent variables Kinetic energy Operators (mathematics) Parameterization Philosophy of Science Physics Physics and Astronomy Plane waves Quantum mechanics Quantum Physics Relativity Theory Separation Statistical Physics and Dynamical Systems Time dependence |
| title | Position as an Independent Variable and the Emergence of the 1/2-Time Fractional Derivative in Quantum Mechanics |
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