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Schrödinger Equations for Delta Potential Barrier in Quantum Phase Space
Available online at www.sciencedirect.com
Procedia Engineering 23 (2011) 95 – 98
Power Electronics and Engineering Application
Schrödinger equations for delta potential barrier in quantum phase space Jun LU* Department of Foundational Science, Beijing Union University, Beijing, 100101, China
Abstract The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords: phase space; delta potential barrier; collision; Schrödinger equation __________________________________________________________________________________________________________
1. Introduction The quantum theory represents a radical change, not only in the content of scientific knowledge, but also in the fundamental conceptual framework in terms of which such knowledge can be expressed. The true extent of this change of conceptual framework has perhaps been obscured by the contrast between the relatively pictorial and easily imagined terms in which classical theory has always been expressed, with the very abstract and mathematical form in which quantum theory obtained it original development. Nevertheless, with the further development of the physical interpretation of the theory, it finally became possible to express the results of the quantum theory in terms of comparatively qualitative and imaginative concepts, which are, however, of a totally different nature from those appearing in the classical theory [1]. Phase space is just one of these concepts. As an important concept in classical statistical mechanics, phase space cannot be utilized directly in quantum theory due to the Heisenberg uncertainty principle. However, since Wigner [2-4] introduced the
*
Corresponding author. Tel.:+86-10-64900221; fax: +86-10-64900217. E-mail address: [email protected].
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.11.2471
96
Jun LU / Procedia Engineering 23 (2011) 95 – 98
first quasi-probability distribution function in phase space for quantum corrections to classical statistical mechanics, a variety of quantum phase-space distribution functions have been proposed, and found to have extensive uses in many areas of physics and chemistry. In 1990s, Torres-Vega and Frederick established a new phase-space representation of quantum mechanics [5, 6]. They define quantum state functions directly in the phase space, and determine corresponding Schrödinger equations and quantum Liouville equations. The most important character of this quantum phase-space theory is that almost all the mathematical properties in the general position or momentum representation of quantum mechanics are retained for quantum state functions in the phase space. Therefore, many useful methods and results can be replanted to the phase-space representation. It should be noted that the mathematical treatment for physical problems in the phase space is much more complicated than that in general position or momentum space, because the number of dimensions in the phase space is twice of that in general position or momentum space. It is well known that the wave functions in position or momentum representation of quantum mechanics can be defined exactly, differing at most by a phase factor. However, the wave functions in the phase-space representation are not unique, just like the quantum phase-space distribution functions. The wave functions in the phase space are also more complicated than those in general position or momentum space in terms of formulation. Therefore, it is of significance to find some solvable physical systems in the phase space, not only for the discussion of general properties of the wave functions but also for practical applications. Recently, a variety of advances have been made in this area [7, 8]. The scattering theory in the phase space has also been developed [9]. At the same time, some equivalent quantum phase-space theories, for example, the relative-state theory, the linear transformation theory, and the displacement-operator theory have also been established [10]. The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. In this paper, we will solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We will define the Schrödinger equations of the delta potential fields in the phase space. 2. Schrödinger equations for a particle in delta potential fields According to the mapping of Torres-Vega and Frederick, the position and momentum operators in the phase space can be written as follows,
q o Qˆ p o Pˆ
q w ˈ i= 2 wp p w i= DŽ 2 wq
(1) (2)
Then we obtain the stationary state Schrödinger equation of the systems as 2 ª 1 § p §q w ·º w · ¨¨ i= ¸¸ Vˆ ¨¨ i= ¸¸»< q, p E< q, p (3) « 2P © 2 wp ¹» wq ¹ ©2 ¼ ¬« § · q w ˆ where μ is the mass of the particle, V ¨¨ i= ¸ the potential energy operator and E the eigenenergy. wp ¸¹ By means of the relations ©2
n
§ w · exp ipq / 2= ¨¨ i= ¸¸ exp ipq / 2= © wp ¹
n
§q w · ¨¨ i= ¸¸ ˈ wp ¹ ©2
97
Jun LU / Procedia Engineering 23 (2011) 95 – 98 n
n
§p w · ¨¨ i= ¸¸ wq ¹ ©2
§ w · exp ipq / 2= ¨¨ i= ¸¸ exp ipq / 2= wq ¹ ©
§ §q w · w · exp -ipq / 2= Vˆ ¨¨ q i= ¸¸ exp ipq / 2= Vˆ ¨¨ i= ¸¸ ˈ wp ¹ wp ¹ © ©2 and by setting
§ ipq · ¸) q, p ˈ © 2= ¹
< q, p exp¨ we can transform Eq. (3) as
ª =2 w2 ˆ §¨ q i= w ·¸º») q, p E) q, p DŽ V « ¨ 2 wp ¸¹¼ © ¬ 2 P wq If we insert the potential energy function V (q) to be the delta potential fields, i.e.,
V ( q)
rJG q q 0
J
> 0 ˈ
where Ȗ is a parameter and q0 the eigenvalue of the position operator, and the delta function is represented as
G q q 0 lim
D of
D 2
exp D q q 0 ˈ
where Į is a parameter, we obtain the potential energy operator as
§ w · Vˆ ¨¨ q i= ¸¸ wp ¹ ©
§ w · rJGˆ¨¨ q q 0 i= ¸¸ wp ¹ ©
J
> 0 DŽ
Then, if regarding
Tˆ
§ w · exp¨¨ i=D ¸¸ ˈ wp ¹ ©
as the momentum displacement operator, and noting that exp D q q0
is commutable with Tˆ i.e.,
ª ˆº «exp D q q 0 , T » 0 ˈ ¬ ¼ if the delta function is represented as Eq. (10), we can easily prove that
98
Jun LU / Procedia Engineering 23 (2011) 95 – 98
§
Gˆ¨¨ q q 0 i= ©
w · ¸ G q q 0 Tˆ DŽ wp ¸¹
With the help of Eqs. (9) and (14), Eq. (8) becomes
=2 w2 ) q, p 2 P wq 2
>E B JG q q Tˆ @) q, p ˈ 0
This is the stationary state Schrödinger equation of a particle in the delta potential fields in the phase space.
Acknowledgements This work was supported by the Science and Technology Research Program of Beijing Municipal Education Commission (Grant No. KM201111417007), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (Grant No. PHR200907120), and the Innovative Talent Foundation of Beijing Union University.
References [1] Bohm D. Quantum Theory. New York: Prentice-Hall; 1951. [2] Wigner EP. Quantum corrections for thermodynamic equilibrium. Phys Rev 1932; 40: 749-59. [3] Glauber RJ. The quantum theory of optical coherence. Phys Rev 1963; 130: 2529-35. [4] Klauder JR, Skagerstum BS. Coherent States. Singapore: World Scientific; 1985. [5] Torres-Vega G, Frederick JH. Quantum mechanics in phase space: new approaches to the correspondence principle. J Chem Phys 1990; 93: 8862-74. [6] Torres-Vega G, Frederick JH. A quantum mechanical representation in phase space. J Chem Phys 1993; 98: 3103-20. [7] Li QS, Hu XG. On the quantum mechanical representation in phase space. Phys Scripta 1995; 51: 417-22. [8] Møller KB, Jorgensen TG, Torres-Vega G. On coherent-state representations of quantum mechanics: wave mechanics in phase space. J Chem Phys 1997; 106: 7228-40. [9] Hu XG, Li QS. Morse oscillator in a quantum phase-space representation: rigorous solutions. J Phys A: Gen Math 1999; 32: 139-46. [10] Ban M. Relative-state formulation of quantum systems. Phys Rev A 1993; 48: 3452-60.
Procedia Engineering 23 (2011) 95 – 98
Power Electronics and Engineering Application
Schrödinger equations for delta potential barrier in quantum phase space Jun LU* Department of Foundational Science, Beijing Union University, Beijing, 100101, China
Abstract The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords: phase space; delta potential barrier; collision; Schrödinger equation __________________________________________________________________________________________________________
1. Introduction The quantum theory represents a radical change, not only in the content of scientific knowledge, but also in the fundamental conceptual framework in terms of which such knowledge can be expressed. The true extent of this change of conceptual framework has perhaps been obscured by the contrast between the relatively pictorial and easily imagined terms in which classical theory has always been expressed, with the very abstract and mathematical form in which quantum theory obtained it original development. Nevertheless, with the further development of the physical interpretation of the theory, it finally became possible to express the results of the quantum theory in terms of comparatively qualitative and imaginative concepts, which are, however, of a totally different nature from those appearing in the classical theory [1]. Phase space is just one of these concepts. As an important concept in classical statistical mechanics, phase space cannot be utilized directly in quantum theory due to the Heisenberg uncertainty principle. However, since Wigner [2-4] introduced the
*
Corresponding author. Tel.:+86-10-64900221; fax: +86-10-64900217. E-mail address: [email protected].
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.11.2471
96
Jun LU / Procedia Engineering 23 (2011) 95 – 98
first quasi-probability distribution function in phase space for quantum corrections to classical statistical mechanics, a variety of quantum phase-space distribution functions have been proposed, and found to have extensive uses in many areas of physics and chemistry. In 1990s, Torres-Vega and Frederick established a new phase-space representation of quantum mechanics [5, 6]. They define quantum state functions directly in the phase space, and determine corresponding Schrödinger equations and quantum Liouville equations. The most important character of this quantum phase-space theory is that almost all the mathematical properties in the general position or momentum representation of quantum mechanics are retained for quantum state functions in the phase space. Therefore, many useful methods and results can be replanted to the phase-space representation. It should be noted that the mathematical treatment for physical problems in the phase space is much more complicated than that in general position or momentum space, because the number of dimensions in the phase space is twice of that in general position or momentum space. It is well known that the wave functions in position or momentum representation of quantum mechanics can be defined exactly, differing at most by a phase factor. However, the wave functions in the phase-space representation are not unique, just like the quantum phase-space distribution functions. The wave functions in the phase space are also more complicated than those in general position or momentum space in terms of formulation. Therefore, it is of significance to find some solvable physical systems in the phase space, not only for the discussion of general properties of the wave functions but also for practical applications. Recently, a variety of advances have been made in this area [7, 8]. The scattering theory in the phase space has also been developed [9]. At the same time, some equivalent quantum phase-space theories, for example, the relative-state theory, the linear transformation theory, and the displacement-operator theory have also been established [10]. The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. In this paper, we will solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We will define the Schrödinger equations of the delta potential fields in the phase space. 2. Schrödinger equations for a particle in delta potential fields According to the mapping of Torres-Vega and Frederick, the position and momentum operators in the phase space can be written as follows,
q o Qˆ p o Pˆ
q w ˈ i= 2 wp p w i= DŽ 2 wq
(1) (2)
Then we obtain the stationary state Schrödinger equation of the systems as 2 ª 1 § p §q w ·º w · ¨¨ i= ¸¸ Vˆ ¨¨ i= ¸¸»< q, p E< q, p (3) « 2P © 2 wp ¹» wq ¹ ©2 ¼ ¬« § · q w ˆ where μ is the mass of the particle, V ¨¨ i= ¸ the potential energy operator and E the eigenenergy. wp ¸¹ By means of the relations ©2
n
§ w · exp ipq / 2= ¨¨ i= ¸¸ exp ipq / 2= © wp ¹
n
§q w · ¨¨ i= ¸¸ ˈ wp ¹ ©2
97
Jun LU / Procedia Engineering 23 (2011) 95 – 98 n
n
§p w · ¨¨ i= ¸¸ wq ¹ ©2
§ w · exp ipq / 2= ¨¨ i= ¸¸ exp ipq / 2= wq ¹ ©
§ §q w · w · exp -ipq / 2= Vˆ ¨¨ q i= ¸¸ exp ipq / 2= Vˆ ¨¨ i= ¸¸ ˈ wp ¹ wp ¹ © ©2 and by setting
§ ipq · ¸) q, p ˈ © 2= ¹
< q, p exp¨ we can transform Eq. (3) as
ª =2 w2 ˆ §¨ q i= w ·¸º») q, p E) q, p DŽ V « ¨ 2 wp ¸¹¼ © ¬ 2 P wq If we insert the potential energy function V (q) to be the delta potential fields, i.e.,
V ( q)
rJG q q 0
J
> 0 ˈ
where Ȗ is a parameter and q0 the eigenvalue of the position operator, and the delta function is represented as
G q q 0 lim
D of
D 2
exp D q q 0 ˈ
where Į is a parameter, we obtain the potential energy operator as
§ w · Vˆ ¨¨ q i= ¸¸ wp ¹ ©
§ w · rJGˆ¨¨ q q 0 i= ¸¸ wp ¹ ©
J
> 0 DŽ
Then, if regarding
Tˆ
§ w · exp¨¨ i=D ¸¸ ˈ wp ¹ ©
as the momentum displacement operator, and noting that exp D q q0
is commutable with Tˆ i.e.,
ª ˆº «exp D q q 0 , T » 0 ˈ ¬ ¼ if the delta function is represented as Eq. (10), we can easily prove that
98
Jun LU / Procedia Engineering 23 (2011) 95 – 98
§
Gˆ¨¨ q q 0 i= ©
w · ¸ G q q 0 Tˆ DŽ wp ¸¹
With the help of Eqs. (9) and (14), Eq. (8) becomes
=2 w2 ) q, p 2 P wq 2
>E B JG q q Tˆ @) q, p ˈ 0
This is the stationary state Schrödinger equation of a particle in the delta potential fields in the phase space.
Acknowledgements This work was supported by the Science and Technology Research Program of Beijing Municipal Education Commission (Grant No. KM201111417007), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (Grant No. PHR200907120), and the Innovative Talent Foundation of Beijing Union University.
References [1] Bohm D. Quantum Theory. New York: Prentice-Hall; 1951. [2] Wigner EP. Quantum corrections for thermodynamic equilibrium. Phys Rev 1932; 40: 749-59. [3] Glauber RJ. The quantum theory of optical coherence. Phys Rev 1963; 130: 2529-35. [4] Klauder JR, Skagerstum BS. Coherent States. Singapore: World Scientific; 1985. [5] Torres-Vega G, Frederick JH. Quantum mechanics in phase space: new approaches to the correspondence principle. J Chem Phys 1990; 93: 8862-74. [6] Torres-Vega G, Frederick JH. A quantum mechanical representation in phase space. J Chem Phys 1993; 98: 3103-20. [7] Li QS, Hu XG. On the quantum mechanical representation in phase space. Phys Scripta 1995; 51: 417-22. [8] Møller KB, Jorgensen TG, Torres-Vega G. On coherent-state representations of quantum mechanics: wave mechanics in phase space. J Chem Phys 1997; 106: 7228-40. [9] Hu XG, Li QS. Morse oscillator in a quantum phase-space representation: rigorous solutions. J Phys A: Gen Math 1999; 32: 139-46. [10] Ban M. Relative-state formulation of quantum systems. Phys Rev A 1993; 48: 3452-60.