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Sahni V. Perspectives on determinism in quantum mechanics: Born, Bohm, and the "Quantal Newtonian" laws. J Chem Phys 2022; 157:244106. [PMID: 36586987 DOI: 10.1063/5.0130945] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
Quantum mechanics has a deterministic Schrödinger equation for the wave function. The Göttingen-Copenhagen statistical interpretation is based on the Born Rule that interprets the wave function as a "probability amplitude." A precept of this interpretation is the lack of determinism in quantum mechanics. The Bohm interpretation is that the wave function is a source of a field experienced by the electrons, thereby attributing determinism to quantum theory. In this paper, we present a new perspective on such determinism. The ideas are based on the equations of motion or "Quantal Newtonian" Laws obeyed by each electron. These Laws, derived from the temporal and stationary-state Schrödinger equation, are interpreted in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. According to the Second Law, each electron experiences an external field-the quantal Coulomb-Lorentz law. It also experiences an internal field representative of properties of the system: correlations due to Coulomb repulsion and Pauli principle; the density; kinetic effects; and an internal magnetic field component. There is a response field. The First Law states that the sum of the external and internal fields experienced by each electron vanishes. These fields are akin to those of classical physics: They pervade all space; their structure is descriptive of the quantum system; the energy of the system is stored in these fields. It is in the classical behavior of these fields, which arise from quantal sources that one may then speak of determinism in quantum mechanics.
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Affiliation(s)
- Viraht Sahni
- Brooklyn College and The Graduate Center of the City University of New York, Brooklyn, New York 11210, USA
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Zanoli S, Roca-Maza X, Colò G, Shen S. Harmonic Potential Theorem: Extension to Spin-, Velocity-, and Density-Dependent Interactions. PHYSICAL REVIEW LETTERS 2019; 123:112501. [PMID: 31573255 DOI: 10.1103/physrevlett.123.112501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2019] [Revised: 07/18/2019] [Indexed: 06/10/2023]
Abstract
One of the few exact results for the description of the time evolution of an inhomogeneous, interacting many-particle system is given by the harmonic potential theorem (HPT). The relevance of this theorem is that it sets a tight constraint on time-dependent many-body approximations. In this contribution, we show that the original formulation of the HPT is valid also for the case of spin-, velocity-, and density-dependent interactions. This result is completely general and relevant, among the rest, for nuclear structure theory both in the case of ab initio and of more phenomenological approaches. As an example, we report on a numerical implementation by testing the small-amplitude limit of the time-dependent Hartree-Fock-also known as the random phase approximation-for the translational frequencies of a neutron system trapped in a harmonic potential.
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Affiliation(s)
- S Zanoli
- Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, 20133 Milano, Italy and INFN, Sezione di Milano, 20133 Milano, Italy
| | - X Roca-Maza
- Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, 20133 Milano, Italy and INFN, Sezione di Milano, 20133 Milano, Italy
| | - G Colò
- Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, 20133 Milano, Italy and INFN, Sezione di Milano, 20133 Milano, Italy
| | - Shihang Shen
- Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, 20133 Milano, Italy and INFN, Sezione di Milano, 20133 Milano, Italy
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Slamet M, Sahni V. Kinetic effects in 2D and 3D quantum dots: Comparison between high and low electron correlation regimes. COMPUT THEOR CHEM 2018. [DOI: 10.1016/j.comptc.2018.06.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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Magnetic field effect on the energy levels of an exciton in a GaAs quantum dot: Application for excitonic lasers. Sci Rep 2018; 8:5073. [PMID: 29567977 PMCID: PMC5864831 DOI: 10.1038/s41598-018-23348-9] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2017] [Accepted: 03/02/2018] [Indexed: 11/12/2022] Open
Abstract
The problem of an exciton trapped in a Gaussian quantum dot (QD) of GaAs is studied in both two and three dimensions in the presence of an external magnetic field using the Ritz variational method, the 1/N expansion method and the shifted 1/N expansion method. The ground state energy and the binding energy of the exciton are obtained as a function of the quantum dot size, confinement strength and the magnetic field and compared with those available in the literature. While the variational method gives the upper bound to the ground state energy, the 1/N expansion method gives the lower bound. The results obtained from the shifted 1/N expansion method are shown to match very well with those obtained from the exact diagonalization technique. The variation of the exciton size and the oscillator strength of the exciton are also studied as a function of the size of the quantum dot. The excited states of the exciton are computed using the shifted 1/N expansion method and it is suggested that a given number of stable excitonic bound states can be realized in a quantum dot by tuning the quantum dot parameters. This can open up the possibility of having quantum dot lasers using excitonic states.
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Slamet M, Sahni V. Electron correlations in an excited state of a quantum dot in a uniform magnetic field. COMPUT THEOR CHEM 2017. [DOI: 10.1016/j.comptc.2017.05.012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Wasserman A, Nafziger J, Jiang K, Kim MC, Sim E, Burke K. The Importance of Being Inconsistent. Annu Rev Phys Chem 2017; 68:555-581. [PMID: 28463652 DOI: 10.1146/annurev-physchem-052516-044957] [Citation(s) in RCA: 74] [Impact Index Per Article: 10.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Adam Wasserman
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907
| | - Jonathan Nafziger
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
| | - Kaili Jiang
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907
| | - Min-Cheol Kim
- Department of Chemistry, Yonsei University, Seoul 03722, South Korea
| | - Eunji Sim
- Department of Chemistry, Yonsei University, Seoul 03722, South Korea
| | - Kieron Burke
- Department of Chemistry, University of California, Irvine, California 92697
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Lai MY, Pan XY. The generalized harmonic potential theorem in the presence of a time-varying magnetic field. Sci Rep 2016; 6:35412. [PMID: 27748461 PMCID: PMC5066324 DOI: 10.1038/srep35412] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2016] [Accepted: 09/27/2016] [Indexed: 11/08/2022] Open
Abstract
We investigate the evolution of the many-body wave function of a quantum system with time-varying effective mass, confined by a harmonic potential with time-varying frequency in the presence of a uniform time-varying magnetic field, and perturbed by a time-dependent uniform electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. In other words, we generalize the harmonic potential theorem to the case when the effective mass, harmonic potential, and the external uniform magnetic field with arbitrary orientation are all time-varying. The results reduce to various special cases obtained in the literature, particulary to that of the harmonic potential theorem wave function when the effective mass and frequency are both static and the external magnetic field is absent.
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Affiliation(s)
- Meng-Yun Lai
- Department of Physics, Ningbo University, Ningbo, 315211, China
| | - Xiao-Yin Pan
- Department of Physics, Ningbo University, Ningbo, 315211, China
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Zhu HM, Chen JW, Pan XY, Sahni V. Wave function for harmonically confined electrons in time-dependent electric and magnetostatic fields. J Chem Phys 2014; 140:024318. [PMID: 24437888 DOI: 10.1063/1.4858463] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We derive via the interaction "representation" the many-body wave function for harmonically confined electrons in the presence of a magnetostatic field and perturbed by a spatially homogeneous time-dependent electric field-the Generalized Kohn Theorem (GKT) wave function. In the absence of the harmonic confinement - the uniform electron gas - the GKT wave function reduces to the Kohn Theorem wave function. Without the magnetostatic field, the GKT wave function is the Harmonic Potential Theorem wave function. We further prove the validity of the connection between the GKT wave function derived and the system in an accelerated frame of reference. Finally, we provide examples of the application of the GKT wave function.
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Affiliation(s)
- Hong-Ming Zhu
- Department of Physics, Ningbo University, Ningbo 315211, China
| | - Jin-Wang Chen
- Department of Physics, Ningbo University, Ningbo 315211, China
| | - Xiao-Yin Pan
- Department of Physics, Ningbo University, Ningbo 315211, China
| | - Viraht Sahni
- Department of Physics, Brooklyn College and The Graduate School of the City University of New York, New York, New York 10016, USA
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Li YQ, Pan XY, Sahni V. Wave function for time-dependent harmonically confined electrons in a time-dependent electric field. J Chem Phys 2013; 139:114301. [PMID: 24070284 DOI: 10.1063/1.4820245] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the Harmonic Potential Theorem wave function for the case of the time-independent harmonic confining potential.
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Affiliation(s)
- Yu-Qi Li
- Department of Physics, Ningbo University, Ningbo 315211, China
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Reuter D, Kailuweit P, Wieck AD, Zeitler U, Wibbelhoff O, Meier C, Lorke A, Maan JC. Coulomb-interaction-induced incomplete shell filling in the hole system of InAs quantum dots. PHYSICAL REVIEW LETTERS 2005; 94:026808. [PMID: 15698214 DOI: 10.1103/physrevlett.94.026808] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2004] [Indexed: 05/24/2023]
Abstract
We have studied the hole charging spectra of self-assembled InAs quantum dots in perpendicular magnetic fields by capacitance-voltage spectroscopy. From the magnetic-field dependence of the individual peaks we conclude that the s-like ground state is completely filled with two holes but that the fourfold degenerate p shell is only half filled with two holes before the filling of the d shell starts. The resulting six-hole ground state is highly polarized. This incomplete shell filling can be explained by the large influence of the Coulomb interaction in this system.
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Affiliation(s)
- D Reuter
- Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44799 Bochum, Germany
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Santer M, Mehlig B, Moseler M. Optical response of two-dimensional electron fluids beyond the kohn regime: strong nonparabolic confinement and intense laser light. PHYSICAL REVIEW LETTERS 2002; 89:286801. [PMID: 12513170 DOI: 10.1103/physrevlett.89.286801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2002] [Indexed: 05/24/2023]
Abstract
We investigate the linear and nonlinear optical response of two-dimensional interacting electron fluids confined by a strong nonparabolic potential. We show that such fluids may exhibit higher-harmonic spectra under realistic experimental conditions. Higher harmonics arise as the electrons explore anharmonicities of the confinement potential (electron-electron interactions reduce this nonlinear effect). This opens the possibility of controlling the optical functionality by engineering the confinement potential. Our results were obtained within time-dependent density-functional theory. A classical hydrodynamical model is in good agreement with the quantum-mechanical results.
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Affiliation(s)
- M Santer
- Theoretical Quantum Dynamics, University of Freiburg, 79104 Freiburg, Germany
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Vignale G, Kohn W. Current-Dependent Exchange-Correlation Potential for Dynamical Linear Response Theory. PHYSICAL REVIEW LETTERS 1996; 77:2037-2040. [PMID: 10061841 DOI: 10.1103/physrevlett.77.2037] [Citation(s) in RCA: 112] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Schaich WL, Geller MR, Vignale G. Classical continuum theory of the dipole-forbidden collective excitations in quantum strips. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:13016-13023. [PMID: 9982979 DOI: 10.1103/physrevb.53.13016] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Mikhailov SA, Volkov VA. Theory of electromagnetic response and collective excitations in antidots. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:17260-17268. [PMID: 9981156 DOI: 10.1103/physrevb.52.17260] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Vignale G. Center of mass and relative motion in time dependent density functional theory. PHYSICAL REVIEW LETTERS 1995; 74:3233-3236. [PMID: 10058145 DOI: 10.1103/physrevlett.74.3233] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Dobson JF. Harmonic-potential theorem: Implications for approximate many-body theories. PHYSICAL REVIEW LETTERS 1994; 73:2244-2247. [PMID: 10057009 DOI: 10.1103/physrevlett.73.2244] [Citation(s) in RCA: 54] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Tanaka T, Arakawa Y, Bauer GW. Magnetoexcitons in quantum wires with an anisotropic parabolic potential. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:7719-7723. [PMID: 9974756 DOI: 10.1103/physrevb.50.7719] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Zhu KD, Gu SW. Cyclotron resonance of magnetopolarons in a parabolic quantum dot in strong magnetic fields. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:12941-12944. [PMID: 10005496 DOI: 10.1103/physrevb.47.12941] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Drexler H, Hansen W, Kotthaus JP, Holland M, Beaumont SP. Nonparabolic confinement in quantum wire superlattices. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:12849-12852. [PMID: 10003225 DOI: 10.1103/physrevb.46.12849] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Haupt R, Wendler L, Pechstedt R. Depolarization shift in quasi-one-dimensional quantum-well wires. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:13635-13640. [PMID: 9999567 DOI: 10.1103/physrevb.44.13635] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Johnson NF, Payne MC. Exactly solvable model of interacting particles in a quantum dot. PHYSICAL REVIEW LETTERS 1991; 67:1157-1160. [PMID: 10045090 DOI: 10.1103/physrevlett.67.1157] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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