On time derivatives for <X^< and <p^<: formal 1D calculations
We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators...
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| Autor*in: |
Salvatore De Vincenzo [verfasserIn] |
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E-Artikel |
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| Sprache: |
Portugiesisch |
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In: Revista Brasileira de Ensino de Física - Sociedade Brasileira de Física, 2004 |
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| DOI / URN: |
10.1590/S1806-11172013000200008 |
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DOAJ019820224 |
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| 520 | |a We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. | ||
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10.1590/S1806-11172013000200008 doi (DE-627)DOAJ019820224 (DE-599)DOAJ4ae2ca8392824800958830a6f36dd9b0 DE-627 ger DE-627 rakwb por QC1-999 Salvatore De Vincenzo verfasserin aut On time derivatives for <X^< and <p^<: formal 1D calculations Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. quantum mechanics schrödinger equation probability density probability density current bohm's quantum potential quantum force Physics In Revista Brasileira de Ensino de Física Sociedade Brasileira de Física, 2004 (DE-627)365134007 (DE-600)2111196-0 18069126 nnns https://doi.org/10.1590/S1806-11172013000200008 kostenfrei https://doaj.org/article/4ae2ca8392824800958830a6f36dd9b0 kostenfrei http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172013000200008&lng=en&tlng=en kostenfrei https://doaj.org/toc/1806-1117 Journal toc kostenfrei https://doaj.org/toc/1806-9126 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ AR |
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10.1590/S1806-11172013000200008 doi (DE-627)DOAJ019820224 (DE-599)DOAJ4ae2ca8392824800958830a6f36dd9b0 DE-627 ger DE-627 rakwb por QC1-999 Salvatore De Vincenzo verfasserin aut On time derivatives for <X^< and <p^<: formal 1D calculations Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. quantum mechanics schrödinger equation probability density probability density current bohm's quantum potential quantum force Physics In Revista Brasileira de Ensino de Física Sociedade Brasileira de Física, 2004 (DE-627)365134007 (DE-600)2111196-0 18069126 nnns https://doi.org/10.1590/S1806-11172013000200008 kostenfrei https://doaj.org/article/4ae2ca8392824800958830a6f36dd9b0 kostenfrei http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172013000200008&lng=en&tlng=en kostenfrei https://doaj.org/toc/1806-1117 Journal toc kostenfrei https://doaj.org/toc/1806-9126 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ AR |
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We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. |
| abstractGer |
We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. |
| abstract_unstemmed |
We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region. |
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| title_short |
On time derivatives for <X^< and <p^<: formal 1D calculations |
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https://doi.org/10.1590/S1806-11172013000200008 https://doaj.org/article/4ae2ca8392824800958830a6f36dd9b0 http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172013000200008&lng=en&tlng=en https://doaj.org/toc/1806-1117 https://doaj.org/toc/1806-9126 |
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