Complex entropy and resultant information measures
Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\va...
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| Autor*in: |
Nalewajski, Roman F. [verfasserIn] |
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Artikel |
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| Sprache: |
Englisch |
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2016 |
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© The Author(s) 2016 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of mathematical chemistry - Springer International Publishing, 1987, 54(2016), 9 vom: 06. Juni, Seite 1777-1782 |
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| Übergeordnetes Werk: |
volume:54 ; year:2016 ; number:9 ; day:06 ; month:06 ; pages:1777-1782 |
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| DOI / URN: |
10.1007/s10910-016-0651-6 |
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| Katalog-ID: |
OLC206042271X |
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| 100 | 1 | |a Nalewajski, Roman F. |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a Complex entropy and resultant information measures |
| 264 | 1 | |c 2016 | |
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| 520 | |a Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ | ||
| 650 | 4 | |a Complex entropy | |
| 650 | 4 | |a Fisher information | |
| 650 | 4 | |a Information theory | |
| 650 | 4 | |a Nonclassical information | |
| 650 | 4 | |a Resultant information measures | |
| 650 | 4 | |a Shannon entropy | |
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10.1007/s10910-016-0651-6 doi (DE-627)OLC206042271X (DE-He213)s10910-016-0651-6-p DE-627 ger DE-627 rakwb eng 510 540 VZ Nalewajski, Roman F. verfasserin aut Complex entropy and resultant information measures 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ Complex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy Enthalten in Journal of mathematical chemistry Springer International Publishing, 1987 54(2016), 9 vom: 06. Juni, Seite 1777-1782 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:54 year:2016 number:9 day:06 month:06 pages:1777-1782 https://doi.org/10.1007/s10910-016-0651-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_70 GBV_ILN_4012 AR 54 2016 9 06 06 1777-1782 |
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10.1007/s10910-016-0651-6 doi (DE-627)OLC206042271X (DE-He213)s10910-016-0651-6-p DE-627 ger DE-627 rakwb eng 510 540 VZ Nalewajski, Roman F. verfasserin aut Complex entropy and resultant information measures 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ Complex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy Enthalten in Journal of mathematical chemistry Springer International Publishing, 1987 54(2016), 9 vom: 06. Juni, Seite 1777-1782 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:54 year:2016 number:9 day:06 month:06 pages:1777-1782 https://doi.org/10.1007/s10910-016-0651-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_70 GBV_ILN_4012 AR 54 2016 9 06 06 1777-1782 |
| allfields_unstemmed |
10.1007/s10910-016-0651-6 doi (DE-627)OLC206042271X (DE-He213)s10910-016-0651-6-p DE-627 ger DE-627 rakwb eng 510 540 VZ Nalewajski, Roman F. verfasserin aut Complex entropy and resultant information measures 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ Complex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy Enthalten in Journal of mathematical chemistry Springer International Publishing, 1987 54(2016), 9 vom: 06. Juni, Seite 1777-1782 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:54 year:2016 number:9 day:06 month:06 pages:1777-1782 https://doi.org/10.1007/s10910-016-0651-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_70 GBV_ILN_4012 AR 54 2016 9 06 06 1777-1782 |
| allfieldsGer |
10.1007/s10910-016-0651-6 doi (DE-627)OLC206042271X (DE-He213)s10910-016-0651-6-p DE-627 ger DE-627 rakwb eng 510 540 VZ Nalewajski, Roman F. verfasserin aut Complex entropy and resultant information measures 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ Complex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy Enthalten in Journal of mathematical chemistry Springer International Publishing, 1987 54(2016), 9 vom: 06. Juni, Seite 1777-1782 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:54 year:2016 number:9 day:06 month:06 pages:1777-1782 https://doi.org/10.1007/s10910-016-0651-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_70 GBV_ILN_4012 AR 54 2016 9 06 06 1777-1782 |
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10.1007/s10910-016-0651-6 doi (DE-627)OLC206042271X (DE-He213)s10910-016-0651-6-p DE-627 ger DE-627 rakwb eng 510 540 VZ Nalewajski, Roman F. verfasserin aut Complex entropy and resultant information measures 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ Complex entropy Fisher information Information theory Nonclassical information Resultant information measures Shannon entropy Enthalten in Journal of mathematical chemistry Springer International Publishing, 1987 54(2016), 9 vom: 06. Juni, Seite 1777-1782 (DE-627)129246441 (DE-600)59132-4 (DE-576)27906036X 0259-9791 nnns volume:54 year:2016 number:9 day:06 month:06 pages:1777-1782 https://doi.org/10.1007/s10910-016-0651-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-CHE SSG-OLC-MAT SSG-OLC-PHA SSG-OLC-DE-84 SSG-OPC-MAT GBV_ILN_70 GBV_ILN_4012 AR 54 2016 9 06 06 1777-1782 |
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The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Complex entropy</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Fisher information</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Information theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nonclassical information</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Resultant information measures</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Shannon entropy</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of mathematical chemistry</subfield><subfield code="d">Springer International Publishing, 1987</subfield><subfield code="g">54(2016), 9 vom: 06. 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Nalewajski, Roman F. |
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complex entropy and resultant information measures |
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Complex entropy and resultant information measures |
| abstract |
Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ © The Author(s) 2016 |
| abstractGer |
Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ © The Author(s) 2016 |
| abstract_unstemmed |
Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$ © The Author(s) 2016 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC206042271X</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230518172320.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200819s2016 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10910-016-0651-6</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC206042271X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s10910-016-0651-6-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="a">540</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Nalewajski, Roman F.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Complex entropy and resultant information measures</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2016</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s) 2016</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Classical and nonclassical contributions to Author’s resultant Shannon- and Fisher-type measures of the information content in general electronic state $$\varphi ( {\varvec{r}} ) =R( {\varvec{r}}) \hbox { exp}[\hbox {i}\phi ( {\varvec{r}} )]$$, due to the state probability density $$p( {\varvec{r}} ) =R( {\varvec{r}} )^{2}$$ and its phase $$\phi ( {\varvec{r}} )$$ or current $${\varvec{j}}( {\varvec{r}} )=\left( \hbar /m \right) p( {\varvec{r}} )\nabla \phi \left( {\varvec{r}} \right) $$ distributions, respectively, are reexamined. The components of the overall entropy, S[φ]≡-∫p(r)[lnp(r)+2ϕ(r)]dr≡S[p]+S[ϕ],$$\begin{aligned} S[\varphi ]\equiv -\int {p({\varvec{r}})[\ln p({\varvec{r}})+ 2\phi ( {\varvec{r}} )] \,d{\varvec{r}}\equiv S[p]+S[\phi ]}, \end{aligned}$$are shown to determine the real and imaginary parts of the state complex Shannon entropy, H[φ]≡-2φ|lnφ|φ=Sp+iS[ϕ],$$\begin{aligned} H[\varphi ]\equiv -2\left\langle {\varphi |\ln \varphi |\varphi } \right\rangle =S\left[ p \right] +\hbox {i}S[\phi ], \end{aligned}$$a natural quantum-amplitude generalization of the classical Shannon entropy. Its contributions are related to the associated terms in the state resultant Fisher information, I[φ]≡-4⟨φ|∇2|φ⟩≡∫p(r){[∇lnp(r)]2+[2∇ϕr]2}dr≡I[p]+I[ϕ]=I[p]+∫p(r)[(2m/ħ)j(r)/p(r)]2dr≡I[p]+I[j],$$\begin{aligned} I[\varphi ]\equiv & {} -4\langle {\varphi |\nabla ^{2}|\varphi } \rangle \equiv \int {p({\varvec{r}})\{[} \nabla \ln p({\varvec{r}})]^{2}+[2\nabla \phi \left( {\varvec{r}} \right) ]^{2}\}\,d{\varvec{r}}\equiv I[p]+I[\phi ] \\= & {} I[p]+\int {p({\varvec{r}})[(2m/\hbar ){\varvec{j}}({\varvec{r}})/p({\varvec{r}})]^{2}\,d{\varvec{r}}\equiv I[p]+I[{\varvec{j}}],} \\ \end{aligned}$$and the gradient entropy: I~[φ]≡⟨φ|[(∇lnp)2+(i2∇ϕ)2]|φ⟩=I[p]-I[ϕ]=I~[p]+I~[ϕ].$$\begin{aligned} \tilde{I}[\varphi ]\equiv \langle {\varphi |[(\nabla \ln p)^{2}+(\hbox {i}2\nabla \phi )^{2}]|\varphi } \rangle =I[p]-I[\phi ]=\tilde{I}[p]+\tilde{I}[\phi ]. \end{aligned}$$</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Complex entropy</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Fisher information</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Information theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nonclassical information</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Resultant information measures</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Shannon entropy</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of mathematical chemistry</subfield><subfield code="d">Springer International Publishing, 1987</subfield><subfield code="g">54(2016), 9 vom: 06. Juni, Seite 1777-1782</subfield><subfield code="w">(DE-627)129246441</subfield><subfield code="w">(DE-600)59132-4</subfield><subfield code="w">(DE-576)27906036X</subfield><subfield code="x">0259-9791</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:54</subfield><subfield code="g">year:2016</subfield><subfield code="g">number:9</subfield><subfield code="g">day:06</subfield><subfield code="g">month:06</subfield><subfield code="g">pages:1777-1782</subfield></datafield><datafield tag="856" ind1="4" ind2="1"><subfield code="u">https://doi.org/10.1007/s10910-016-0651-6</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_OLC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-CHE</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-PHA</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-DE-84</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OPC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_70</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_4012</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">54</subfield><subfield code="j">2016</subfield><subfield code="e">9</subfield><subfield code="b">06</subfield><subfield code="c">06</subfield><subfield code="h">1777-1782</subfield></datafield></record></collection>
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