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Integral Forms for Quantum-Mechanical Momentum Operators
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| Zeitschriftentitel: | Journal of Mathematical Physics |
|---|---|
| Personen und Körperschaften: | |
| In: | Journal of Mathematical Physics, 7, 1966, 11, S. 2060-2065 |
| Medientyp: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
AIP Publishing
|
| Schlagwörter: |
| author_facet |
Robinson, Peter D. Robinson, Peter D. |
|---|---|
| author |
Robinson, Peter D. |
| spellingShingle |
Robinson, Peter D. Journal of Mathematical Physics Integral Forms for Quantum-Mechanical Momentum Operators Mathematical Physics Statistical and Nonlinear Physics |
| author_sort |
robinson, peter d. |
| spelling |
Robinson, Peter D. 0022-2488 1089-7658 AIP Publishing Mathematical Physics Statistical and Nonlinear Physics http://dx.doi.org/10.1063/1.1704889 <jats:p>The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.</jats:p> Integral Forms for Quantum-Mechanical Momentum Operators Journal of Mathematical Physics |
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10.1063/1.1704889 |
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Mathematik Physik |
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AIP Publishing, 1966 |
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AIP Publishing, 1966 |
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0022-2488 1089-7658 |
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0022-2488 1089-7658 |
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1966 |
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AIP Publishing |
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Journal of Mathematical Physics |
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49 |
| title |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_unstemmed |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_full |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_fullStr |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_full_unstemmed |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_short |
Integral Forms for Quantum-Mechanical Momentum Operators |
| title_sort |
integral forms for quantum-mechanical momentum operators |
| topic |
Mathematical Physics Statistical and Nonlinear Physics |
| url |
http://dx.doi.org/10.1063/1.1704889 |
| publishDate |
1966 |
| physical |
2060-2065 |
| description |
<jats:p>The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.</jats:p> |
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| author | Robinson, Peter D. |
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| container_title | Journal of Mathematical Physics |
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| description | <jats:p>The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.</jats:p> |
| doi_str_mv | 10.1063/1.1704889 |
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| imprint | AIP Publishing, 1966 |
| imprint_str_mv | AIP Publishing, 1966 |
| institution | DE-D275, DE-Bn3, DE-Brt1, DE-D161, DE-Zi4, DE-Gla1, DE-15, DE-Pl11, DE-Rs1, DE-14, DE-105, DE-Ch1, DE-L229 |
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| physical | 2060-2065 |
| publishDate | 1966 |
| publishDateSort | 1966 |
| publisher | AIP Publishing |
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| series | Journal of Mathematical Physics |
| source_id | 49 |
| spelling | Robinson, Peter D. 0022-2488 1089-7658 AIP Publishing Mathematical Physics Statistical and Nonlinear Physics http://dx.doi.org/10.1063/1.1704889 <jats:p>The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.</jats:p> Integral Forms for Quantum-Mechanical Momentum Operators Journal of Mathematical Physics |
| spellingShingle | Robinson, Peter D., Journal of Mathematical Physics, Integral Forms for Quantum-Mechanical Momentum Operators, Mathematical Physics, Statistical and Nonlinear Physics |
| title | Integral Forms for Quantum-Mechanical Momentum Operators |
| title_full | Integral Forms for Quantum-Mechanical Momentum Operators |
| title_fullStr | Integral Forms for Quantum-Mechanical Momentum Operators |
| title_full_unstemmed | Integral Forms for Quantum-Mechanical Momentum Operators |
| title_short | Integral Forms for Quantum-Mechanical Momentum Operators |
| title_sort | integral forms for quantum-mechanical momentum operators |
| title_unstemmed | Integral Forms for Quantum-Mechanical Momentum Operators |
| topic | Mathematical Physics, Statistical and Nonlinear Physics |
| url | http://dx.doi.org/10.1063/1.1704889 |