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Physical problems solved by the phase-integral method / / Nanny Fröman and Per Olof Fröman [[electronic resource]]
| Physical problems solved by the phase-integral method / / Nanny Fröman and Per Olof Fröman [[electronic resource]] |
| Autore | Fröman Nanny |
| Pubbl/distr/stampa | Cambridge : , : Cambridge University Press, , 2002 |
| Descrizione fisica | 1 online resource (xiii, 214 pages) : digital, PDF file(s) |
| Disciplina | 530.12/4 |
| Soggetto topico |
WKB approximation
Wave equation |
| ISBN |
1-107-12544-8
0-511-02050-3 1-280-43394-9 9786610433940 0-511-17681-3 0-511-15771-1 0-511-30463-3 0-511-53508-2 0-511-04525-5 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
1. Historical survey -- 1.1. Development from 1817 to 1926 -- 1.2. Development after 1926 -- 2. Description of the phase-integral method -- 2.1. Form of the wave function and the q-equation -- 2.2. Phase-integral approximation generated from an unspecified base function -- 2.3. F-matrix method -- 2.4. F-matrix connecting points on opposite sides of a well-isolated turning point, and expressions for the wave function in these regions -- 2.5. Phase-integral connection formulas for a real, smooth, single-hump potential barrier -- 3. Problems with solutions -- 3.1. Base function for the radial Schrodinger equation when the physical potential has at the most a Coulomb singularity at the origin -- 3.2. Base function and wave function close to the origin when the physical potential is repulsive and strongly singular at the origin -- 3.3. Reflectionless potential -- 3.4. Stokes and anti-Stokes lines -- 3.5. Properties of the phase-integral approximation along an anti-Stokes line -- 3.6. Properties of the phase-integral approximation along a path on which the absolute value of exp[iw(z)] is monotonic in the strict sense, in particular along a Stokes line -- 3.7. Determination of the Stokes constants associated with the three anti-Stokes lines that emerge from a well isolated, simple transition zero -- 3.8. Connection formula for tracing a phase-integral wave function from a Stokes line emerging from a simple transition zero t to the anti-Stokes line emerging from t in the opposite direction -- 3.9. Connection formula for tracing a phase-integral wave function from an anti-Stokes line emerging from a simple transition zero t to the Stokes line emerging from t in the opposite direction -- 3.10. Connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region -- 3.11. One-directional nature of the connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region -- 3.12. Connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region -- 3.13. One-directional nature of the connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region -- 3.14. Value at the turning point of the wave function associated with the connection formula for tracing a phase-integral wave function from the classically forbidden to the classically allowed region -- 3.15. Value at the turning point of the wave function associated with a connection formula for tracing the phase-integral wave function from the classically allowed to the classically forbidden region -- 3.16. Illustration of the accuracy of the approximate formulas for the value of the wave function at a turning point -- 3.17. Expression for the a-coefficients associated with the Airy functions -- 3.18. Expressions for the parameters [alpha], [beta] and [gamma] when Q[superscript 2](z) = R(z) = -z -- 3.19. Solutions of the Airy differential equation that at a fixed point on one side of the turning point are represented by a single, pure phase-integral function, and their representation on the other side of the turning point -- 3.20. Connection formulas and their one-directional nature demonstrated for the Airy differential equation -- 3.21. Dependence of the phase of the wave function in a classically allowed region on the value of the logarithmic derivative of the wave function at a fixed point X[subscript 1] in an adjacent classically forbidden region -- 3.22. Phase of the wave function in the classically allowed regions adjacent to a real, symmetric potential barrier, when the logarithmic derivative of the wave function is given at the centre of the barrier -- 3.23. Eigenvalue problem for a quantal particle in a broad, symmetric potential well between two symmetric potential barriers of equal shape, with boundary conditions imposed in the middle of each barrier -- 3.24. Dependence of the phase of the wave function in a classically allowed region on the position of the point [chi][subscript 1] in an adjacent classically forbidden region where the boundary condition [psi][chi][subscript 1]) = 0 is imposed -- 3.25. Phase-shift formula -- 3.26. Distance between near-lying energy levels in different types of physical systems, expressed either in terms of the frequency of classical oscillations in a potential well or in terms of the derivative of the energy with respect to a quantum number -- 3.27. Arbitrary-order quantization condition for a particle in a single-well potential, derived on the assumption that the classically allowed region is broad enough to allow the use of a connection formula -- 3.28. Arbitrary-order quantization condition for a particle in a single-well potential, derived without the assumption that the classically allowed region is broad -- 3.29. Displacement of the energy levels due to compression of an atom (simple treatment) -- 3.30. Displacement of the energy levels due to compression of an atom (alternative treatment) -- 3.31. Quantization condition for a particle in a smooth potential well, limited on one side by an impenetrable wall and on the other side by a smooth, infinitely thick potential barrier, and in particular for a particle in a uniform gravitational field limited from below by an impenetrable plane surface -- 3.32. Energy spectrum of a non-relativistic particle in a potential proportional to cot[superscript 2]([chi]/a[subscript 0]), where 0 < [chi]/a[subscript 0] < [pi] and a[subscript 0] is a quantity with the dimension of length, e.g. the Bohr radius -- 3.33. Determination of a one-dimensional, smooth, single-well potential from the energy spectrum of the bound states -- 3.34. Determination of a radical, smooth, single-well potential from the energy spectrum of the bound states -- 3.35. Determination of the radial, single-well potential, when the energy eigenvalues are -mZ[superscript 2]e[superscript 4]/[2[actual symbol not reproducible][superscript 2]([iota]+s+1)[superscript 2]], where [iota] is the angular momentum quantum number, and s is the radial quantum number -- 3.36. Exact formula for the normalization integral for the wave function pertaining to a bound state of a particle in a radial potential -- 3.37. Phase-integral formula for the normalized radial wave function pertaining to a bound state of a particle in a radial single-well potential -- 3.38. Radial wave function [psi](z) for as s-electron in a classically allowed region containing the origin, when the potential near the origin is dominated by a strong, attractive Coulomb singularity, and the normalization factor is chosen such that, when the radial variable z is dimensionless, [psi](z)/z tends to unity as z tends to zero -- 3.39. Quantization condition, and value of the normalized wave function at the origin expressed in terms of the level density, for as s-electron in a single-well potential with a strong attractive Coulomb singularity at the origin -- 3.40. Expectation value of an unspecified function f(z) for a non-relativistic particle in a bound state -- 3.41. Some cases in which the phase-integral expectation value formula yields the expectation value exactly in the first-order approximation -- 3.42. Expectation value of the kinetic energy of a non-relativistic particle in a bound state. Verification of the virial theorem -- 3.43. Phase-integral calculation of quantal matrix elements -- 3.44. Connection formula for a complex potential barrier -- 3.45. Connection formula for a real, single-hump potential barrier -- 3.46. Energy levels of a particle in a smooth double-well potential, when no symmetry requirement is imposed -- 3.47. Energy levels of a particle in a smooth, symmetric, double-well potential -- 3.48. Determination of the quasi-stationary energy levels of a particle in a radial potential with a thick single-hump barrier -- 3.49.
Transmission coefficient for a particle penetrating a real single-hump potential barrier -- 3.50. Transmission coefficient for a particle penetrating a real, symmetric, superdense double-hump potential barrier. |
| Record Nr. | UNINA-9910455574203321 |
Fröman Nanny
|
||
| Cambridge : , : Cambridge University Press, , 2002 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Physical problems solved by the phase-integral method / / Nanny Fröman and Per Olof Fröman [[electronic resource]]
| Physical problems solved by the phase-integral method / / Nanny Fröman and Per Olof Fröman [[electronic resource]] |
| Autore | Fröman Nanny |
| Pubbl/distr/stampa | Cambridge : , : Cambridge University Press, , 2002 |
| Descrizione fisica | 1 online resource (xiii, 214 pages) : digital, PDF file(s) |
| Disciplina | 530.12/4 |
| Soggetto topico |
WKB approximation
Wave equation |
| ISBN |
1-107-12544-8
0-511-02050-3 1-280-43394-9 9786610433940 0-511-17681-3 0-511-15771-1 0-511-30463-3 0-511-53508-2 0-511-04525-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
1. Historical survey -- 1.1. Development from 1817 to 1926 -- 1.2. Development after 1926 -- 2. Description of the phase-integral method -- 2.1. Form of the wave function and the q-equation -- 2.2. Phase-integral approximation generated from an unspecified base function -- 2.3. F-matrix method -- 2.4. F-matrix connecting points on opposite sides of a well-isolated turning point, and expressions for the wave function in these regions -- 2.5. Phase-integral connection formulas for a real, smooth, single-hump potential barrier -- 3. Problems with solutions -- 3.1. Base function for the radial Schrodinger equation when the physical potential has at the most a Coulomb singularity at the origin -- 3.2. Base function and wave function close to the origin when the physical potential is repulsive and strongly singular at the origin -- 3.3. Reflectionless potential -- 3.4. Stokes and anti-Stokes lines -- 3.5. Properties of the phase-integral approximation along an anti-Stokes line -- 3.6. Properties of the phase-integral approximation along a path on which the absolute value of exp[iw(z)] is monotonic in the strict sense, in particular along a Stokes line -- 3.7. Determination of the Stokes constants associated with the three anti-Stokes lines that emerge from a well isolated, simple transition zero -- 3.8. Connection formula for tracing a phase-integral wave function from a Stokes line emerging from a simple transition zero t to the anti-Stokes line emerging from t in the opposite direction -- 3.9. Connection formula for tracing a phase-integral wave function from an anti-Stokes line emerging from a simple transition zero t to the Stokes line emerging from t in the opposite direction -- 3.10. Connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region -- 3.11. One-directional nature of the connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed region -- 3.12. Connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region -- 3.13. One-directional nature of the connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden region -- 3.14. Value at the turning point of the wave function associated with the connection formula for tracing a phase-integral wave function from the classically forbidden to the classically allowed region -- 3.15. Value at the turning point of the wave function associated with a connection formula for tracing the phase-integral wave function from the classically allowed to the classically forbidden region -- 3.16. Illustration of the accuracy of the approximate formulas for the value of the wave function at a turning point -- 3.17. Expression for the a-coefficients associated with the Airy functions -- 3.18. Expressions for the parameters [alpha], [beta] and [gamma] when Q[superscript 2](z) = R(z) = -z -- 3.19. Solutions of the Airy differential equation that at a fixed point on one side of the turning point are represented by a single, pure phase-integral function, and their representation on the other side of the turning point -- 3.20. Connection formulas and their one-directional nature demonstrated for the Airy differential equation -- 3.21. Dependence of the phase of the wave function in a classically allowed region on the value of the logarithmic derivative of the wave function at a fixed point X[subscript 1] in an adjacent classically forbidden region -- 3.22. Phase of the wave function in the classically allowed regions adjacent to a real, symmetric potential barrier, when the logarithmic derivative of the wave function is given at the centre of the barrier -- 3.23. Eigenvalue problem for a quantal particle in a broad, symmetric potential well between two symmetric potential barriers of equal shape, with boundary conditions imposed in the middle of each barrier -- 3.24. Dependence of the phase of the wave function in a classically allowed region on the position of the point [chi][subscript 1] in an adjacent classically forbidden region where the boundary condition [psi][chi][subscript 1]) = 0 is imposed -- 3.25. Phase-shift formula -- 3.26. Distance between near-lying energy levels in different types of physical systems, expressed either in terms of the frequency of classical oscillations in a potential well or in terms of the derivative of the energy with respect to a quantum number -- 3.27. Arbitrary-order quantization condition for a particle in a single-well potential, derived on the assumption that the classically allowed region is broad enough to allow the use of a connection formula -- 3.28. Arbitrary-order quantization condition for a particle in a single-well potential, derived without the assumption that the classically allowed region is broad -- 3.29. Displacement of the energy levels due to compression of an atom (simple treatment) -- 3.30. Displacement of the energy levels due to compression of an atom (alternative treatment) -- 3.31. Quantization condition for a particle in a smooth potential well, limited on one side by an impenetrable wall and on the other side by a smooth, infinitely thick potential barrier, and in particular for a particle in a uniform gravitational field limited from below by an impenetrable plane surface -- 3.32. Energy spectrum of a non-relativistic particle in a potential proportional to cot[superscript 2]([chi]/a[subscript 0]), where 0 < [chi]/a[subscript 0] < [pi] and a[subscript 0] is a quantity with the dimension of length, e.g. the Bohr radius -- 3.33. Determination of a one-dimensional, smooth, single-well potential from the energy spectrum of the bound states -- 3.34. Determination of a radical, smooth, single-well potential from the energy spectrum of the bound states -- 3.35. Determination of the radial, single-well potential, when the energy eigenvalues are -mZ[superscript 2]e[superscript 4]/[2[actual symbol not reproducible][superscript 2]([iota]+s+1)[superscript 2]], where [iota] is the angular momentum quantum number, and s is the radial quantum number -- 3.36. Exact formula for the normalization integral for the wave function pertaining to a bound state of a particle in a radial potential -- 3.37. Phase-integral formula for the normalized radial wave function pertaining to a bound state of a particle in a radial single-well potential -- 3.38. Radial wave function [psi](z) for as s-electron in a classically allowed region containing the origin, when the potential near the origin is dominated by a strong, attractive Coulomb singularity, and the normalization factor is chosen such that, when the radial variable z is dimensionless, [psi](z)/z tends to unity as z tends to zero -- 3.39. Quantization condition, and value of the normalized wave function at the origin expressed in terms of the level density, for as s-electron in a single-well potential with a strong attractive Coulomb singularity at the origin -- 3.40. Expectation value of an unspecified function f(z) for a non-relativistic particle in a bound state -- 3.41. Some cases in which the phase-integral expectation value formula yields the expectation value exactly in the first-order approximation -- 3.42. Expectation value of the kinetic energy of a non-relativistic particle in a bound state. Verification of the virial theorem -- 3.43. Phase-integral calculation of quantal matrix elements -- 3.44. Connection formula for a complex potential barrier -- 3.45. Connection formula for a real, single-hump potential barrier -- 3.46. Energy levels of a particle in a smooth double-well potential, when no symmetry requirement is imposed -- 3.47. Energy levels of a particle in a smooth, symmetric, double-well potential -- 3.48. Determination of the quasi-stationary energy levels of a particle in a radial potential with a thick single-hump barrier -- 3.49.
Transmission coefficient for a particle penetrating a real single-hump potential barrier -- 3.50. Transmission coefficient for a particle penetrating a real, symmetric, superdense double-hump potential barrier. |
| Record Nr. | UNINA-9910780252103321 |
|
Fröman Nanny
|
||
| Cambridge : , : Cambridge University Press, , 2002 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Stark effect in a hydrogenic atom or ion [[electronic resource] ] : treated by the phase-integral method / / Nanny Fröman, Per Olof Fröman ; with adjoined papers by A. Hökback and P.O. Fröman
| Stark effect in a hydrogenic atom or ion [[electronic resource] ] : treated by the phase-integral method / / Nanny Fröman, Per Olof Fröman ; with adjoined papers by A. Hökback and P.O. Fröman |
| Autore | Fröman Nanny |
| Pubbl/distr/stampa | London, : Imperial College Press |
| Descrizione fisica | 1 online resource (164 p.) |
| Disciplina | 530.12 |
| Altri autori (Persone) | FrömanPer Olof |
| Soggetto topico |
Stark effect
Optical spectroscopy Quantum theory Schrödinger equation |
| Soggetto genere / forma | Electronic books. |
| ISBN |
1-281-86768-3
9786611867683 1-86094-925-8 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Contents; Preface; 1 Introduction; Brief review of different aspects studied and various methods used; Brief account of the background of this book; Publications with relevance to this book; Treatment in this book; Brief account of the contents of this book; 2 Schrödinger Equation, its Separation and its Exact Eigenfunctions; 2.1 Separation of the time-independent Schrödinger equation for the internal motion; 2.2 Properties of the eigenfunctions of the time-independent Schrödinger equation for the internal motion; 3 Development in Time of the Probability Amplitude for a Decaying State
4 Phase-Integral Method 4.1 Phase-integral approximation generated from an unspecified base function; 4.2 Connection formulas associated with a single transition point; 4.2.1 Connection formulas pertaining to a first-order transition zero on the real axis; 4.2.2 Connection formula pertaining to a first-order transition pole at the origin; 4.3 Connection formula for a real, smooth, single-hump potential barrier; 4.3.1 Wave function given as a standing wave; 4.3.2 Supplementary quantity φ; 4.4 Quantization conditions for single-well potentials 5 Derivation of Phase-Integral Formulas for Profiles, Energies and Half-Widths of Stark Levels5.1 Positions of the Stark levels; 5.2 Formulas for the calculation of dL/dE, dK2n/dE and dK/dE; 5.3 Half-widths of the Stark levels; 6 Procedure for Transformation of the Phase-Integral Formulas into Formulas Involving Complete Elliptic Integrals; Adjoined Papers by Anders Hökback and Per Olof Fröman; 7 Phase-Inegral Quantities and Their Partial Derivatives with Respect to E and Z1 Expressed in Terms of Complete Elliptic Integrals; 7.1 The ξ-equation; 7.2 The η-equation in the sub-barrier case 7.3 The η-equation in the super-barrier case8 Numerical Results; References; Name Index; Subject Index |
| Record Nr. | UNINA-9910453176303321 |
|
Fröman Nanny
|
||
| London, : Imperial College Press | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Stark effect in a hydrogenic atom or ion [[electronic resource] ] : treated by the phase-integral method / / Nanny Fröman, Per Olof Fröman ; with adjoined papers by A. Hökback and P.O. Fröman
| Stark effect in a hydrogenic atom or ion [[electronic resource] ] : treated by the phase-integral method / / Nanny Fröman, Per Olof Fröman ; with adjoined papers by A. Hökback and P.O. Fröman |
| Autore | Fröman Nanny |
| Pubbl/distr/stampa | London, : Imperial College Press |
| Descrizione fisica | 1 online resource (164 p.) |
| Disciplina | 530.12 |
| Altri autori (Persone) | FrömanPer Olof |
| Soggetto topico |
Stark effect
Optical spectroscopy Quantum theory Schrödinger equation |
| ISBN |
1-281-86768-3
9786611867683 1-86094-925-8 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Contents; Preface; 1 Introduction; Brief review of different aspects studied and various methods used; Brief account of the background of this book; Publications with relevance to this book; Treatment in this book; Brief account of the contents of this book; 2 Schrödinger Equation, its Separation and its Exact Eigenfunctions; 2.1 Separation of the time-independent Schrödinger equation for the internal motion; 2.2 Properties of the eigenfunctions of the time-independent Schrödinger equation for the internal motion; 3 Development in Time of the Probability Amplitude for a Decaying State
4 Phase-Integral Method 4.1 Phase-integral approximation generated from an unspecified base function; 4.2 Connection formulas associated with a single transition point; 4.2.1 Connection formulas pertaining to a first-order transition zero on the real axis; 4.2.2 Connection formula pertaining to a first-order transition pole at the origin; 4.3 Connection formula for a real, smooth, single-hump potential barrier; 4.3.1 Wave function given as a standing wave; 4.3.2 Supplementary quantity φ; 4.4 Quantization conditions for single-well potentials 5 Derivation of Phase-Integral Formulas for Profiles, Energies and Half-Widths of Stark Levels5.1 Positions of the Stark levels; 5.2 Formulas for the calculation of dL/dE, dK2n/dE and dK/dE; 5.3 Half-widths of the Stark levels; 6 Procedure for Transformation of the Phase-Integral Formulas into Formulas Involving Complete Elliptic Integrals; Adjoined Papers by Anders Hökback and Per Olof Fröman; 7 Phase-Inegral Quantities and Their Partial Derivatives with Respect to E and Z1 Expressed in Terms of Complete Elliptic Integrals; 7.1 The ξ-equation; 7.2 The η-equation in the sub-barrier case 7.3 The η-equation in the super-barrier case8 Numerical Results; References; Name Index; Subject Index |
| Record Nr. | UNINA-9910782271003321 |
|
Fröman Nanny
|
||
| London, : Imperial College Press | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||