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Back-of-the-envelope quantum mechani...

Olshanii, M.

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Back-of-the-envelope quantum mechanics = with extensions to many-body systems, integrable PDEs, and rare and exotic methods /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Back-of-the-envelope quantum mechanics/ Maxim Olshanii.
Reminder of title:	with extensions to many-body systems, integrable PDEs, and rare and exotic methods /
Author:	Olshanii, M.
Published:	Singapore :World Scientific, : c2024.,
Description:	1 online resource (223 p.) :ill.
Notes:	Includes index.
[NT 15003449]:	Intro -- Contents -- Preface -- 1. Ground State Energy of a Hybrid Harmonic-Quartic Oscillator: a Case Study -- Introduction -- 1.1 Solved problems -- 1.1.1 Dimensional analysis and why it fails in this case -- 1.1.1.1 Side comment: dimensional analysis and approximations -- 1.1.1.2 Side comment: how to recast input equations in a dimensionless form -- 1.1.2 Dimensional analysis: the harmonic oscillator alone -- 1.1.3 Order-of-magnitude estimate: full solution -- 1.1.3.1 Order-of-magnitude estimates vis-a-vis dimensional analysis -- 1.1.3.2 Harmonic vs. quartic regimes -- 1.1.3.3 The harmonic oscillator alone -- 1.1.3.4 The quartic oscillator alone -- 1.1.3.5 The boundary between the regimes and the final result -- 1.1.4 An afterthought: boundary between regimes from dimensional considerations -- 1.1.5 A Gaussian variational solution -- 2. A dimensional estimate for the Planck temperature: a Case Study -- 2.1 Solved problems -- 2.1.1 Estimating the Planck temperature -- 3. Bohr-Sommerfeld Quantization -- 3.1 Solved problems -- 3.1.1 Ground state energy of a harmonic oscillator -- 3.1.2 Spectrum of a harmonic oscillator -- 3.1.3 WKB treatment of a "straightened" harmonic oscillator -- 3.1.4 Ground state energy of power-law potentials -- 3.1.5 Spectrum of power-law potentials -- 3.1.6 The number of bound states of a diatomic molecule. -- 3.1.7 Coulomb problem at zero angular momentum -- 3.1.8 Quantization of angular momentum from WKB -- 3.1.9 From WKB quantization of 4D angular momentum to quantization of the Coulomb problem -- 3.1.10 Ground state energy of a logarithmic potential, a WKB analysis -- 3.2 Problems without provided solutions -- 3.2.1 Size of a neutral meson in Schwinger's toy model of quark confinement -- 3.2.2 Bohr-Sommerfeld quantization for periodic boundary conditions.
[NT 15003449]:	3.2.3 Ground state energy of multi-dimensional power-law potentials -- 3.2.4 1D box as a limit of power-law potentials -- 3.2.5 Ground state energy of a logarithmic potential, an estimate -- 3.2.6 Spectrum of a logarithmic potential -- 3.2.7 Closest approach to a logarithmic hill and to power-law hills -- 3.2.8 Spin-1/2 in the field of a wire -- 3.2.9 Dimensional analysis of the time-dependent Schrödinger equation for a hybrid harmonic-quartic oscillator -- 3.3 Background -- 3.3.1 Bohr-Sommerfeld quantization -- 3.3.2 Multi-dimensional WKB -- 3.4 Problems linked to the "Background" -- 3.4.1 Bohr-Sommerfeld quantization for one soft turning point and a hard wall -- 3.4.2 Bohr-Sommerfeld quantization for two hard walls -- 4. "Halved" Harmonic Oscillator: a Case Study -- 4.1 Solved problems -- 4.1.1 Dimensional analysis -- 4.1.2 Order-of-magnitude estimate -- 4.1.3 Another order-of-magnitude estimate -- 4.1.4 Straightforward WKB -- 4.1.5 Exact solution -- 5. Semi-Classical Matrix Elements of Observables and Perturbation Theory -- 5.1 Solved problems -- 5.1.1 Quantum expectation value of x6 in a harmonic oscillator -- 5.1.2 Expectation value of r2 for a circular Coulomb orbit -- 5.1.3 WKB approximation for some integrals involving spherical harmonics -- 5.1.4 Ground state wavefunction of a one-dimensional box -- 5.1.5 Eigenstates of the harmonic oscillator at the origin: how a factor of two can restore a quantum-classical correspondence -- 5.1.6 Probability density distribution in a "straightened" harmonic oscillator -- 5.1.7 Eigenstates of a quartic potential at the origin -- 5.1.8 Perturbation theory with exact and semi-classical matrix elements for a harmonic oscillator perturbed by a quartic correction or . . . -- 5.1.9 . . . or by a cubic correction -- 5.1.10 Shift of the energy of the first excited state -- 5.1.11 Impossible potentials.
[NT 15003449]:	5.1.12 Correction to the frequency of a harmonic oscillator as a perturbation -- 5.1.13 Outer orbital of sodium atom -- 5.1.14 Relative contributions of the expectation values of the unperturbed Hamiltonian and the perturbation to the first and the second order perturbation theory correction to energy -- 5.2 Problems without provided solutions -- 5.2.1 A perturbation theory estimate -- 5.2.2 Eigenstates of a two-dimensional harmonic oscillator at the origin -- 5.2.3 Approximate WKB expressions for matrix elements of observables in a harmonic oscillator -- 5.2.4 Off-diagonal matrix elements of the spatial coordinate for a particle in a box -- 5.2.5 Harmonic oscillator perturbed by a δ-potential, . . . -- 5.2.6 . . . and by a uniform field -- 5.2.7 Perturbative expansion of the expectation value of the perturbation itself and the virial theorem -- 5.2.8 A little theorem -- 5.3 Background -- 5.3.1 Matrix elements of operators in the WKB approximation -- 5.3.2 Perturbation theory: a brief summary -- 5.3.3 Non-positivity of the second order perturbation theory shift of the ground state energy -- 6. Variational Problems -- 6.1 Solved problems -- 6.1.1 Inserting a wall -- 6.1.2 Parity of the eigenstates -- 6.1.3 Simple variational estimate for the ground state energy of a harmonic oscillator -- 6.1.4 A property of variational estimates -- 6.1.5 Absence of nodes in the ground state -- 6.1.6 Absence of degeneracy of the ground state energy level -- 6.2 Problems without provided solutions -- 6.2.1 Do stronger potentials always lead to higher ground state energies? -- 6.2.2 Variational analysis meets perturbation theory -- 6.2.3 Another variational estimate for the ground state energy of a harmonic oscillator -- 6.2.4 . . . and yet another -- 6.2.5 Gaussian- and wedge- variational ground state energy of a quartic oscillator -- 6.3 Background.
[NT 15003449]:	6.3.1 Variational analysis -- 6.4 Problems linked to the "Background" -- 6.4.1 Complex vs. real variational spaces -- 6.4.2 A proof that the (ψ')2 energy functional does not have minima with discontinuous derivatives -- 7. Gravitational Well: a Case Study -- Introduction -- 7.1 Solved problems -- 7.1.1 Bohr-Sommerfeld quantization -- 7.1.2 A WKB-based order-of-magnitude estimate for the spectrum -- 7.1.3 A WKB-based dimensional estimate for the spectrum -- 7.1.4 A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The first order -- 7.1.5 A dimensional estimate for the perturbative correction to the spectrum -- 7.1.6 A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The second order -- 7.1.7 A simple variational treatment of the ground state of a gravitational well -- 8. Miscellaneous -- 8.1 Solved problems -- 8.1.1 A dimensional approach to the question of the number of bound states in δ-potential well . . . -- 8.1.2 . . . and in a Pöschl-Teller potential -- 8.1.3 Existence of lossless eigenstates in the 1/x2-potential -- 8.1.4 On the absence of the unitary limit in two dimensions -- 9. The Hellmann-Feynman Theorem -- 9.1 Solved problems -- 9.1.1 Lieb-Liniger model -- 9.1.2 Expectation values of 1/r2 and 1/r in the Coulomb problem, using the Hellmann-Feynman theorem -- 9.1.3 Expectation value of the trapping energy in the ground state of the Calogero system -- 9.1.4 Virial theorem from the Hellmann-Feynman theorem -- 9.2 Problems without provided solutions -- 9.2.1 Virial theorem for the logarithmic potential and its corollaries -- 9.3 Background -- 9.3.1 The Hellmann-Feynman theorem -- 10. Local Density Approximation Theories -- 10.1 Solved problems -- 10.1.1 A Thomas-Fermi estimate for the atom size and total ionization energy.
[NT 15003449]:	10.1.2 The size of an ion -- 10.1.3 Time-dependent Thomas-Fermi model for cold bosons -- 10.2 Problems without provided solutions -- 10.2.1 The quantum dot -- 10.2.2 Dimensional analysis of an atom beyond the Thomas-Fermi model -- 11. Integrable Partial Differential Equations -- 11.1 Solved problems -- 11.1.1 Solitons of the Korteweg-de Vries equation -- 11.1.2 Breathers of the nonlinear Schrödinger equation -- 11.1.3 Healing length -- 11.1.4 Dimensional analysis of the projectile problem as a prelude to a discussion on the Kadomtsev-Petviashvili solitons -- 11.1.5 Kadomtsev-Petviashvili equation -- 11.1.6 The nonlinear transport equation -- 11.1.7 Burgers equation -- 11.2 Problems without provided solutions -- 11.2.1 Stationary solitons of the Burgers equation -- 11.2.2 Stationary solitons of the nonlinear Schrödinger equation -- 11.2.3 Solitons of the sine-Gordon equation -- 12. Rare and exotic methods in elementary quantum mechanics and beyond -- 12.1 Solved problems -- 12.1.1 Quantum-mechanical supersymmetry (QM-SUSY): Pöschl-Teller as an example -- 12.1.2 What the supersymmetric structure alone implies for the scattering states of the Pöschl-Teller potential -- 12.1.3 Power-index method. Example of the nonlinear Schrödinger equation, with the Pöschl-Tellerproblem as a byproduct -- 12.1.4 A stationary-kink solution of the Burgers equation through the power-index method -- 12.1.5 Scale invariance: quantum Calogero potential as an example -- 12.1.6 Classical Calogero potential: a posteriori manifestations of the scale invariance -- 12.1.7 Classical Calogero potential: a priori manifestations of scale invariance at the Maupertuis-Jacobi level. Finding the zero-energy orbit from symmetries alone -- 12.1.8 Circle inversion, quantum: zero-energy eigenstates in a 1/r4 potential.
[NT 15003449]:	12.1.9 Self-similar tilings: moment of inertia of an equilateral triangle as a paradigm.
Subject:	Differential equations. -
Online resource:	https://www.worldscientific.com/worldscibooks/10.1142/13680#t=toc
ISBN:	9789811286384

Back-of-the-envelope quantum mechanics = with extensions to many-body systems, integrable PDEs, and rare and exotic methods /
Olshanii, M.

Back-of-the-envelope quantum mechanicswith extensions to many-body systems, integrable PDEs, and rare and exotic methods /[electronic resource] :Maxim Olshanii. - 2nd ed. - Singapore :World Scientific,c2024. - 1 online resource (223 p.) :ill.

Includes index.

Intro -- Contents -- Preface -- 1. Ground State Energy of a Hybrid Harmonic-Quartic Oscillator: a Case Study -- Introduction -- 1.1 Solved problems -- 1.1.1 Dimensional analysis and why it fails in this case -- 1.1.1.1 Side comment: dimensional analysis and approximations -- 1.1.1.2 Side comment: how to recast input equations in a dimensionless form -- 1.1.2 Dimensional analysis: the harmonic oscillator alone -- 1.1.3 Order-of-magnitude estimate: full solution -- 1.1.3.1 Order-of-magnitude estimates vis-a-vis dimensional analysis -- 1.1.3.2 Harmonic vs. quartic regimes -- 1.1.3.3 The harmonic oscillator alone -- 1.1.3.4 The quartic oscillator alone -- 1.1.3.5 The boundary between the regimes and the final result -- 1.1.4 An afterthought: boundary between regimes from dimensional considerations -- 1.1.5 A Gaussian variational solution -- 2. A dimensional estimate for the Planck temperature: a Case Study -- 2.1 Solved problems -- 2.1.1 Estimating the Planck temperature -- 3. Bohr-Sommerfeld Quantization -- 3.1 Solved problems -- 3.1.1 Ground state energy of a harmonic oscillator -- 3.1.2 Spectrum of a harmonic oscillator -- 3.1.3 WKB treatment of a "straightened" harmonic oscillator -- 3.1.4 Ground state energy of power-law potentials -- 3.1.5 Spectrum of power-law potentials -- 3.1.6 The number of bound states of a diatomic molecule. -- 3.1.7 Coulomb problem at zero angular momentum -- 3.1.8 Quantization of angular momentum from WKB -- 3.1.9 From WKB quantization of 4D angular momentum to quantization of the Coulomb problem -- 3.1.10 Ground state energy of a logarithmic potential, a WKB analysis -- 3.2 Problems without provided solutions -- 3.2.1 Size of a neutral meson in Schwinger's toy model of quark confinement -- 3.2.2 Bohr-Sommerfeld quantization for periodic boundary conditions.

ISBN: 9789811286384Subjects--Topical Terms:

517952
Differential equations.

LC Class. No.: QA371

Dewey Class. No.: 515.35

Back-of-the-envelope quantum mechanics = with extensions to many-body systems, integrable PDEs, and rare and exotic methods /
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505 0 $a Intro -- Contents -- Preface -- 1. Ground State Energy of a Hybrid Harmonic-Quartic Oscillator: a Case Study -- Introduction -- 1.1 Solved problems -- 1.1.1 Dimensional analysis and why it fails in this case -- 1.1.1.1 Side comment: dimensional analysis and approximations -- 1.1.1.2 Side comment: how to recast input equations in a dimensionless form -- 1.1.2 Dimensional analysis: the harmonic oscillator alone -- 1.1.3 Order-of-magnitude estimate: full solution -- 1.1.3.1 Order-of-magnitude estimates vis-a-vis dimensional analysis -- 1.1.3.2 Harmonic vs. quartic regimes -- 1.1.3.3 The harmonic oscillator alone -- 1.1.3.4 The quartic oscillator alone -- 1.1.3.5 The boundary between the regimes and the final result -- 1.1.4 An afterthought: boundary between regimes from dimensional considerations -- 1.1.5 A Gaussian variational solution -- 2. A dimensional estimate for the Planck temperature: a Case Study -- 2.1 Solved problems -- 2.1.1 Estimating the Planck temperature -- 3. Bohr-Sommerfeld Quantization -- 3.1 Solved problems -- 3.1.1 Ground state energy of a harmonic oscillator -- 3.1.2 Spectrum of a harmonic oscillator -- 3.1.3 WKB treatment of a "straightened" harmonic oscillator -- 3.1.4 Ground state energy of power-law potentials -- 3.1.5 Spectrum of power-law potentials -- 3.1.6 The number of bound states of a diatomic molecule. -- 3.1.7 Coulomb problem at zero angular momentum -- 3.1.8 Quantization of angular momentum from WKB -- 3.1.9 From WKB quantization of 4D angular momentum to quantization of the Coulomb problem -- 3.1.10 Ground state energy of a logarithmic potential, a WKB analysis -- 3.2 Problems without provided solutions -- 3.2.1 Size of a neutral meson in Schwinger's toy model of quark confinement -- 3.2.2 Bohr-Sommerfeld quantization for periodic boundary conditions.
505 8 $a 3.2.3 Ground state energy of multi-dimensional power-law potentials -- 3.2.4 1D box as a limit of power-law potentials -- 3.2.5 Ground state energy of a logarithmic potential, an estimate -- 3.2.6 Spectrum of a logarithmic potential -- 3.2.7 Closest approach to a logarithmic hill and to power-law hills -- 3.2.8 Spin-1/2 in the field of a wire -- 3.2.9 Dimensional analysis of the time-dependent Schrödinger equation for a hybrid harmonic-quartic oscillator -- 3.3 Background -- 3.3.1 Bohr-Sommerfeld quantization -- 3.3.2 Multi-dimensional WKB -- 3.4 Problems linked to the "Background" -- 3.4.1 Bohr-Sommerfeld quantization for one soft turning point and a hard wall -- 3.4.2 Bohr-Sommerfeld quantization for two hard walls -- 4. "Halved" Harmonic Oscillator: a Case Study -- 4.1 Solved problems -- 4.1.1 Dimensional analysis -- 4.1.2 Order-of-magnitude estimate -- 4.1.3 Another order-of-magnitude estimate -- 4.1.4 Straightforward WKB -- 4.1.5 Exact solution -- 5. Semi-Classical Matrix Elements of Observables and Perturbation Theory -- 5.1 Solved problems -- 5.1.1 Quantum expectation value of x6 in a harmonic oscillator -- 5.1.2 Expectation value of r2 for a circular Coulomb orbit -- 5.1.3 WKB approximation for some integrals involving spherical harmonics -- 5.1.4 Ground state wavefunction of a one-dimensional box -- 5.1.5 Eigenstates of the harmonic oscillator at the origin: how a factor of two can restore a quantum-classical correspondence -- 5.1.6 Probability density distribution in a "straightened" harmonic oscillator -- 5.1.7 Eigenstates of a quartic potential at the origin -- 5.1.8 Perturbation theory with exact and semi-classical matrix elements for a harmonic oscillator perturbed by a quartic correction or . . . -- 5.1.9 . . . or by a cubic correction -- 5.1.10 Shift of the energy of the first excited state -- 5.1.11 Impossible potentials.
505 8 $a 5.1.12 Correction to the frequency of a harmonic oscillator as a perturbation -- 5.1.13 Outer orbital of sodium atom -- 5.1.14 Relative contributions of the expectation values of the unperturbed Hamiltonian and the perturbation to the first and the second order perturbation theory correction to energy -- 5.2 Problems without provided solutions -- 5.2.1 A perturbation theory estimate -- 5.2.2 Eigenstates of a two-dimensional harmonic oscillator at the origin -- 5.2.3 Approximate WKB expressions for matrix elements of observables in a harmonic oscillator -- 5.2.4 Off-diagonal matrix elements of the spatial coordinate for a particle in a box -- 5.2.5 Harmonic oscillator perturbed by a δ-potential, . . . -- 5.2.6 . . . and by a uniform field -- 5.2.7 Perturbative expansion of the expectation value of the perturbation itself and the virial theorem -- 5.2.8 A little theorem -- 5.3 Background -- 5.3.1 Matrix elements of operators in the WKB approximation -- 5.3.2 Perturbation theory: a brief summary -- 5.3.3 Non-positivity of the second order perturbation theory shift of the ground state energy -- 6. Variational Problems -- 6.1 Solved problems -- 6.1.1 Inserting a wall -- 6.1.2 Parity of the eigenstates -- 6.1.3 Simple variational estimate for the ground state energy of a harmonic oscillator -- 6.1.4 A property of variational estimates -- 6.1.5 Absence of nodes in the ground state -- 6.1.6 Absence of degeneracy of the ground state energy level -- 6.2 Problems without provided solutions -- 6.2.1 Do stronger potentials always lead to higher ground state energies? -- 6.2.2 Variational analysis meets perturbation theory -- 6.2.3 Another variational estimate for the ground state energy of a harmonic oscillator -- 6.2.4 . . . and yet another -- 6.2.5 Gaussian- and wedge- variational ground state energy of a quartic oscillator -- 6.3 Background.
505 8 $a 6.3.1 Variational analysis -- 6.4 Problems linked to the "Background" -- 6.4.1 Complex vs. real variational spaces -- 6.4.2 A proof that the (ψ')2 energy functional does not have minima with discontinuous derivatives -- 7. Gravitational Well: a Case Study -- Introduction -- 7.1 Solved problems -- 7.1.1 Bohr-Sommerfeld quantization -- 7.1.2 A WKB-based order-of-magnitude estimate for the spectrum -- 7.1.3 A WKB-based dimensional estimate for the spectrum -- 7.1.4 A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The first order -- 7.1.5 A dimensional estimate for the perturbative correction to the spectrum -- 7.1.6 A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The second order -- 7.1.7 A simple variational treatment of the ground state of a gravitational well -- 8. Miscellaneous -- 8.1 Solved problems -- 8.1.1 A dimensional approach to the question of the number of bound states in δ-potential well . . . -- 8.1.2 . . . and in a Pöschl-Teller potential -- 8.1.3 Existence of lossless eigenstates in the 1/x2-potential -- 8.1.4 On the absence of the unitary limit in two dimensions -- 9. The Hellmann-Feynman Theorem -- 9.1 Solved problems -- 9.1.1 Lieb-Liniger model -- 9.1.2 Expectation values of 1/r2 and 1/r in the Coulomb problem, using the Hellmann-Feynman theorem -- 9.1.3 Expectation value of the trapping energy in the ground state of the Calogero system -- 9.1.4 Virial theorem from the Hellmann-Feynman theorem -- 9.2 Problems without provided solutions -- 9.2.1 Virial theorem for the logarithmic potential and its corollaries -- 9.3 Background -- 9.3.1 The Hellmann-Feynman theorem -- 10. Local Density Approximation Theories -- 10.1 Solved problems -- 10.1.1 A Thomas-Fermi estimate for the atom size and total ionization energy.
505 8 $a 10.1.2 The size of an ion -- 10.1.3 Time-dependent Thomas-Fermi model for cold bosons -- 10.2 Problems without provided solutions -- 10.2.1 The quantum dot -- 10.2.2 Dimensional analysis of an atom beyond the Thomas-Fermi model -- 11. Integrable Partial Differential Equations -- 11.1 Solved problems -- 11.1.1 Solitons of the Korteweg-de Vries equation -- 11.1.2 Breathers of the nonlinear Schrödinger equation -- 11.1.3 Healing length -- 11.1.4 Dimensional analysis of the projectile problem as a prelude to a discussion on the Kadomtsev-Petviashvili solitons -- 11.1.5 Kadomtsev-Petviashvili equation -- 11.1.6 The nonlinear transport equation -- 11.1.7 Burgers equation -- 11.2 Problems without provided solutions -- 11.2.1 Stationary solitons of the Burgers equation -- 11.2.2 Stationary solitons of the nonlinear Schrödinger equation -- 11.2.3 Solitons of the sine-Gordon equation -- 12. Rare and exotic methods in elementary quantum mechanics and beyond -- 12.1 Solved problems -- 12.1.1 Quantum-mechanical supersymmetry (QM-SUSY): Pöschl-Teller as an example -- 12.1.2 What the supersymmetric structure alone implies for the scattering states of the Pöschl-Teller potential -- 12.1.3 Power-index method. Example of the nonlinear Schrödinger equation, with the Pöschl-Tellerproblem as a byproduct -- 12.1.4 A stationary-kink solution of the Burgers equation through the power-index method -- 12.1.5 Scale invariance: quantum Calogero potential as an example -- 12.1.6 Classical Calogero potential: a posteriori manifestations of the scale invariance -- 12.1.7 Classical Calogero potential: a priori manifestations of scale invariance at the Maupertuis-Jacobi level. Finding the zero-energy orbit from symmetries alone -- 12.1.8 Circle inversion, quantum: zero-energy eigenstates in a 1/r4 potential.
505 8 $a 12.1.9 Self-similar tilings: moment of inertia of an equilateral triangle as a paradigm.
588 $a Description based on print version record.
650 0 $a Differential equations. $3 517952
650 0 $a Quantum theory. $3 516552
856 4 0 $u https://www.worldscientific.com/worldscibooks/10.1142/13680#t=toc