Back-of-the-Envelope Quantum Mechanics

The following sections are included:

• Dedication
• Preface
• Contents
• Some notations

The following sections are included:

• Introduction
• Solved problems
  • Dimensional analysis and why it fails in this case
    • Side comment: dimensional analysis and approximations
    • Side comment: how to recast input equations in a dimensionless form
  • Dimensional analysis: the harmonic oscillator alone
  • Order-of-magnitude estimate: full solution
    • Order-of-magnitude estimates vis-a-vis dimensional analysis
    • Harmonic vs. quartic regimes
    • The harmonic oscillator alone
    • The quartic oscillator alone
    • The boundary between the regimes and the final result
  • An afterthought: boundary between regimes from dimensional considerations
  • A Gaussian variational solution

The following sections are included:

• Solved problems
  • Estimating the Planck temperature

The following sections are included:

• Solved problems
  • Ground state energy of a harmonic oscillator
  • Spectrum of a harmonic oscillator
  • WKB treatment of a “straightened” harmonic oscillator
  • Ground state energy of power-law potentials
  • Spectrum of power-law potentials
  • The number of bound states of a diatomic molecule
  • Coulomb problem at zero angular momentum
  • Quantization of angular momentum from WKB
  • From WKB quantization of 4D angular momentum to quantization of the Coulomb problem
  • Ground state energy of a logarithmic potential, a WKB analysis
• Problems without provided solutions
  • Size of a neutral meson in Schwinger’s toy model of quark confinement
  • Bohr-Sommerfeld quantization for periodic boundary conditions
  • Ground state energy of multi-dimensional power-law potentials
  • 1D box as a limit of power-law potentials
  • Ground state energy of a logarithmic potential, an estimate
  • Spectrum of a logarithmic potential
  • Closest approach to a logarithmic hill and to power-law hills
  • Spin-1/2 in the field of a wire
  • Dimensional analysis of the time-dependent Schrödinger equation for a hybrid harmonic-quartic oscillator
• Background
  • Bohr-Sommerfeld quantization
  • Multi-dimensional WKB
• Problems linked to the “Background”
  • Bohr-Sommerfeld quantization for one soft turning point and a hard wall
  • Bohr-Sommerfeld quantization for two hard walls

The following sections are included:

• Introduction
• Solved problems
  • Dimensional analysis
  • Order-of-magnitude estimate
  • Another order-of-magnitude estimate
  • Straightforward WKB
  • Exact solution

The following sections are included:

• Solved problems
  • Quantum expectation value of x⁶ in a harmonic oscillator
  • Expectation value of r² for a circular Coulomb orbit
  • WKB approximation for some integrals involving spherical harmonics
  • Ground state wavefunction of a one-dimensional box
  • Eigenstates of the harmonic oscillator at the origin: how a factor of two can restore a quantum-classical correspondence
  • Probability density distribution in a “straightened” harmonic oscillator
  • Eigenstates of a quartic potential at the origin
  • Perturbation theory with exact and semi-classical matrix elements for a harmonic oscillator perturbed by a quartic correction or
  • … or by a cubic correction
  • Shift of the energy of the first excited state
  • Impossible potentials
  • Correction to the frequency of a harmonic oscillator as a perturbation
  • Outer orbital of sodium atom
  • 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
• Problems without provided solutions
  • A perturbation theory estimate
  • Eigenstates of a two-dimensional harmonic oscillator at the origin
  • Approximate WKB expressions for matrix elements of observables in a harmonic oscillator
  • Off-diagonal matrix elements of the spatial coordinate for a particle in a box
  • Harmonic oscillator perturbed by a δ-potential, …
  • … and by a uniform field
  • Perturbative expansion of the expectation value of the perturbation itself and the virial theorem
  • A little theorem
• Background
  • Matrix elements of operators in the WKB approximation
  • Perturbation theory: a brief summary
  • Non-positivity of the second order perturbation theory shift of the ground state energy

The following sections are included:

• Solved problems
  • Inserting a wall
  • Parity of the eigenstates
  • Simple variational estimate for the ground state energy of a harmonic oscillator
  • A property of variational estimates
  • Absence of nodes in the ground state
  • Absence of degeneracy of the ground state energy level
• Problems without provided solutions
  • Do stronger potentials always lead to higher ground state energies?
  • Variational analysis meets perturbation theory
  • Another variational estimate for the ground state energy of a harmonic oscillator
  • … and yet another
  • Gaussian- and wedge- variational ground state energy of a quartic oscillator
• Background
  • Variational analysis
• Problems linked to the “Background”
  • Complex vs. real variational spaces
  • A proof that the (ψ′)² energy functional does not have minima with discontinuous derivatives

The following sections are included:

• Introduction
• Solved problems
  • Bohr-Sommerfeld quantization
  • A WKB-based order-of-magnitude estimate for the spectrum
  • A WKB-based dimensional estimate for the spectrum
  • A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The first order
  • A dimensional estimate for the perturbative correction to the spectrum
  • A perturbative calculation of the shift of the energy levels under a small change in the coupling constant. The second order
  • A simple variational treatment of the ground state of a gravitational well

The following sections are included:

• Solved problems
  • A dimensional approach to the question of the number of bound states in δ-potential well
  • … and in a Pöschl-Teller potential
  • Existence of lossless eigenstates in the 1/x²-potential
  • On the absence of the unitary limit in two dimensions

The following sections are included:

• Solved problems
  • Lieb-Liniger model
  • Expectation values of 1/r² and 1/r in the Coulomb problem, using the Hellmann-Feynman theorem
  • Expectation value of the trapping energy in the ground state of the Calogero system
  • Virial theorem from the Hellmann-Feynman theorem
• Problems without provided solutions
  • Virial theorem for the logarithmic potential and its corollaries
• Background
  • The Hellmann-Feynman theorem

The following sections are included:

• Solved problems
  • A Thomas-Fermi estimate for the atom size and total ionization energy
  • The size of an ion
  • Time-dependent Thomas-Fermi model for cold bosons
• Problems without provided solutions
  • The quantum dot
  • Dimensional analysis of an atom beyond the Thomas-Fermi model

The following sections are included:

• Solved problems
  • Solitons of the Korteweg-de Vries equation
  • Breathers of the nonlinear Schrödinger equation
  • Healing length
  • Dimensional analysis of the projectile problem as a prelude to a discussion on the Kadomtsev-Petviashvili solitons
  • Kadomtsev-Petviashvili equation
  • The nonlinear transport equation
  • Burgers equation
• Problems without provided solutions
  • Stationary solitons of the Burgers equation
  • Stationary solitons of the nonlinear Schrödinger equation
  • Solitons of the sine-Gordon equation

The following sections are included:

• Solved problems
  • Quantum-mechanical supersymmetry (QM-SUSY): Pöschl-Teller as an example
  • What the supersymmetric structure alone implies for the scattering states of the Pöschl-Teller potential.
  • Power-index method. Example of the nonlinear Schrödinger equation, with the Pöschl-Teller problem as a byproduct
  • A stationary-kink solution of the Burgers equation through the power-index method
  • Scale invariance: quantum Calogero potential as an example
  • Classical Calogero potential: a posteriori manifestations of the scale invariance
  • Classical Calogero potential: a priori manifestations of scale invariance at the Maupertuis-Jacobi level. Finding the zero-energy orbit from symmetries alone
  • Circle inversion, quantum: zero-energy eigenstates in a 1/r⁴ potential
  • Self-similar tilings: moment of inertia of an equilateral triangle as a paradigm
  • Properties of the spectrum of an equilateral triangular quantum billiards that follow from the tiling self-similarity
  • Center of mass of a filled “golden b” shape, through self-similar tilings
• Problems without provided solutions
  • Power-index method applied to the Pöschl-Teller problem directly
  • Quasi-integrability: the overall idea and the example of the sixtic oscillator
  • A quasi-integrable instance of two nonlinearly coupled oscillators
  • Moment of inertia of a filled “golden b” shape, through the self-similar tilings
• Background
  • Quasi-integrability: the overall idea and the example of the sixtic oscillator
  • Relationship between various methods
  • Further reading: Bethe ansatz

The following sections are included:

• Further Reading
• General Index
• Author Index