[3rd Kroll Lectures]
Special Topics in Theoretical Physics:
Exactly Solvable Quantum Mechanics
【第三屆克洛爾講座】理論物理特論：可精確解之量子力學


由Prof. Ryu Sasaki (Yukawa Institute, Kyoto University, Japan)、黃偉彥開授


1.Introduction:
Birth: Heisenberg, Born, Jordan
Schrodinger, Dirac.
Overview of the 3rd Kroll Lectures
2.General Formulation
Factorized Hamiltonian hermiticity & orthogonality, Crum’s theorem, Krein-Adler deformation Darboux transformation, Absaham-Moses tr.
3.Shape invariance, sufficient condition for exact solvability;
*harmonic oscillator
*radial oscillator
*Poschl-Teller
Exact eigenvalues and eigenfunctions with the classical orthogonal polynomials Hermite, Laguerre, and Jacob polynomiuls
4.Heisenberg operator solutions creation and annihilation operators for the three potentials
*harmonic oscillator
*radial oscillator
*Poschl-Teller
Coherent states and eigenvectors of the annihilation operator
Scattering Problems
Morse, Eckart and other potentials having finitely many discrete eigenstates reflection and transmission amplitudes reflectionless potentials and KdV solitons.
Deformations:
From exactly solvable models to infinitely many exactly solvable models
Seed solutions obtained by discrete symmetry of the system
Deformations of harmonic oscillator
by using the eigenfunctions (Krein-Adler)
by using the pseudo-virtual state wavefunctions
their duality and identities
Deformations of radial oscillator by using the virtual state wavefunctions
exceptional and multi-indexed orthogonal polynonrials
by eigenfunctions, pseudo-virtual
Deformations of Poschl-Teller
5.Outlooks: discrete quantum mechanics