1,347 research outputs found
Equivalent power law potentials
It is shown that the radial Schroedinger equation for a power law potential
and a particular angular momentum may be transformed using a change of variable
into another Schroedinger equation for a different power law potential and a
different angular momentum. It is shown that this leads to a mapping of the
spectra of the two related power law potentials. It is shown that a similar
correspondence between the classical orbits in the two related power law
potentials exists. The well known correspondence of the Coulomb and oscillator
spectra is a special case of a more general correspondence between power law
potentials.Comment: 10 pages. Typographical mistakes in the earlier version are correcte
Phase equivalent potentials, Complex coordinates and Supersymmetric Quantum Mechanics
Supersymmetric Quantum Mechanics may be used to construct reflectionless
potentials and phase-equivalent potentials. The exactly solvable case of the
potential is used to show that for certain values of the
strength the phase-equivalent singular potential arising from the
elimination of all the boundstates is identical to the original potential
evaluated at a point shifted in the complex cordinate space. This equivalence
has the consequence that certain general relations valid for reflectionless
potentials and phase-equivalent potentials lead to hitherto unknown identities
satisfied by the Associated Legendre functions. This exactly solvable probelm
is used to demonstrate some aspects of scattering theory.Comment: 11page
Sum rules and the domain after the last node of an eigenstate
It is shown that it is possible to establish sum rules that must be satisfied
at the nodes and extrema of the eigenstates of confining potentials which are
functions of a single variable. At any boundstate energy the Schroedinger
equation has two linearly independent solutions one of which is normalisable
while the other is not. In the domain after the last node of a boundstate
eigenfunction the unnormalisable linearly independent solution has a simple
form which enables the construction of functions analogous to Green's functions
that lead to certain sum rules. One set of sum rules give conditions that must
be satisfied at the nodes and extrema of the boundstate eigenfunctions of
confining potentials. Another sum rule establishes a relation between an
integral involving an eigenfunction in the domain after the last node and a sum
involving all the eigenvalues and eigenstates. Such sum rules may be useful in
the study of properties of confining potentials. The exactly solvable cases of
the particle in a box and the simple harmonic oscillator are used to illustrate
the procedure. The relations between one of the sum rules and two-particle
densities and a construction based on Supersymmetric Quantum Mechanics are
discussed.Comment: 17 page
SUSY transformation of the Green function and a trace formula
An integral relation is established between the Green functions corresponding
to two Hamiltonians which are supersymmetric (SUSY) partners and in general may
possess both discrete and continuous spectra. It is shown that when the
continuous spectrum is present the trace of the difference of the Green
functions for SUSY partners is a finite quantity which may or may not be equal
to zero despite the divergence of the traces of each Green function. Our
findings are illustrated by using the free particle example considered both on
the whole real line and on a half line
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