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Semi-unitary tran sformations for supe-s ymmetric harm onic oscillator

Manu Pratap Singh

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Constr ucting the operators connecting the states of same energy for superpartner Hamilt onians and superpartner potent ials for a li near harmonic osci lla tor, a semi -unit ary transformation has been obtained and it has been shown to destroy the ground state of a supersymmetric partner Hamilt onian with its eigen functions constitu ting a compl ete orthonormal set in a subspace of the full Hilh ert space.

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In recent years, some interes t has been raised by the problem of the quan tum harmonic-osci l la tor wi th a coupling cons tant K(t), which is a funct ion of t ime (1.2). The impor tance of this p roblem has been stressed by the analysis of the mot ion of a charged part ic le in the presence of a magnet ic field H~(t), which has a fixed direct ion bu t changes i ts in tens i ty wi th t ime (1). The reason of this note is tha t , among these recent studies, the most na tura l approach is surprisingly absent. We will describe i t here, and we will analyse some easily solvable potent ials . W e say t h a t i t is a ve ry na tura l approach, because i t consists in a ve ry na ive general ization of Dirae 's method. I n order to in t roduce it, we remark tha t the Dirae creat ion opera tor (in the t imeindependen t case) could be considered as the opera tor t ha t gives a solution once applied to another solution (for ins tance the vacuum). So we in t roduce an opera tor A ( x , ~ /~x , t) with the p rope r ty tha t , once appl ied to a solut ion of the SehrSdinger equat ion , i t gives another solution. The expl ici t form of this opera tor A is obta ined by means of a simple differential equat ion wi th one and only one solution. W e then find an explici t solution of the t ime-dependen t SchrSdinger equat ion , as a simple general izat ion of the vacuum wave function. App ly ing i tc ra t ive ly the A opera to r we obta in an explici t form for a set of solutions. Noth ing can be said, however , abou t the general case.

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Constr ucting the operators connecting the states of same energy for superpartner Hamilt onians and superpartner potent ials for a li near harmonic osci lla tor, a semi -unit ary transformation has been obtained and it has been shown to destroy the ground state of a supersymmetric partner Hamilt onian with its eigen functions constitu ting a compl ete orthonormal set in a subspace of the full Hilh ert space.
It is shown that in supersymmetric quantum mechanics the superpartner Hamiltonian H+{\mathit{H}}_{+}H+​ is related to the Hamiltonian ${\mathit{H}}_{\mathrm{\ensuremath{-}}}$ by a semiunitary transformation. The properties of the semiunitary transformation are discussed. \textcopyright{} 1996 The American Physical Society.
In recent years, some interes t has been raised by the problem of the quan tum harmonic-osci l la tor wi th a coupling cons tant K(t), which is a funct ion of t ime (1.2). The impor tance of this p roblem has been stressed by the analysis of the mot ion of a charged part ic le in the presence of a magnet ic field H~(t), which has a fixed direct ion bu t changes i ts in tens i ty wi th t ime (1). The reason of this note is tha t , among these recent studies, the most na tura l approach is surprisingly absent. We will describe i t here, and we will analyse some easily solvable potent ials . W e say t h a t i t is a ve ry na tura l approach, because i t consists in a ve ry na ive general ization of Dirae 's method. I n order to in t roduce it, we remark tha t the Dirae creat ion opera tor (in the t imeindependen t case) could be considered as the opera tor t ha t gives a solution once applied to another solution (for ins tance the vacuum). So we in t roduce an opera tor A ( x , ~ /~x , t) with the p rope r ty tha t , once appl ied to a solut ion of the SehrSdinger equat ion , i t gives another solution. The expl ici t form of this opera tor A is obta ined by means of a simple differential equat ion wi th one and only one solution. W e then find an explici t solution of the t ime-dependen t SchrSdinger equat ion , as a simple general izat ion of the vacuum wave function. App ly ing i tc ra t ive ly the A opera to r we obta in an explici t form for a set of solutions. Noth ing can be said, however , abou t the general case.