Supersymmetry in quantum mechanics

An elementary introduction is given to the subject of supersymmetry in quantum mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states,...

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Bibliographic Details
Published inAIP conference proceedings Vol. 744; no. 1; pp. 133 - 165
Main Author Khare, Avinash
Format Journal Article
LanguageEnglish
Published United States 2005
Subjects
BOUND STATE
EIGENFUNCTIONS
EIGENVALUES
EXACT SOLUTIONS
GROUND STATES
HAMILTONIANS
HARMONIC OSCILLATORS
ONE-DIMENSIONAL CALCULATIONS
PERIODICITY
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
POTENTIALS
QUANTUM MECHANICS
QUANTUM NUMBERS
S MATRIX
SCATTERING
SUPERSYMMETRY
WKB APPROXIMATION
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Summary:An elementary introduction is given to the subject of supersymmetry in quantum mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct n new exactly solvable Hamiltonians having n - 1, n - 2,, 0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape-invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape-invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.
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ISBN:0735402280
9780735402287
ISSN:0094-243X
1551-7616
DOI:10.1063/1.1853201