Supersymmetry and quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multisoliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

[1]  Shastry.  Taking the square root of the discrete 1/r2 model. , 1992, Physical review letters.

[2]  William H. Miller,et al.  Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants , 1974 .

[3]  F. Berezin,et al.  Method of Second Quantization , 1966 .

[4]  L. Lathouwers.  The Hamiltonian H = (−1/2) d2/dx2 + x2/2 + λ/x2 reobserved , 1975 .

[5]  A. Khare,et al.  Supersymmetry, Shape Invariance and Exactly Solvable Potentials , 1988 .

[6]  Khare,et al.  Semiclassical approach to quantum-mechanical problems with broken supersymmetry. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[7]  B. Sutherland.  Quantum many body problem in one-dimension: Ground state , 1971 .

[8]  A. Khare,et al.  Supersymmetry and the Dirac equation , 1988 .

[9]  U. Sukhatme,et al.  Bound States of the Yukawa Potential via the Shifted 1/$N$ Expansion Technique , 1984 .

[10]  Khare,et al.  New exactly solvable Hamiltonians: Shape invariance and self-similarity. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[11]  G. Stanciu.  Further Exact Solutions of the Dirac Equation , 1967 .

[12]  J. B. Krieger,et al.  Use of the WKB Method for Obtaining Energy Eigenvalues , 1967 .

[13]  Laurence G. Yaffe,et al.  Large N limits as classical mechanics , 1982 .

[14]  A. Vainshtein,et al.  On the Hilbert space of supersymmetric quantum systems , 1988 .

[15]  Khare,et al.  Countable infinity of isospectral potential families. , 1989, Physical review. A, General physics.

[16]  W. Miller.  Semiclassical limit of quantum mechanical transition state theory for nonseparable systems , 1975 .

[17]  M. Gell-Mann,et al.  Theory of the Fermi Interaction , 1958 .

[18]  Edward Witten,et al.  Dynamical Breaking of Supersymmetry , 1981 .

[19]  J. B. Krieger,et al.  Application of a Higher-Order WKB Approximation to Radial Problems , 1967 .

[20]  L. Vinet,et al.  Parasupersymmetry and truncated supersymmetry in quantum mechanics , 1991 .

[21]  A. Khare,et al.  A quasi-exactly solvable problem without Si(2) symmetry , 1989 .

[22]  Y. Alhassid,et al.  Potential scattering, transfer matrix, and group theory , 1983 .

[23]  F. Calogero.  Solution of a three-body problem in one-dimension , 1969 .

[24]  O. W. Greenberg,et al.  Selection rules for parafields and the absence of para particles in nature , 1965 .

[25]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[26]  E. Berdermann,et al.  On the momentum correlation of (e+e−) pairs observed in U + U and U + Pb collisions , 1989 .

[27]  A. Wipf,et al.  The U(1)-anomaly, the non-compact index theorem, and the (supersymmetric) BA-effect , 1986 .

[28]  Haldane,et al.  Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-(1/2 antiferromagnetic Heisenberg chain with 1/r2 exchange. , 1988, Physical review letters.

[29]  J. Schrieffer,et al.  Solitons with Fermion Number 1/2 in Condensed Matter and Relativistic Field Theories , 1981 .

[30]  F. L. Yost,et al.  Coulomb wave‐functions , 1935 .

[31]  Zhao,et al.  Relativistic extension of the Kay-Moses method for constructing transparent potentials in quantum mechanics. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[32]  Bernd Thaller,et al.  The Dirac Equation , 1992 .

[33]  A. Rau,et al.  Supersymmetry in Quantum Mechanics , 1986 .

[34]  J. Ginocchio.  A class of exactly solvable potentials II. The three-dimensional Schrödinger equation , 1985 .

[35]  Y. Golfand,et al.  EXTENSION OF THE ALGEBRA OF POINCARE GROUP GENERATORS AND VIOLATION OF P INVARIANCE. , 1971 .

[36]  D. Freedman,et al.  An Exactly Solvable $N$ Particle System in Supersymmetric Quantum Mechanics , 1990 .

[37]  Amado Rd.  Phase-equivalent supersymmetric quantum-mechanical partners of the Coulomb potential. , 1988 .

[38]  I. M. Singer,et al.  Recent Developments in Gauge Theories , 1980 .

[39]  N. Debergh,et al.  ON PARASUPERSYMMETRIC COHERENT STATES , 1989 .

[40]  M. Shifman.  NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM) , 1989 .

[41]  A. Comtet,et al.  Supersymmetric WKB approach to scattering problems , 1991 .

[42]  A. Khare.  How good is the supersymmetry-inspired WKB quantization condition? , 1985 .

[43]  A. Andrianov,et al.  From supersymmetric quantum mechanics to a parasupersymmetric one , 1991 .

[44]  A. Khare,et al.  Supersymmetry in quantum mechanics , 1985, math-ph/0409003.

[45]  A. Niemi,et al.  Fractionization of the witten index , 1984 .

[46]  R. H. Good,et al.  A WKB-Type Approximation to the Schrödinger Equation , 1953 .

[47]  R. E. Raab,et al.  Operator Methods in Quantum Mechanics , 1992 .

[48]  C. Itzykson,et al.  DENSITY OF STATES IN THE PRESENCE OF A STRONG MAGNETIC FIELD AND RANDOM IMPURITIES , 1984 .

[49]  A. Khare,et al.  Solvable quantum mechanical examples of broken supersymmetry , 1993 .

[50]  Bethge,et al.  Observation of correlated narrow-peak structures in positron and electron spectra from superheavy collision systems. , 1986, Physical review letters.

[51]  G. Stedman,et al.  Simple supersymmetry: I. Basic examples , 1985 .

[52]  A. Neveu,et al.  FACTORIZABLE DUAL MODEL OF PIONS. , 1971 .

[53]  G. Ghosh,et al.  Rotating harmonic oscillator: Violation of an equipartition theorem , 1984 .

[54]  L. Vinet,et al.  Conformal parasuperalgebras and their realizations on the line , 1989 .

[55]  Polychronakos.  Exchange operator formalism for integrable systems of particles. , 1992, Physical review letters.

[56]  S. Rajeev.  Quantum electrodynamics as a supersymmetric theory of loops , 1987 .

[57]  A. Khare,et al.  Orthosupersymmetric quantum mechanics , 1993 .

[58]  L. Simmons,et al.  Limiting spectra from confining potentials , 1979 .

[59]  G. Schön,et al.  Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions , 1990 .

[60]  N. Debergh.  Coherent states and parasupersymmetric quantum mechanics , 1989 .

[61]  H. Backe,et al.  Correlated e+ e- peaks observed in heavy ion collisions , 1990 .

[62]  L. Gendenshtein.  Derivation of Exact Spectra of the Schrodinger Equation by Means of Supersymmetry , 1984 .

[63]  A. Khare,et al.  Supersymmetry-inspired WKB approximation in quantum mechanics , 1991 .

[64]  Spiridonov.  Exactly solvable potentials and quantum algebras. , 1992, Physical review letters.

[65]  Baye.  Supersymmetry between deep and shallow nucleus-nucleus potentials. , 1987, Physical review letters.

[66]  N. Meyer‐Vernet.  Strange bound states in the Schrödinger wave equation: When usual tunneling does not occur , 1982 .

[67]  Ian Affleck,et al.  Quantum Statistical Metastability , 1981 .

[68]  Montemayor,et al.  Supersymmetry shape invariance and solubility in quantum mechanics. , 1989, Physical review. A, General physics.

[69]  A. Khare,et al.  Supersymmetry and nearest neighbour x-y model with random transverse magnetic field , 1984 .

[70]  Panigrahi,et al.  Supersymmetry and tunneling in an asymmetric double well. , 1993, Physical Review A. Atomic, Molecular, and Optical Physics.

[71]  Sukhatme,et al.  Supersymmetric quantum mechanics and large-N expansions. , 1985, Physical review letters.

[72]  Y. Alhassid,et al.  Group theory approach to scattering , 1983 .

[73]  H. S. Green.  A Generalized Method of Field Quantization , 1953 .

[74]  Thomas,et al.  Tunneling exchange, supersymmetry, and Riccati equations. , 1986, Physical review letters.

[75]  K. Efetov.  Supersymmetry and theory of disordered metals , 1983 .

[76]  J. W. Holten,et al.  Fermionic coordinates and supersymmetry in quantum mechanics , 1982 .

[77]  Amado,et al.  Supersymmetric quantum mechanics and superdeformed nuclei. , 1991, Physical review letters.

[78]  S. Chern,et al.  Lie groups and KdV equations , 1979 .

[79]  A. Leggett,et al.  Quantum tunnelling in a dissipative system , 1983 .

[80]  A. Andrianov,et al.  Nonstationary approach scattering theory for supersymmetric Hamiltonians in quantum mechanics and supersymmetry of nuclear interactions , 1986 .

[81]  L. Vinet,et al.  Higher order parasupersymmetric quantum mechanics , 1991 .

[82]  Julian Schwinger,et al.  On gauge invariance and vacuum polarization , 1951 .

[83]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[84]  R. E. PEIERLS,et al.  Lectures on Theoretical Physics , 1951, Nature.

[85]  A. Andrianov,et al.  Supersymmetric origin of equivalent quantum systems , 1985 .

[86]  M. Nieto.  HYDROGEN ATOM AND RELATIVISTIC PI MESIC ATOM IN N SPACE DIMENSIONS , 1979 .

[87]  N. Fröman,et al.  JWKB approximation : contributions to the theory , 1965 .

[88]  J. Ginocchio.  A class of exactly solvable potentials. I. One-dimensional Schrödinger equation☆ , 1984 .

[89]  C. Pajares,et al.  On the large-N limits of relativistic equations , 1984 .

[90]  F. B. Hildebrand.  Advanced Calculus for Applications , 1962 .

[91]  A. Shabat.  The infinite-dimensional dressing dynamical system , 1992 .

[92]  Edward Witten,et al.  Constraints on Supersymmetry Breaking , 1982 .

[93]  B. Zumino,et al.  Supergauge Transformations in Four-Dimensions , 1974 .

[94]  D. Volkov,et al.  Is the Neutrino a Goldstone Particle , 1973 .

[95]  A. Khare.  Parasupersymmetry in quantum mechanics , 1993 .

[96]  J. Wolfes.  On the three‐body linear problem with three‐body interaction , 1974 .

[97]  S. Coleman,et al.  Aspects of Symmetry , 1985 .

[98]  V. Rittenberg,et al.  Supersymmetric Quantum Mechanics , 1983 .

[99]  S. Kamefuchi,et al.  Quantum field theory and parastatistics , 1982 .

[100]  Sodano,et al.  Supersymmetry and bistable soft potentials. , 1988, Physical review letters.

[101]  Anthony J Leggett,et al.  Influence of Dissipation on Quantum Tunneling in Macroscopic Systems , 1981 .

[102]  Wiktor Eckhaus.  The Inverse Scattering Transformation and the Theory of Solitons: An Introduction , 1981 .

[103]  Lowell S. Brown,et al.  Supersymmetry and the Bistable Fokker-Planck equation , 1984 .

[104]  P. Vecchia,et al.  Supersymmetry and instantons , 1978 .

[105]  Michael Atiyah,et al.  The index of elliptic operators on compact manifolds , 1963 .

[106]  P. K. Roy,et al.  Supersymmetry in Quantum Mechanics , 1990 .

[107]  G. Rajasekaran,et al.  Algebra for fermions with a new exclusion principle , 1991 .

[108]  L. Mlodinow,et al.  Pseudo-spin structure and large N expansion for a class of generalized helium Hamiltonians , 1981 .

[109]  E. Witten,et al.  Supersymmetry algebras that include topological charges , 1978 .

[110]  D. Lancaster.  Supersymmetry breakdown in supersymmetric quantum mechanics , 1984 .

[111]  Greenberg.  Example of infinite statistics. , 1990, Physical review letters.

[112]  Bound states in the continuum from supersymmetric quantum mechanics. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[113]  M. Nieto.  Relationship Between Supersymmetry and the Inverse Method in Quantum Mechanics , 1984 .

[114]  Shastry,et al.  Super Lax pairs and infinite symmetries in the 1/r2 system. , 1992, Physical review letters.

[115]  V. Spiridonov.  DEFORMED CONFORMAL AND SUPERSYMMETRIC QUANTUM MECHANICS , 1992 .

[116]  A. Comtet,et al.  EXACTNESS OF SEMICLASSICAL BOUND STATE ENERGIES FOR SUPERSYMMETRIC QUANTUM MECHANICS , 1985 .

[117]  Nogami,et al.  Supersymmetry aspects of the Dirac equation in one dimension with a Lorentz scalar potential. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[118]  和達 三樹,et al.  F. Calogero and A. Degasperis: Spectral Transform and Solitons; Tools to Solve and Investigate Nonlinear Evolution Equations, Vol. 1, North-Holland, Amsterdam and New York, 1982, xvi+516ページ, 23×16cm, 33,600円 (Studies in Mathematics and Its Applications, Vol. 13). , 1983 .

[119]  A. Khare,et al.  Exactness of supersymmetric WKB spectra for shape-invariant potentials , 1986 .

[120]  Shang‐keng Ma,et al.  Critical dynamics of ferromagnets in 6 − ε dimensions: General discussion and detailed calculation , 1975 .

[121]  J. Vary,et al.  Electron-positron scattering resonances from relativistic two-body wave equations , 1991 .

[122]  A. Andrianov,et al.  The factorization method and quantum systems with equivalent energy spectra , 1984 .

[123]  Richard J. Hughes,et al.  Supersymmetry in a first-order Dirac equation for a Landau-Hall system , 1986 .

[124]  A. Perelomov,et al.  Classical integrable finite-dimensional systems related to Lie algebras , 1981 .

[125]  W. Keung,et al.  SOLITONS FROM SUPERSYMMETRY , 1990 .

[126]  Nieto,et al.  Evidence from alkali-metal-atom transition probabilities for a phenomenological atomic supersymmetry. , 1985, Physical review. A, General physics.

[127]  P. Panigrahi,et al.  Inter-Relations of Solvable Potentials , 1993, hep-th/9310172.

[128]  M. Nieto,et al.  SUSY-WKB is neither exact nor never worse than WKB for all solvable potentials , 1990 .

[129]  B. Zumino,et al.  Supergauge Invariant Extension of Quantum Electrodynamics , 1974 .

[130]  Friedrich,et al.  Interfering resonances and bound states in the continuum. , 1985, Physical review. A, General physics.

[131]  G. Natanzon.  General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functions , 1979 .

[132]  L. Ballentine,et al.  Quantum mechanics , 1989 .

[133]  G. Stedman.  Simple supersymmetry: II. Factorisation method in quantum mechanics , 1985 .

[134]  T. E. Hull,et al.  The factorization method , 1951 .

[135]  A. Bray,et al.  Influence of Dissipation on Quantum Coherence , 1982 .

[136]  L. Vinet,et al.  Supersymmetry of the Pauli equation in the presence of a magnetic monopole , 1984 .

[137]  A. Turbiner.  Quasi-exactly-solvable problems andsl(2) algebra , 1988 .

[138]  A. Balantekin.  Accidental Degeneracies and Supersymmetric Quantum Mechanics , 1985 .

[139]  L. Girardello,et al.  On Constant Configurations and the Evaluation of the Witten Index , 1983 .

[140]  S. Coleman,et al.  All Possible Symmetries of the S Matrix , 1967 .

[141]  R. Willey,et al.  Quantum mechanical metastability: When and why? , 1992 .

[142]  Francesco Calogero,et al.  Solution of the One‐Dimensional N‐Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials , 1971 .

[143]  L. Mlodinow,et al.  Solving the Schrödinger equation with use of 1/N perturbation theory , 1984 .

[144]  A. Khare,et al.  Supersymmetric quantum mechanics in one, two and three dimensions , 1984 .

[145]  C. Sukumar.  Supersymmetry, potentials with bound states at arbitrary energies and multi-soliton configurations , 1986 .

[146]  G. Parisi,et al.  Supersymmetric field theories and stochastic differential equations , 1982 .

[147]  Raymond J. Seeger,et al.  Lectures in Theoretical Physics , 1962 .

[148]  L. Girardello,et al.  Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry , 1983 .

[149]  J. Rosner,et al.  Supersymmetric Quantum Mechanics and Inverse Scattering , 1986 .

[150]  Y. Murayama.  Supersymmetric Wentzel-Kramers-Brilluoin (SWKB) method , 1989 .

[151]  W. Reid,et al.  Riccati Differential Equations , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[152]  A. Khare,et al.  Parasupersymmetric Quantum Mechanics and its Infinite Order Limit , 1993 .

[153]  K. Laidler,et al.  The theory of rate processes : the kinetics of chemical reactions, viscosity, diffusion and electrochemical phenomena , 1941 .

[154]  G. Lamb.  Elements of soliton theory , 1980 .

[155]  A. Khare,et al.  How good is the supersymmetry-inspired higher-order WKB approximation? , 1987 .

[156]  S. Chakravarty.  Quantum Fluctuations in the Tunneling between Superconductors , 1982 .

[157]  Shastry,et al.  Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions. , 1988, Physical review letters.

[158]  N. Kampen,et al.  A soluble model for diffusion in a bistable potential , 1977 .

[159]  Nieto,et al.  Analytical wave functions for atomic quantum-defect theory. , 1985, Physical review. A, General physics.

[160]  J. Ader.  Moment method and the Schrödinger equation in the large N limit , 1983 .

[161]  K. Raghunathan,et al.  On the exactness of the SUSY semiclassical quantization rule , 1987 .

[162]  Supersymmetry and Kramers' reaction rate , 1990 .

[163]  Sidney D. Drell,et al.  Relativistic Quantum Mechanics , 1965 .

[164]  J. Casahorrán,et al.  Singular Superpotentials and Explicit Breaking of Supersymmetry , 1991 .

[165]  S. Berman,et al.  Nuovo Cimento , 1983 .

[166]  A. Turbiner,et al.  Spectral singularities and quasi-exactly solvable quantal problem , 1987 .

[167]  Frank H. Stillinger,et al.  Bound states in the continuum , 1975 .

[168]  A. Khare,et al.  IS SHAPE INVARIANCE ALSO NECESSARY FOR LOWEST ORDER SUPERSYMMETRIC WKB TO BE EXACT , 1989 .

[169]  J. Faist,et al.  Observation of an electronic bound state above a potential well , 1992, Nature.

[170]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[171]  A. Leggett,et al.  Dynamics of the two-state system with ohmic dissipation , 1984 .

[172]  Y. P. Varshni,et al.  Energies of s Eigenstates in a Static Screened Coulomb Potential , 1971 .

[173]  V. Kostelecký,et al.  Evidence for a Phenomenological Supersymmetry in Atomic Physics , 1984 .

[174]  P. Panigrahi,et al.  Singular superpotentials in supersymmetric quantum mechanics , 1993 .

[175]  Fred Cooper,et al.  Aspects of supersymmetric quantum mechanics , 1983 .

[176]  F. Iachello.  DYNAMICAL SUPERSYMMETRIES IN NUCLEI , 1980 .

[177]  P. Sabatier,et al.  Inverse Problems in Quantum Scattering Theory , 1977 .

[178]  L. Mlodinow,et al.  SO(2,1) Algebra and the Large $N$ Expansion in Quantum Mechanics , 1980 .

[179]  A. Perelomov,et al.  Quantum Integrable Systems Related to Lie Algebras , 1983 .

[180]  A. Wipf,et al.  Derivation of the S-matrix using supersymmetry , 1988 .

[181]  C. Bender,et al.  Semiclassical perturbation theory for the hydrogen atom in a uniform magnetic field , 1982 .

[182]  Kovács,et al.  Supersymmetry and double-well potentials. , 1988, Physical review letters.