Supersymmetric quantum mechanics of one-dimensional systems

Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % pepucqWFtepvaaa!46A4! $$ \mathcal{P}\mathcal{

“…lower) signs correspond to V A,B+1 (x) (resp. V A,B−1 (x)) in (8). We note that for the former choice, φ(x) decreases exponentially for x → ∞, but increases as x −(B−A) for x → 0, while for the latter choice it vanishes as x B−A−1 for x → 0, but increases exponentially for x → ∞, thus showing in both cases that neither φ nor φ −1 is normalizable, as is required by the general theory [8].…”

“…SUSY systems are appropriately described by a superpotential descendant of the given Schrödinger potential, which in turn furnishes the so-called SUSY partner Hamiltonians. In one-dimensional systems, strict isospectrality may be present in the first-order systems (broken SUSY) when the factorization energy E is smaller than the ground-state energy of the starting Hamiltonian [7,8]. In such a context, the potential and eigenfunctions of the partner Hamiltonian are known in terms of analytic expressions, which, however, may be a good deal more complicated than the corresponding ones of the initial Hamiltonian [9].…”

Oscilador Harmônico Quântico 1D Confinado e sua Energia Variacional obtida com Auxílio da Supersimetria

“…O processo descrito acima para a solução do estado fundamental da equação de Schrödinger pode ser estendido para estados excitados através da hierarquia de Hamiltonianos, [2,4,5,12,13,20,21]. Entretanto, para os propósitos deste trabalho basta a descrição do tratamento para o estado fundamental.…”

“…O formalismo resultante da Mecânica Quântica Supersimétrica, MQS, [4,5,12,13], tem sido usado com muito sucesso na análise de problemas quânticos; vale destacar o tratamento de potenciais exatamente, [2,20,21], e parcialmente solúveis, [6,7,16]. Usando este formalismoé possível encontrar um potencial efetivo e, a partir dele, a função de onda teste a ser usada no método variacional.…”

Supersymmetry and the Hartmann potential of theoretical chemistry

“…Since V 1 (x) in equation 11 is chosen such that its ground state wave function ψ 0 (1) has eigenvalue equal to zero, equation 10 gives…”