We present some essential results for the Hamiltonian of a particle in a box. We discuss the invariance of this operator under time-reversal T, the possibility of choosing real eigenfunctions for it and the degeneracy of its energy eigenvalues. Once these results have been presented, we introduce the usual nondegeneracy theorem and discuss some issues surrounding it. We find that the nondegeneracy theorem is true if the boundary conditions are T-invariant but "confining" (i.e., the particle is in a real impenetrable box). If the boundary conditions are not T-invariant (belonging to a family of so-called "not confining" boundary conditions), the respective eigenfunctions are strictly complex and there is no degeneracy. Consistently, we verify the validity of the theorem also in this case. Finally, if the boundary conditions are also T-invariant, but "not confining", then we can have degeneracy in the energy levels only if the respective eigenfunctions can be specifically written as complex. We find that the nondegeneracy theorem fails in these cases. If the respective eigenfunctions can be written as only real, then we do not have degeneracy and the nondegeneracy theorem is true.
Some physical restrictions and their influences on the boundary conditions for a particle in a one-dimensional box are pointed out. We discuss the invariance of the corresponding Hamiltonian under time-reversal T ̂ and its relation with the de-generacy of the energy eigenvalues. We also discuss the effect on the boundary conditions of requiring the invariance of the Hamiltonian under parity P ̂ and si-multaneous space and time reflection P ̂ T ̂. A condition, which depends only on the boundary values of an eigenfunction of the Hamiltonian and its derivative, deter-mines if the corresponding eigenvalue satisfies the inequality E> V (x) = const, V (x) being the potential inside the box. Once these results have been presented, we choose various representative analytically solvable examples of boundary conditions (Hamiltonians) and supersymmetrize the corresponding problems. We find real and complex valued supersymmetric partner potentials among all these systems.
In
PT-symmetric quantum mechanics a fundamental principle of quantum mechanics, that the Hamiltonian must be Hermitian, is replaced by another set of requirements, including notably symmetry under
PT, where
P denotes parity and
T denotes time reversal. Here we study the role of boundary conditions in
PT-symmetric quantum mechanics by constructing a simple model that is the
PT-symmetric analog of a particle in a box. The model has the usual particle-in-a-box Hamiltonian but boundary conditions that respect
PT symmetry rather than Hermiticity. We find that for a broad class of
PT-symmetric boundary conditions the model respects the condition of unbroken
PT symmetry, namely, that the Hamiltonian and the symmetry operator
PT have simultaneous eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the
PT-symmetric inner product. Thus we obtain a simple soluble model that fulfills all the requirements of
PT-symmetric quantum mechanics. In the second part of this paper we formulate a variational principle for
PT-symmetric quantum mechanics that is the analog of the textbook Rayleigh-Ritz principle. Finally we consider electromagnetic analogs of the
PT-symmetric particle in a box. We show that the isolated particle in a box may be realized as a Fabry-Perot cavity between an absorbing medium and its conjugate gain medium. Coupling the cavity to an external continuum of incoming and outgoing states turns the energy levels of the box into sharp resonances. Remarkably we find that the resonances have a Breit-Wigner line shape in transmission and a Fano line shape in reflection; by contrast, in the corresponding Hermitian case the line shapes always have a Breit-Wigner form in both transmission and reflection.
Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.
In this paper we show that the point-group (geometrical) symmetry is insufficient to account for the degeneracy of the energy levels of the particle in a cubic box. The discrepancy is due to hidden (dynamical symmetry). We obtain the operators that commute with the Hamiltonian one and connect eigenfunctions of different symmetries. We also show that the addition of a suitable potential inside the box breaks the dynamical symmetry but preserves the geometrical one.The resulting degeneracy is that predicted by point-group symmetry.
In this chapter we discuss the properties of the time-reversal operator (introduced into quantum mechanics by Wigner in 1932) for particles without spin, as well as taking into account the spin. There are given Wigner criteria ( a ), ( b ), and ( c ) concerning the absence or presence of an additional degeneracy of energy levels due to the symmetry with respect to time reversal, both in the case when the spin of particles that form the quantum system is taken into account and in the case it is not. Wigner corepresentations of symmetry groups associated with the presence of time-reversal symmetry are also discussed.