Particle in a box in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric quantum mechanics and an electromagnetic analog

In $\mathcal{PT}$-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 $\mathcal{PT}$, where $\mathcal{P}$ denotes parity and $\mathcal{T}$ denotes time reversal. Here we study the role of boundary conditions in $\mathcal{PT}$-symmetric quantum mechanics by constructing a simple model that is the $\mathcal{PT}$-symmetric analog of a particle in a box. The model has the usual particle-in-a-box Hamiltonian but boundary conditions that respect $\mathcal{PT}$ symmetry rather than Hermiticity. We find that for a broad class of $\mathcal{PT}$-symmetric boundary conditions the model respects the condition of unbroken $\mathcal{PT}$ symmetry, namely, that the Hamiltonian and the symmetry operator $\mathcal{PT}$ have simultaneous eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the $\mathcal{PT}$-symmetric inner product. Thus we obtain a simple soluble model that fulfills all the requirements of $\mathcal{PT}$-symmetric quantum mechanics. In the second part of this paper we formulate a variational principle for $\mathcal{PT}$-symmetric quantum mechanics that is the analog of the textbook Rayleigh-Ritz principle. Finally we consider electromagnetic analogs of the $\mathcal{PT}$-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.

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