The operator in -symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, , , and . These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate . The usual approach has been to express the Hamiltonian as H = H 0 + H 1 , and to seek a solution for in the form , where Q = Q ( q , p ) is odd in the momentum p , even in the coordinate q , and has a perturbation expansion of the form Q = Q 1 + 3 Q 3 + 5 Q 5 + ⋅⋅⋅. (In previous work it has always been assumed that the coefficients of even powers of in this expansion would be absent because their presence would violate the condition that Q ( p , q ) is odd in p .) In an earlier paper it was argued that the operator is not unique because the perturbation coefficient Q 1 is nonunique. Here, the nonuniqueness of is demonstrated at a more fundamental level: it is shown that the perturbation expansion for Q actually has the more general form Q = Q 0 + Q 1 + 2 Q 2 + ⋅⋅⋅ in which all powers and not just odd powers of appear. For the case in which H 0 is the harmonic-oscillator Hamiltonian, Q 0 is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q .
It is necessary to calculate the
C operator for the non-Hermitian
PT-symmetric Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$ in order to demonstrate that
H defines a consistent unitary theory of quantum mechanics. However, the
C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}+ig{x}^{3}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$, whose
C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for
C having a nontrivial dependence on the dynamical variables
x and
p and on the parameter $\ensuremath{\lambda}$.
In quantum mechanics the time operator Θ satisfies the commutation relation [Θ, H] = i, and thus it may be thought of as being formally canonically conjugate to the Hamiltonian H. The time operator associated with a given Hamiltonian H is not unique because one can replace Θ by Θ + Θhom, where Θhom satisfies the homogeneous condition [Θhom, H] = 0. To study this nonuniqueness the matrix elements of Θ for the harmonic-oscillator Hamiltonian are calculated in the eigenstate basis. This calculation requires the summation of divergent series, and the summation is accomplished by using zeta-summation techniques. It is shown that by including appropriate homogeneous contributions, the matrix elements of Θ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.
It is shown that if the operator for a -symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. The fact that the operator is unbounded is significant because, while there is a formal equivalence between a -symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that -symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a
c-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
If a HamiltonianH has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT . The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT -symmetric Hamiltonian H = 12p + 1 2x 2 + i x using perturbative techniques. It is also shown how to construct the operator C for H = 12p 2 + 12x 2 − x using nonperturbative methods [1].