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Stefan Weigert

Publications and source records attributed to Stefan Weigert.

At least 55 records · Page 3Linked to original sources
arXiv2003

Completeness and Orthonormality in PT-symmetric Quantum Systems

Some PT-symmetric non-hermitean Hamiltonians have only real eigenvalues. There is numerical evidence that the associated PT-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced `charge operator' is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that PT-symmetry provides an additional link between the eigenstates of the Hamiltonian and those of its adjoint, which form a dual pair of bases. The `charge operator' emerges naturally upon expressing the properties of the dual bases in terms of one basis only.

quant-ph↗
arXiv2003

Quantum Search for Zeros of Polynomials

A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved.

quant-ph↗
arXiv2003

The quantum way to diagonalize hermitean matrices

An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. Here, the genuinely quantum mechanical approach is considered in detail for (2x2) matrices. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. The simple low-dimensional case allows one to illustrate the conceptual features of the general method which applies to (NxN) hermitean matrices.

quant-ph↗
arXiv2002

PT-symmetry and its spontaneous breakdown explained by anti-linearity

The impact of an anti-unitary symmetry on the spectrum of non-hermitean operators is studied. Wigner's normal form of an anti-unitary operator is shown to account for the spectral properties of non-hermitean, PT-symmetric Hamiltonians. Both the occurrence of single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one- or two-dimensional irreducible representations of the symmetry PT. In this framework, the concept of a spontaneously broken PT-symmetry is not needed.

quant-ph↗
arXiv2001

Commensurate Harmonic Oscillators: Classical Symmetries

The symmetry properties of a classical N-dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phase-space functions form the algebra su(N) with respect to the Poisson bracket. In the isotropic case, the phase-space flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group SU(N). For a commensurate oscillator, however, the group SU(N) of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.

math-ph↗
arXiv2001

Quantum Parametric Resonance

The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically instable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.

quant-ph↗
arXiv2000

Contracting the Wigner-Kernel of a Spin to the Wigner-Kernel of a Particle

A general relation between the Moyal formalisms for a spin and a particle is established. Once the formalism has been set up for a spin, the phase-space description of a particle is obtained from the `contraction' of the group of rotations to the group of translations. This is shown by explicitly contracting a spin Wigner-kernel to the Wigner kernel of a particle. In fact, only one out of 2^{2s} different possible kernels for a spin shows this behaviour.

quant-ph↗
arXiv2000

Discrete Moyal-type Representations for a Spin

In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e. by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P- or Q-symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels which give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)(2s+1) points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism, and it is illustrated by the example of a spin 1/2.

quant-ph↗
arXiv2000

Quantum Diagonalization of Hermitean Matrices

To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource. It allows one, in principle, to diagonalize hermitean (N by N) matrices by quantum mechanical measurements only. To do so, one considers the given matrix as an observable of a single spin with appropriate length s=(N-1)/2, which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As it is based on the `collapse of the wave function' associated with a measurement, the procedure is neither a digital nor an analog calculation--it defines thus a new quantum mechanical method of computation.

quant-ph↗
arXiv1999

Expanding Hermitean Operators in a Basis of Projectors on Coherent Spin States

The expectation values of a hermitean operator A in (2s+1)(2s+1) specific coherent states of a spin are known to determine the operator unambiguously. As shown here, (almost) any other (2s+1)(2s+1) coherent states also provide a basis for self-adjoint operators. This is proven by considering the determinant of the Gram matrix associated with the coherent state projectors as a Hamiltonian of a fictitious classical spin system.

quant-ph↗
arXiv1999

Discrete Q- and P-symbols for spin s

Non-orthogonal bases of projectors on coherent states are introduced to expand hermitean operators acting on the Hilbert space of a spin s. It is shown that the expectation values of a hermitean operator A in a family of (2s+1)(2s+1) spin-coherent states determine the operator unambiguously. In other words, knowing the Q-symbol of A at (2s+1)(2s+1) points on the unit sphere is already sufficient in order to recover the operator. This provides a straightforward method to reconstruct the mixed state of a spin since its density matrix is explicitly parametrized in terms of expectation values. Furthermore, a discrete P-symbol emerges naturally which is related to a basis dual to the original one.

quant-ph↗
arXiv1999

A Discrete Phase-Space Calculus for Quantum Spins based on a Reconstruction Method using Coherent States

To reconstruct a mixed or pure quantum state of a spin s is possible through coherent states: its density matrix is fixed by the probabilities to measure the value s along 4s(s+1) appropriately chosen directions in space. Thus, after inverting the experimental data, the statistical operator is parametrized entirely by expectation values. On this basis, a symbolic calculus for quantum spins is developed, the `expectation-value representation.' It resembles the Moyal representation for SU(2) but two important differences exist. On the one hand, the symbols take values on a discrete set of points in phase space only. On the other hand, no quasi-probabilities - that is, phase-space distributions with negative values - are encountered in this approach.

quant-ph↗
arXiv1999

Coherent States and the Reconstruction of Pure Spin States

Coherent states provide an appealing method to reconstruct efficiently a pure state of a quantum mechanical spin s. A Stern-Gerlach apparatus is used to measure (4s+1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeroes of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stern-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase-space which are smaller than hbar.

quant-ph↗
arXiv1999

Quantum Time Evolution in Terms of Nonredundant Expectation Values

Each scheme of state reconstruction comes down to parametrize the state of a quantum system by expectation values or probabilities directly measurable in an experiment. It is argued that the time evolution of these quantities provides an unambiguous description of the quantal dynamics. This is shown explicitly for a single spin s, using a quorum of expectation values which contains no redundant information. The quantum mechanical time evolution of the system is rephrased in terms of a closed set of linear first-order differential equations coupling (2s+1)^2 expectation values. This `realization' of the dynamical law refers neither to the wavefunction of the system nor to its statistical operator.

quant-ph↗
arXiv1999

Reconstructing the density matrix of a spin s through Stern-Gerlach measurements (II)

The density matrix of a spin s is fixed uniquely if the probabilities to obtain the value s upon measuring n.S are known for 4s(s+1) appropriately chosen directions n in space. These numbers are just the expectation values of the density operator in coherent spin states, and they can be determined in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data can be inverted providing thus a parametrization of the statistical operator by 4s(s+1) positive parameters.

quant-ph↗
arXiv1998

Gauge transformations for a driven quantum particle in an infinite square well

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations. Implications for the quantization procedure of a particle in a box are pointed out.

quant-ph↗
arXiv1998

Reconstruction of spin states and its conceptual implications

State reconstruction for quantum spins is reviewed. Emphasis is on non-tomographic approaches which are based on measurements performed with a Stern-Gerlach apparatus. Two consequences of successfully implemented state reconstruction are pointed out. First, it allows one to determine experimentally the expectation value of an arbitrary operator without a device measuring it. Second, state reconstruction suggests a reformulation of Schroedinger's equation in terms of expectation values only, without explicit reference to a wave function or a density operator.

quant-ph↗