Symmetry In Quantum Mechanics Research Articles

Discrete Symmetry in Relativistic Quantum Mechanics

EPR experiment on $K^0-\bar{K}^0$ system in 1998\cite{1} strongly hints that one should use operators $\hat{E}_c=-i\hbar\frac{\partial}{\partial t}$ and $\hat{\bf p}_c=i\hbar\nabla$ for the wavefunction (WF) of antiparticle. Further analysis on Klein-Gordon (KG) equation reveals that there is a discrete symmetry hiding in relativistic quantum mechanics (RQM) that ${\cal P}{\cal T}={\cal C}$. Here ${\cal P}{\cal T}$ means the (newly defined) combined space-time inversion (with ${\bf x}\to -{\bf x}, t\to-t$), while ${\cal C}$ the transformation of WF $\psi$ between particle and its antiparticle whose definition is just residing in the above symmetry. After combining with Feshbach-Villars (FV) dissociation of KG equation ($\psi=\phi+\chi$)\cite{2}, this discrete symmetry can be rigorously reformulated by the invariance of coupling equation of $\phi$ and $\chi$ under either the combined space-time inversion ${\cal P}{\cal T}$ or the mass inversion ($m\to -m$), which makes the KG equation a self-consistent theory. Dirac equation is also discussed accordingly. Various applications of this discrete symmetry are discussed, including the prediction of antigravity between matter and antimatter as well as the reason why we believe neutrinos are likely the tachyons.

Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics

By adding an imaginary interacting term proportional to $i{p}_{1}{p}_{2}$ to the Hamiltonian of a free anisotropic planar oscillator, we construct a model which is described by the $PT$-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our $PT$-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positive and negative energy levels in our model. Moreover, we find that the permutation symmetry of two dimensions in our Hamiltonian corresponds to the identity (not in magnitude but in attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, and reveal that the unequal-frequency condition imposed as a prerequisite upon the Pais-Uhlenbeck oscillator can reasonably be explained as the spontaneous breaking of this identity.

States of physical systems in classical and quantum mechanics

We discuss the descriptions of states of physical systems in classical and quantum mechanics. We show that while it is possible to evolve a terminology common to both, the differences in the underlying mathematical structures lead to significant points of departure between the two descriptions both at mathematical and conceptual levels. We analyse the state spaces associated with physical systems described by two and three dimensional complex Hilbert spaces in considerable detail to illustrate how subtle this question can in general be. We highlight the role the Bargmann invariants play in the passage from the Hilbert space to the ray space, the space of states in quantum mechanics, and also in the context of Wigner’s theorem on symmetries in quantum mechanics where they originally appeared.

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.

Example of a Quantum Anomaly in the Physics of Ultracold Gases

In this Letter, we propose an experimental scheme for the observation of a quantum anomaly--quantum-mechanical symmetry breaking--in a two-dimensional harmonically trapped Bose gas. The anomaly manifests itself in a shift of the monopole excitation frequency away from the value dictated by the Pitaevskii-Rosch dynamical symmetry [L. P. Pitaevskii and A. Rosch, Phys. Rev. A 55, R853 (1997)]. While the corresponding classical Gross-Pitaevskii equation and the hydrodynamic equations derived from it do exhibit this symmetry, it is--as we show in our paper--violated under quantization. The resulting frequency shift is of the order of 1% of the carrier, well in reach for modern experimental techniques. We propose using the dipole oscillations as a frequency gauge.

On Global and Nonlinear Symmetries in Quantum Mechanics

(3) and furthermore nonlinear Schroedinger equations with given potentials which were also derived in another context in [1, 2].

Coupling constant metamorphosis and Nth-order symmetries in classical and quantum mechanics

We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the momenta to polynomials in the momenta. We study specializations of these actions which preserve polynomials and also the structure of the symmetry algebras in both the classical and quantum cases. We give several examples of non-constant curvature third- and fourth-order superintegrable systems in two space dimensions obtained via CCM, with some details on the structure of the symmetry algebras preserved by the transform action.

E. C. G. Sudarshan and symmetry in classical dynamics, optics and quantum mechanics

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed.Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase.The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described.In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

Shape Invariant and Rodrigues Solution of the Dirac-shifted Oscillator and Dirac-Morse Potentials

We show that the Dirac equation for a charged spinor in spherically symmetric electromagnetic potentials as Dirac-shifted oscillator and Dirac-Morse potentials have the conditions of shape invariant symmetry in non-relativistic quantum mechanics. The relativistic spectra of the bound states and spinor wavefunctions can be obtained by the Rodrigues polynomials of one associated differential equation.

Symmetry and degeneracy in quantum mechanics. Self-duality in finite spin systems

The symmetry of self-duality (Savit 1980 Rev. Mod. Phys. 52 453) of some models of statistical mechanics and quantum field theory is discussed for finite spin blocks of the Ising chain in a transverse magnetic field. The existence of this symmetry in a specific type of these blocks, and not in others, is manifest by the degeneracy of their energy levels. This illustrates the fact that in quantum mechanics, more symmetry in the Hamiltonian induces more degeneracy in the energy spectrum.

Gauging non-Hermitian Hamiltonians

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in X implies minimal substitution in the form P → P − eA(X). We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not.

Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.

Discrete accidental symmetry for a particle in a constant magnetic field on a torus

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the 1 / r and r 2 potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge–Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the θ -vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters θ x and θ y explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner's theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner's theorem is not fully appreciated in general. It is Wigner's theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner's theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.

Gauge fixing and residual symmetries in gauge/gravity theories with extra dimensions

We study compactified pure gauge/gravitational theories with gauge-fixing terms and show that these theories possess quantum mechanical supersymmetriclike symmetries between unphysical degrees of freedom. These residual symmetries are global symmetries and generated by quantum mechanical $\mathcal{N}=2$ supercharges. Also, we establish a new one-parameter family of gauge choices for higher-dimensional gravity and calculate as a check of its validity one graviton exchange amplitude in the lowest tree-level approximation. We confirm that the result is indeed $\ensuremath{\xi}$ independent and the cancellation of the $\ensuremath{\xi}$ dependence is ensured by the residual symmetries. We also give a simple interpretation of the van Dam-Veltman-Zakharov discontinuity, which arises in the lowest tree-level approximation, from the supersymmetric point of view.

Lissajous curves and semiclassical theory: The two-dimensional harmonic oscillator

The semiclassical treatment of the two-dimensional harmonic oscillator provides an instructive example of the relation between classical motion and the quantum mechanical energy spectrum. We extend previous work on the anisotropic oscillator with incommensurate frequencies and the isotropic oscillator to the case with commensurate frequencies for which the Lissajous curves appear as classical periodic orbits. Because of the three different scenarios depending on the ratio of its frequencies, the two-dimensional harmonic oscillator offers a unique way to explicitly analyze the role of symmetries in classical and quantum mechanics.

H-Solid State NMR Studies of Tunneling Phenomena and Isotope Effects in Transition Metal Dihydrides

In many transition metal dihydrides and dihydrogen complexes the hydrogens are relatively weakly bound and exhibit a fairly high mobility, in particular with respect to their mutual exchange. Part of this high mobility is due to the exchange symmetry of the two hydrogens, which causes an energy splitting into even and odd spatial energy eigenfunctions, resulting in the typical coherent tunneling of a two-level system. Owing to the quantum mechanical symmetry selection principles the eigenfunctions are connected to the possible nuclear spin states of the system. If the tunneling frequency is in the proper frequency window it is thus possible to observe these tunneling transitions by NMR at very low temperatures, where no thermally induced exchange reactions overshadow the tunneling. The first part of this review gives an introduction into the interplay of chemical kinetics and tunneling phenomena in general, rotational tunneling of dihydrogen in a two-fold potential in particular and the Bell tunnel model, followed by a summary of solid state NMR techniques for the observation of these tunnel processes. Then a discussion of the effects of these processes on the 2H NMR line shape is given. The second part of the review reports results of a 2H-solid state NMR spectroscopy and T1 relaxatiometry study of trans-[Ru(D2)Cl(PPh2CH2CH2PPh2)2]PF6, in the temperature regime from 5.4 to 320 K. In the Ru-D2 sample coherent tunneling and incoherent exchange processes on the time scale of the quadrupolar interaction are observed. From the spectra and T1-data the height of the tunneling barrier is determined. Next results of 2H-spin–lattice relaxation measurements for a selectively η2 − D2 labeled isotopomer of the complex W(PCy3)2(CO)3)(η2 − D2) are presented and discussed. The relaxation measurements are analyzed in terms of a simple one dimensional Bell tunnel model and comparison to incoherent neutron scattering (INS) data from the H2 complex. The comparison reveals a strong isotope effect of 2 × 103 for the exchange rates of the deuterons versus hydrons.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Classification of atomic states by geometrical and quantum-mechanical symmetries

The eigenstates of nonrelativistic two-, three-, four-, and six-valence-electron atoms are classified according to their transformation properties under symmetry operations. In energy-favorable geometrical arrangements of the electrons, the spatial distribution of the wave function is constrained by the demand of invariance under rotation, inversion and permutation of particles. For a given term, the symmetry constraints may leave the energy-favorable arrangement accessible or inaccessible. In the latter case the term is not expected to belong to the low-energy part of the spectrum. Contrary, in the former case with no symmetry-induced inherent nodal surface in the multidimensional coordinate space of the multielectron wave function, the term is expected to belong to the low-energy part of the spectrum. The symmetry-based classification is confirmed by recent calculations.

Scale symmetry in classical and quantum mechanics

In this Letter we address again the issue of the scale anomaly in quantum-mechanical models with inverse square potential. In particular, we examine the interplay between the classical and quantum aspects of the system using in both cases an operatorial approach.