Complex and spinorial techniques of general relativity are used to determine all the states of SU(2) invariant quantum mechanical systems in which the equality holds in uncertainty relations for the components of the angular momentum vector operator in two given directions. The expectation values depend on a discrete quantum number and two parameters, one of them is the angle between two angular momentum components and the other is the quotient of two standard deviations. Allowing the angle between two angular momentum components to be arbitrary, a new genuine quantum mechanical phenomenon emerges: it is shown that although standard deviations change continuously, one of the expectation values changes discontinuously on this parameter space. Since physically neither of the angular momentum components is distinguished over the other, this discontinuity suggests that the genuine parameter space must be a double cover of this classical one: it must be diffeomorphic to a Riemann surface known in connection with the complex function z. Moreover, the angle between angular momentum components plays the role of the parameter of an interpolation between the continuous range of expectation values in the special case of orthogonal angular momentum components and the discrete point spectrum of one angular momentum component. The consequences in the simultaneous measurements of these angular momentum components are also discussed briefly.
Complex and spinorial techniques of general relativity are used to determine all the states of SU(2) invariant quantum mechanical systems in which the equality holds in uncertainty relations for the components of the angular momentum vector operator in two given directions. The expectation values depend on a discrete quantum number and two parameters, one of them is the angle between two angular momentum components and the other is the quotient of two standard deviations. Allowing the angle between two angular momentum components to be arbitrary, a new genuine quantum mechanical phenomenon emerges: it is shown that although standard deviations change continuously, one of the expectation values changes discontinuously on this parameter space. Since physically neither of the angular momentum components is distinguished over the other, this discontinuity suggests that the genuine parameter space must be a double cover of this classical one: it must be diffeomorphic to a Riemann surface known in connection with the complex function z. Moreover, the angle between angular momentum components plays the role of the parameter of an interpolation between the continuous range of expectation values in the special case of orthogonal angular momentum components and the discrete point spectrum of one angular momentum component. The consequences in the simultaneous measurements of these angular momentum components are also discussed briefly.
Complex techniques of general relativity are used to determine all the states in two- and three-dimensional momentum spaces in which the equality holds in uncertainty relations for non-commuting basic observables of Euclidean invariant elementary quantum mechanical systems, even with non-zero intrinsic spin. It is shown that while there is a 1-parameter family of such states for any two components of the angular momentum vector operator with any angle between them, such states exist for a component of the linear and angular momenta only if these components are orthogonal to each other, and hence, the problem is reduced to the two-dimensional Euclidean invariant case. We also show that analogous states exist for a component of the linear momentum and of the center-of-mass vector only if the angle between them is zero or an acute angle. No such state (represented by a square integrable and differentiable wave function) can exist for any pair of components of the center-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra but on the choice of its generators as well.
In this work we study various notions of uncertainty for angular momentum in the spin- s representation of SU(2). We characterize the ‘uncertainty regions’ given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s . Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s , since they are saturated by states taken from the Holstein–Primakoff approximation. We show that, for all s , all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius s ( s + 1 ) . Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s . For large s the Maassen–Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r ( s ) , where r ( s ) / s → 1 as s → ∞ .
In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.
The Mathieu-Schrödinger equation, which describes the behavior of a quantum pendulum, depending on the value of the parameter l (pendulum filament length), can have the symmetry of the Klein's four-group or its invariant subgroups. The paper shows that the mean values of z-components of the angular momentum of nondegenerate quantum states (the symmetry region of the four-group) tend to zero and their root mean square fluctuations are non-zero. Consequently, in this region of parameter values, the fluctuations overlap the mean values of the angular momentum and they become indistinguishable. Therefore, it can be argued that if, with an increase in the parameter, the system goes into a non-degenerate state, then after the inversion of the parameter change and the transition to the region of degenerate states, the initial states will not be restored. This behavior of the average values of angular momenta is caused by the combined actions of two factors: discontinuous change in the system at the points of change of its symmetry and the presence of quantum fluctuations in nondegenerate states.
The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.