Relativistic hamiltonian equations for any spin

We discuss 2(2J + 1)-component Poincaré-invariant Hamiltonian theories that describe free particles of definite mass and spin and that are subject to the conditions (a) every observable $\text{O}$ is either Hermitian or pseudo-Hermitian (i.e., O = ?₃O⁺?₃) and (b) the theory is invariant under the discrete symmetries. Our treatment is based on the Heisenberg equations of motion and on the Lie algebra of the Poincaré group. Explicit formulas are found for the generators of this algebra, including the Hamiltonian H, and all relations between the operators Γ and H that are both necessary and sufficient for $\text{K =}\text{1}\text{2}\text{[x}\text{, H]}{}_{\mathrm{+}}\text{+ Γ}$ to generate Lorentz boosts are found. To illustrate the utility of our results, we apply them to obtain explicit generalizations of the Dirac equation to any spin, by requiring that Γ = 0, and of the Sakata-Taketani spin-0 and spin-1 equations to any spin, by requiring that $\text{Γ = ??}{}_3\text{(}\text{1}\text{2m}\text{)}\text{S × p}$.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

Collective properties of A = 117−127 odd-A I nuclei

Systematic ΔJ = 2 bands have been observed on the $\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$, $\text{g}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ and $\text{h}{}_{\mathrm{<mtext>11</mtext><mtext>2</mtext>}}$ quasi-proton states in A = 117?127 odd-I nuclei. The ΔJ = 2 energy spacings decrease (relative to the Te cores) as A decreases for the $\text{h}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ band but not for the others. Also ΔJ = 1 bands on unsually low-lying $\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$ proton-hole states exist.     

A search for integrable two-dimensional hamiltonian systems with polynomial potential

We report the results of a search for all integrable hamiltonian systems of type $\text{H = (}\text{1}\text{2}\text{)p}{}_{\mathrm{x}}{}^2\text{+ (}\text{1}\text{2}\text{)p}{}_{\mathrm{y}}{}^2\text{+ V(x,y)}$, where V is a polynomial in x and y of degree 5 or less and the second invariant is a polynomial in p_x and p_y of order 4 or less. Both classical and quantum integrability are discussed.     

The measurement of the rotational relaxation times of OCS from the nutation signals

The time dependence of microwave absorption was measured for the J = 2-1 and J = 3-2 transitions of OCS under on- and off-resonant conditions utilizing Stark and source modulation, respectively. The two effective pressure parameters obtained under the two conditions, which correspond to $\text{(T}{}_2{}^{\mathrm{?1}}\text{+ T}{}_1{}^{\mathrm{?1}}\text{)}\text{4πP}$ and $\text{(2πT}{}_2\text{P)}{}^{\mathrm{?1}}$, respectively, according to the Bloch equation, are different beyond experimental error; the difference $\text{(T}{}_2{}^{\mathrm{?1}}\text{? T}{}_1{}^{\mathrm{?1}}\text{)}\text{2πP}$ is 0.94 ± 0.38 (2.5σ) MHz/Torr for J = 2?1. This difference was also determined to be 1.19 ± 0.30 MHz/Torr from the dependence of the nutation frequency on the microwave power.     

Exact approach to equilibrium for the glauber model of a one-dimensional Ising system with random ±J bonds

The (discrete-time) Glauber model is considered for a one-dimensional system of spins s_j = ±1 with nearest-neighbor Ising interaction $\text{H}\text{= -Σ}{}_{\mathrm{j}}\text{J}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j+1}}$. The J_j = ±J are treated as random variables with an arbitrary joint probability p(J). The exact time-dependent average 〈s_j〉_t is determined, and from it the “quenched” average $\text{〈〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}\text{〉}{}_{\mathrm{<mtext>av</mtext>}}\text{=Σ}{}_{\mathrm{<mtext>J</mtext>}}\text{p}\text{(}\text{J}\text{)〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}$ is explicitly found.     

High-spin states in 38K

High-spin states in ³⁸K are investigated with the ²⁴Mg(¹⁶O, pnγ)³⁸K reaction at $\text{E(}{}^{16}\text{O}\text{) = 36–44}\text{MeV}$. A recently developed Compton-suppression spectrometer with 120 msr solid angle and a pulsed beam are employed to study their γ-decay. For the E4 transition from the isomeric level at E_x = 3458 keV to the ground state a branching ratio of (0.15 ± 0.02)% is found. On the basis of angular distribution and polarization measurements, in which the delayed feeding component is eliminated, spin-parity assignments are obtained of J^π(2646 keV) = (2, 4)^?, J^π(3420 keV) = (4, 6)^? and J^π(3458 keV) = (5, 7)⁺. Prompt-delayed and prompt γγ coincidence experiments are performed to locate high-spin levels above the isomer. Hitherto unobserved levels of high spin are found at E_x = 5254, 7397, 8693, 8747 and 10980 keV and assignments of J^π = (9⁺), (10^?), (12^?), (11^?) and (13^?) respectively, are suggested by weak-coupling considerations. The experimental results are compared with a large-scale shell-model calculation performed in a configuration space with a ²⁸Si core and ten active particles distributed over the ($\text{2}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, 1}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{, 1}\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{, 2}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{)}$ shells. The high-spin states appear to have a rather simple shell-model structure.     

l-type doubling transitions in Δ states of OCS

Molecular beam electric resonance spectroscopy has been used to measure the l-type doubling transitions in carbonyl sulfide. Transitions in J = 4 to J = 20 have been observed for the (02²0) vibrational state and for J = 1 and J = 2 for the (03¹0) state. The data has been analyzed to give the v = 2 energy separation $\text{E}{}_{\mathrm{<mtext>\Delta </mtext>}}{}^0\text{- E}{}_{\mathrm{<mtext>\Sigma </mtext>}}{}^0\text{= ?5.7861(2) + 8.36(1) × 10}{}^{\mathrm{?5}}\text{J(J + 1)}\text{cm}{}^{\mathrm{?1}}$, and the vibrational dependence of q to be 86.52(9)(v₂ + 1) KHz. The dipole moment of the (02²0) vibrational state is 0.6936(3) D.     

15N(p,po)15N and the levels of 16O for Ex = 14.8–18.6 MeV

Angular distributions of cross section and analyzing power for elastic scattering of protons from ¹⁵N have been measured for E_p = 2.7–7.0 MeV. A phase-shift analysis of the data yields spin-parity assignments and level parameters for seventeen states of ¹⁶O in the excitation energy region 14.8–18.6MeV. Three of the resonances have not previously been identified, among them being a broad J^π = 2^? level at E_p = 6.1 MeV which is almost certainly the analog of the 2^? 1p1h state with configuration ($\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{,}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$) at E_x ∽ 5.0 MeV in ¹⁶N. The broad level previously reported near E_p = 5.0 MeV has been observed and its parameters determined. A resonance analysis of the phase shifts yielded values of E_r, Γ and Γ_p for all of the levels. The J^π assignments are in agreement with previously reported values. For resonances having J = l, the data can usually be fit with a resonant phase shift corresponding to either J = l + 1 or J = l ? 1, in addition to the phase shift for J = l. Which of the two spurious-J solutions occurs seems to depend on whether the partial wave through which the resonant state is formed is $\text{J = l +}\text{1}\text{2}\text{or}\text{J = l ?}\text{1}\text{2}$.     

Predissociation and Λ-doubling in the even-parity Rydberg states of the nitrogen molecule

Predissociations in the y¹Π_g and $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ Rydberg states of N₂ (configurations $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{4pσ}$ and $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{3pπ}$, respectively) and their likely causes, are discussed. Peaking of rotational intensity at unusually low J values, without sharp breaking off, is interpreted as due to case c^? or case cⁱ predissociation. Λ doubling in the y state, attributed to interactions with the $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state and with another, ¹Σ⁺, state of the same electron configuration as x, is analyzed. From this analysis the location of the (unobserved) ${}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ state, here labeled x′, is obtained. It is concluded that the predissociation in the Π⁺ levels of the y state is an indirect one mediated by the interaction with x′ coupled with predissociation of x′ by a ${}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state dissociating to ${}^4\text{S +}{}^2\text{P}$ atoms: combined, however, with perturbation of the y state by the k¹Π_g Rydberg state (configuration $\text{3σ}{}_{\mathrm{g}}{}^{\mathrm{?1}}\text{4dπ}$), whose Π⁺ levels are completely predissociated.     

On the boundedness of the eigenvalue spectrum of phonon collision operator

Anharmonicity in lattice potential leads to boundedness of the eigenvalue Spectrum of the phonon collision operator. Considering the deviation, ψ_q, in the distribution function of the phonons $\text{N}{}_{\mathrm{q}}\text{, N}{}_{\mathrm{q}}\text{=}\text{N}\text{?}{}_{\mathrm{q}}\text{+}\text{N}\text{?}{}_{\mathrm{q\psi q}}\text{(}\text{N}\text{?}\text{+ 1)}$. bar denoting equilibrium value, as an odd function of the phonon wave vector q it has been possible to obtain a lower bound, μ, on the eigenvalue spectrum of the phonon collision operator P. This satisfies the inequelity relation 0?μ?p_i?λ, where p_i are the eigenvalue of P, and λ is an upper bound on it (as given by Benin). The occurence of μ ensures for the possibility of obtaining a sequence of upper bounds on the lattice thermal transport coefficient.     

Formaldehyde as a semirigid bender: The rotation-inversion spectrum in its Ã1A2 excited state

The semirigid bender Hamiltonian [Bunker and Landsberg, J. Mol. Spectrosc.67, 374–385 (1977)] was used to fit the rotation-inversion energy level separations in the $\text{A}\text{?}{}^1\text{A}{}_2$ excited state of formaldehyde. We fix the r⁰(CH) bond length and allow the R(CO) bond length and (H?H) bond angle to vary with the inversion angle ρ. The fit to 64 rotation-inversion energies (with v₄ and J < 4) is significantly better with a standard deviation of 0.199 cm^?1 than when the rigid bender [Bunker and Stone, J. Mol. Spectrosc.41, 310–332 (1972)] is used. The barrier height to planarity is 358 cm^?1 and the equilibrium ρ_e = 34.7°. The CO bond length is found to decrease by 0.034 from 1.3670 Å and the H?H angle by about 6 from 122.4° as the molecular configuration changes from planar to pyramidal. The rigid bender model developed earlier by Moule and Rao for formaldehyde [J. Mol. Spectrosc.45, 120–141 (1973)] is then used to fit the 32 rotation-(out-of-plane) bending energy levels (with v₄ = 0 and 1) of the $\text{X}\text{?}{}^1\text{A}{}_1$ ground electronic state of H₂CO. For this, a simple potential consisting of quadratic and quartic terms is used and the standard deviation of the fit is 0.148 cm^?1.     

A bound for the existence of invariant circles in a class of two-dimensional dissipative maps

By generalizing results due to Mather we obtain an upper bound for the existence of smooth invariant circles in a class of dissipative two-dimensional maps. In particular the map $\text{x}{}_{\mathrm{n+1}}\text{= x}{}_{\mathrm{n}}\text{+ Ω + by}{}_{\mathrm{n}}\text{? (k/2π)sin 2πx}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{n+1}}\text{= by}{}_{\mathrm{n}}\text{? (k/2π) X sin 2πx}{}_{\mathrm{n}}$ can have no smooth invariant circle for |k| > 2(1 + b)/(2 + b).     

Generalized renormalization group equation for period-doubling bifurcations

Given a mapping $\text{x → ?(λ,x)}$, whose bifurcation points accumulate at λ = Λ_∞, it is shown that the iterates $\text{(?α)}{}^{\mathrm{p\times }}\text{[?}{}^{\mathrm{2p}}\text{(Λ}{}_{\mathrm{\infty }}\text{?μ/δ}{}^{\mathrm{p}}\text{, X}{}_{\mathrm{\infty }}\text{+y/(?α)}{}^{\mathrm{p}}\text{)?X}{}_{\mathrm{\infty }}\text{]}$ converge to a function ψ(μ,y)asp→∞, where X_∞ maximizes $\text{?(Λ}{}_{\mathrm{\infty ,x}}\text{)}$. The function ψ is universal up to scaling in μ and y, and satisfies ψ(μ/δ,ψ(μ/δ,?y/α)) = ?α^?1ψ(μ,y). This result generalizes the well-known Feigenbaum universal function g(y) with g(g(?y/α)) = ?g(y/α, which is the special case for μ = 0.     

A further contribution to the theory of the transient hot-wire technique for thermal conductivity measurements

We study the density of states of a one-dimensional tightbinding electron model with random hopping elements. The Hamiltonian is $\text{H = -∑}{}_{\mathrm{i}}\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i+1}}\text{+a}{}^{\mathrm{+}}{}_{\mathrm{i+1}}\text{a}{}_{\mathrm{i}}\text{)}$, where the $\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}$'s are independent identically distributed random variables. It is proved that the single particle density of states D(E) diverges near E = 0 as $\text{1}\text{|(E}\text{log}{}^3\text{|E|)|}$.     

The evolution of a cluster of particles and the invariants of a time-dependent oscillator

A simple physical realization is found for the time-dependent harmonic oscillator equation $\text{q}\text{?}\text{+ φ(t)q = 0}$, for its auxiliary equation $\text{g}\text{?}\text{9 + φ(t)? = h}{}^2\text{?}{}^{\mathrm{?3}}$, and for the Ermakov invariant. Through generalization of the model, a set of invariants is obtained for a system of equations of the form .     

Infrared diode laser spectroscopy of the CF radical

The fundamental bands of the CF radical in the $\text{X}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{X}{}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ electronic states were observed by using an infrared tunable diode laser as a source. Zeeman modulation could be used in detecting lines not only in the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ state, but also in ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, because the CF radical deviates considerably from Hund's case (a). From the least-squares analysis of the observed spectra, the following molecular constants were obtained: B_e = 1.416 704 (37) cm^?1, α_e = 0.018 419 (50) cm^?1, $\text{r}{}_{\mathrm{e}}\text{= 1.271 977 (17)}\text{A}\text{?}$, D_e = 6.68 (15) × 10^?6cm^?1, p₀ = 0.008 580 (21) cm^?1, p₁ = 0.008 52 (11) cm^?1, and $\text{ν}{}_0\text{= 1286.1281 (5)}\text{cm}{}^{\mathrm{?1}}$, with three standard errors in parentheses.     

Dielectric susceptibility and correlation length of one-dimensional CDW with impurties. II. weak disorder

Dependence of static dielectric susceptibility and correlation length of charge density waves (CDW) with weak defects on parameter of incommensurability with lattice is investigated. In almost commensurate phase (h?h_ch_c), $\text{χ ～ (h?h}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{In}{}^{\mathrm{<mtext>-4</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h?h}{}_{\mathrm{c}}$ and R_c ～ (h ? h_c)^{$\text{2}\text{3}$}. In${}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h ? h}{}_{\mathrm{c}}$. Far from commensurability $\text{(h?h}{}_{\mathrm{c}}\text{) χ～ (a+h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, $\text{R}{}_{\mathrm{c}}\text{～ (a + h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, where a is the dimensionless ratio of random potential intensities, corresponding to backward and forward scattering impurities.     

“Forbidden” rotational spectra of symmetric-top molecules: PH3 and PD3

A millimeter-wave spectrometer having a sensitivity of 4 × 10^?10 cm^?1 in the 2-mm region has been constructed for observation of extremely weak millimeter-wave spectra of gases. It has been used to measure J → J, K = 0 ← 3 transitions in PH₃ and J → J, K = 0 ← 3 as well as K = ±1 ← ±4 transitions in PD₃. The B₀ and C₀ spectral constants (in MHz) are: for PH₃, B₀ = 133 480.15 ± 0.12 and C₀ = 117 488.85 ± 0.16; for PD₃, B₀ = 69 471.10 ± 0.03 and C₀ = 58 974.37 ± 0.05. The effective ground-state values obtained for the bond angle and bond length are: for PH₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.420}{}_0$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.34}{}_5$; for PD₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.417}{}_6$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.35}{}_9$. The corresponding zero-point-average values were calculated to be: for PH₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4269}{}_9\text{± 0.0002}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.228}{}_7$; for PD₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4226}{}_5\text{± 0.0001}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.256}{}_7\text{± 0.004}$. For both species, the equilibrium values are $\text{r}{}_{\mathrm{e}}\text{(}\text{A}\text{?}\text{) = 1.4115}{}_9\text{± 0.0006}$ and $\text{α}{}_{\mathrm{e}}\text{(}{}^{\mathrm{o}}\text{) = 93.32}{}_8\text{± 0.02}$.