The scattering structure function SN (q) of a large flexible ring polymer with N monomers in a good solvent is calculated as a universal expansion in the variable q' , where denotes the mean square radius of gyration of the ring. To solve the interacting polymer problem, the methods of renormalized field theory are applied directly to Fixman's cluster expansion. The results are obtained using the approximation of expanding in E;: 4 -(dimension of space) and indicate that the self-repulsion has a greater influence on the large distance behavior of a ring polymer when compared with a linear polymer. In particular, we find that the radius of gyration ratio < R'/; > 'in/ < R'/; > Ii" is a universal quantity and to first order in E is equal to 0.568. For ideal polymers, the ratio is known to be 0.5. 1574

The interpenetration of two excluded-volume chain molecules of different size in dilute solution is studieci via scaling and renormalization methods. The chains are found to interpenetrate much more strongly than smoothed-density models suggest, in accordance with recent work by Khokhlov. The pair correlation funtion g(r) goes to zero at the origin only as a weak power of r. This power is related to Des Cloizeaux's exponents 0, describing intra chain correlations. The power is also related to the scaling exponents of star polymers. The mutual excluded volume MSL of two chains with greatly different length is proportional to the volume of the smaller chain and to the mass of the larger. Thus MSL is much smaller than a smoothed density model would predict. We discuss which chain correlations give rise to this small M sL ' The universal coefficient relating MSL to the radius of gyration of the smaller chain is strongly dependent on the dimension d of space, according to our second-order expansion in 4-d. The interpenetration behavior predicted here affects measurable thermodynamic, scattering, and physical-chemical properties of the solution.

Spontaneous symmetry breakdown in non-relativistic quantum mechanics Am. J. Phys. 80, 891 (2012) Understanding the damping of a quantum harmonic oscillator coupled to a two-level system using analogies to classical friction Am. J. Phys. 80, 810 (2012) Relation between Poisson and Schrödinger equations Am. J. Phys. 80, 715 (2012) Comment on "Exactly solvable models to illustrate supersymmetry and test approximation methods in quantum mechanics," Am. J. Phys. 79, 755-761 (2011) Am. J. Phys. 80, 734 (2012) The uncertainty product of position and momentum in classical dynamics Am.The 1925 paper "On quantum mechanics" by M. Born and P. Jordan, and the sequel "On quantum mechanics II" by M. Born, W. Heisenberg, and P. Jordan, developed Heisenberg's pioneering theory into the first complete formulation of quantum mechanics. The Born and Jordan paper is the subject of the present article. This paper introduced matrices to physicists. We discuss the original postulates of quantum mechanics, present the two-part discovery of the law of commutation, and clarify the origin of Heisenberg's equation. We show how the 1925 proof of energy conservation and Bohr's frequency condition served as the gold standard with which to measure the validity of the new quantum mechanics.

We present experiments designed to illustrate the basic concepts of statistical mechanics using a gas of “motorized molecules.” Two molecular motion machines are constructed. The pressure fluctuation machine (mechanical interaction simulator) is a working model of two gases separated by a movable piston. The Boltzmann machine (canonical simulator) is a working model of a two-level quantum system in a temperature bath. Dynamical probabilities (fraction of time) are measured using mechanical devices, such as stop watches and motion sensors. Statistical probabilities (fraction of states) are calculated using physical statistics, such as microcanonical and canonical statistics. The experiments enable one to quantitatively test the fundamental principles of statistical mechanics, including the fundamental postulate, the ergodic hypothesis, and the statistics of Boltzmann.

We present an introduction to quantum mechanics based on the formal correspondence between the atomic properties of quantum jumps and the classical harmonics of the electron’s periodic motion. By adding a simple quantum condition to the classical Fourier analysis, we readily find the energies of the stationary states, calculate the transition probabilities between the states, and construct the line spectrum of the emitted light. We provide examples to illustrate the asymptotic, and sometimes exact, agreement between the classical-quantum results (Fourier harmonics) and the exact quantum results (Heisenberg harmonics).