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Canadian Journal of Physics

Tipping time of a holonomic quantum cylinder

Mark R. A. Shegelski, Jamie Sanchez-Fortún Stoker, Ian Kellett

Published Sep 1, 2011• 0 citations
Publisher

Abstract

The classical and quantum mechanical Hamiltonians for a cylinder subject to holonomic constraints are derived. The quantum mechanical Hamiltonian is simplified and cast into a dimensionless form. The tipping time of a quantum mechanical cylinder subject to gravity is calculated. Numerical solutions for an appropriate initial wave function are obtained. We find that the tipping time is given by 〈t〉tip = t0C1 exp [C2(r/r0)9], where t0 is the time scale, C1 and C2 are constants of order unity, r is the radius of the cylinder, and r0 is the length scale for the tipping. We compare our results with those found in previous works.

DOI
10.1139/p11-086

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The classical and quantum mechanical Hamiltonians for a cylinder subject to holonomic constraints are derived. The quantum mechanical Hamiltonian is simplified and cast into a dimensionless form. The tipping time of a quantum mechanical cylinder subject to gravity is calculated. Numerical solutions for an appropriate initial wave function are obtained. We find that the tipping time is given by 〈t〉tip = t0C1 exp [C2(r/r0)9], where t0 is the time scale, C1 and C2 are constants of order unity, r is the radius of the cylinder, and r0 is the length scale for the tipping. We compare our results with those found in previous works.
We study the tipping time of a quantum mechanical rod that is constrained to move in a plane in a gravitational potential. The initial state of the center of mass of the rod is localized within the base of support of the rod. The tipping time is found to increase exponentially with the height H of the rod as C1t0 exp[C2(H/H0)9], where C1 and C2 are dimensionless constants of order one, t0 is the time scale of the motion, and H0 is the length scale of the rod. We show that the tipping time cannot be obtained using the uncertainty principle alone, and compare our result to that obtained using the Wentzel–Kramers–Brillouin approximation.
We calculate the tipping time of a quantum rod that has a height several times that of the edge length of its square base. We use an expression for the tipping time that has heuristic value, and gives the average time at which, upon measurement, the initially balanced rod is found to tip. We use two methods to calculate the tipping time. One method is to examine the "late time" behaviour of the quantum state of the center of mass of the rod by using an equation that has the form of the time-independent Schrödinger equation except that it involves a "complex energy." The other method uses energy resonances in the eigenstates of the Hamiltonian to determine the tipping time. We use the well-known Wentzel–Kramers–Brillouin approximation to calculate the energy eigenstates. With these methods, we obtain expressions for the tipping time that are valid for very long tipping times. PACS Nos.: 03.65.–w, 03.65.Xp
Abstract Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q(0)r, then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q(0)r.
The statistical mechanics of quantum-classical systems with holonomic constraints is formulated rigorously by unifying the classical Dirac bracket and the quantum-classical bracket in matrix form. The resulting Dirac quantum-classical theory, which conserves the holonomic constraints exactly, is then used to formulate time evolution and statistical mechanics. The correct momentum-jump approximation for constrained systems arises naturally from this formalism. Finally, in analogy with what was found in the classical case, it is shown that the rigorous linear-response function of constrained quantum-classical systems contains nontrivial additional terms which are absent in the response of unconstrained systems.
We study the Laughlin wave function on the cylinder. We find it only describes an incompressible fluid when the two lengths of the cylinder are comparable. As the radius is made smaller at fixed area, we observe a continuous transition to the charge-density-wave Tao-Thouless state. We also present some exact properties of the wave function in its polynomial form. We then study the edge excitations of the quantum Hall incompressible fluid modeled by the Laughlin wave function. The exponent describing the fluctuation of the edge predicted by recent theories is shown to be identical with numerical calculations. In particular, we obtain the occupation amplitudes of edge state n(k) for four to ten electron-size systems. When plotted as a function of the scaled wave vector they become essentially free of finite-size effects. The resulting curve obtains a very good agreement with the appropriate infinite-size Calogero-Sutherland model occupation numbers. Finally, we numerically obtain n(k) of the edge excitations for some pairing states which may be relevant to the \ensuremath{\nu}=5/2 incompressible Hall state.
The problem of a photon reflecting or refracting at an n1-n2 index of refraction is solved in (1) using the notion of continuity of the electric field of a photon and its d/dx derivative for the n1-n2 junction at x=0. The photon electric field behaves as cos(-wt+kx) or cos(-Et+px). We argue first of all that the photon is a quantum object and that this problem is really a quantum mechanical one as it may also be solved for a particle with rest mass using two different constant potentials V1, V2 instead of n1,n2. In this note, we consider the photon case for convenience. Given that the problem is really a quantum mechanical one (in our opinion), we ask: How far can one go by trying to solve it classically? We first point out that for a static uniform arrangement of objects in space, P(x)dx =dx/L, where L is an arbitrary length. In the case of particles moving at constant v, one may argue that each particle spends the same amount of time dt in each dx and that this also leads to a constant density in space. We suggest that for this to hold, one must consider a steady state problem and the notion of flux for physical reasons. This leads to writing the equation 1= Probability(reflect)+Probability(refract) as AA/c = BB/c + CC/c2 where AA, BB, CC are the incident, reflected and refracted fluxes and c1=c, c2=c/n2. We have shown in previous notes that one may mathematically write this single equation as: A+B=C and Ap-Bp = Cp2 using E=pc (with E the same for the incident, reflected and refracted photon) and then solve the two equations for the two unknowns B,C, setting A=1. This approach yields a result which matches experiment, but from the standpoint of classical physics, there is no a priori reason to take the square root of flux. It simply arises as a math trick so to speak, we argue, and there seems to be no physical motivation. Here, we suggest approaching the problem differently. We note that if c1-c2<<1, then there should be almost no reflection, while if c2<<1, there should be almost no refraction. We ask: What happens if one suggests that in the c1-c2<<1 case that one insists on BB dropping as (c1-c2)*(c1-c2), i.e. to second power as in a stable equilibrium and for CC dropping as c2*c2 for c2<<1 in the other extreme. This is a possible approach that seems to be linked to some kind of physical principle, and one may consider the simplest solution and compare it with experiment and find that it holds. In other words, it yields the same solution as the above approaches. Moreover, this approach gives a link with the square root approach of the above paragraph, because now one is linking c-c2 and c2 directly with square root fluxes A,B,C so as to obtain the quadratic feature automatically for c-c2<<1 or c<<1 when one ultimately uses AA, BB, CC. In other words, the assumption of quadratic vanishing of BB and CC, automatically leads to c-c2 or c2 vanishing of B and C, suggesting that one may automatically achieve the quadratic vanishing form by working with A,B,C square root flux variables. We suggest that this idea of quadratic vanishing of BB or CC possibly places the approach of using A+B = C, pA-pB=p2C on more physical grounds and points to a new kind of physics which uses A,B,C, namely quantum mechanics.