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Physica A Statistical Mechanics and its Applications

Exact classical and quantum mechanics of a particle coupled to a membrane

H. Dekker

Published Feb 1, 1985• 5 citations
Publisher
DOI
10.1016/0378-4371(85)90182-7

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Abstract Both classical mechanics and quantum mechanics explain the Brownian motion. However, it is unclear whether they are compatible with each other as the physical and mathematical identity of the wavefunction in quantum mechanics has been elusive. Here, a continuum theory using grammars in classical mechanics modeling but compatible with the quantum wavefunction is introduced. The theory explains the confined Brownian motion of cell membrane inclusions interacting with extracellular matrices or cytoskeletons via elastic molecular crosslinkers. This crosslinker theory is combined into the Canham-Helfrich-Evans model for fluid membranes. Calculations through the provision of a finite element method for the combined theory reproduced measured data from adhesion molecular machineries and cell membranes. Overall, by providing physical and mathematical interpretations of the quantum wavefunction, the presented theoretical model provides improved capabilities for the realistic simulation of classical and quantum biomechanical aspects of cell membranes and membrane linker proteins.
Abstract We use an exact solution of the elastic membrane shape equation, representing the curvature, which will serve as a quantum potential in the quantum mechanical two dimensional Schrödinger equation for a (quasi-) particle on the surface of the membrane. Surface curvature in the quasi one-dimensional case is related to an unexpected static formation: on one hand the elastic energy has a maximum where surface curvature has a maximum and on the other hand the concentration of the expectation value to find the (quasi-) particle is again where the elastic energy is concentrated, namely where surface curvature has a maximum. This represents a particular form of a conformon.
This chapter gives an introduction to optomechanics with ultracold atoms. The opening half deals with optomechanical atom–light interactions. Section 9.2 introduces atom trapping. Section 9.3 discusses the properties of trapped atoms as mechanical oscillators. Section 9.4 describes optomechanical interactions, treating the atoms as polarizable particles, a model used in section 9.5 to derive optomechanical coupling of atoms and a cavity field and briefly review cavity optomechanics experiments with atoms in the quantum regime. The second half deals with hybrid mechanical-atomic systems. We start with an overview of different coupling mechanisms, then focus on light-mediated interactions and derive the coupling of a membrane to an ensemble of laser-cooled atoms. Section 9.8 reviews experiments on sympathetic cooling of a membrane with cold atoms, with perspectives for mechanical quantum control discussed in section 9.9. Section 9.10 introduces the possibilities that arise if the mechanical oscillator is coupled to the atomic internal state.