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| QUANTUM HARD SPHERES
| | == Path integral formulation == |
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| The great usefulness of the hard-sphere model for representing particles in classical statistical mechanics is very well known and its study has provided guidance in the understanding of classical fluids and solids. This model assumes that pairwise interactions between particles are singular in that they become an infinite repulsion for distances smaller than the diameter of the spheres, being identically zero otherwise. Perhaps, the most remarkable feature of classical hard spheres is that they show a fluid-solid transition, which was first predicted with computer simulation [Wood & Jacobson, J. CHEM. PHYS. 27, 1207 (1957); Alder & Wainwright, J. CHEM. PHYS. 27, 1208 (1957)], and confirmed later with experiments on colloidal particles (Pusey & van Megen, NATURE 320, 340 (1986). From the thermodynamic point of view the states of this model only need one parameter to be characterized: the (number) density. This classical state of affairs implies that the quantum thermal de Broglie wavelength of the particles is zero. With the use of reduced units (unit length= hard-sphere diameter) the results arising from this singular interaction potential (hard core) can be transferred between systems differing in the size of their spheres. Among the many interesting features displayed by this model one should mention that there is “contact” between particles at distances = diameter (+) (they are like hard billiards), which reflects in the fact that the main peak of the pair radial correlation function is located just at that “contact” point.
| | Dear 62.204.197.227, |
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| Nevertheless, the switching on of the quantum conditions upon this system (i.e. nonzero de Broglie wavelengths) changes dramatically the classical properties. To illustrate this three examples will suffice. First, the characterization of the state points requires an additional parameter, the thermal wavelength, which contains the temperature, the mass of the particles, and Planck’s constant (once again, using reduced units allows one to transfer results between situations at the same values of the density and of the de Broglie wavelength). Secondly, the above classical “contact” is forbidden, as quantum hard spheres repel each other before getting into (classical) contact. And thirdly, the fluid-solid phase transition is driven by energy in the quantum limit of zero temperature, this being different from the classical case which is driven by entropy. Furthermore, quantum hard spheres seem appropriate to understand the low temperature properties of ultrahard materials and colloids. In this endeavour Feynman’s path-integrals combined with computer simulations provide a very powerful tool to undertake the pertinent calculations. Thus, apart from being an appealing mathematical problem, quantum hard spheres can be very useful from a practical standpoint for the design of new materials.
| | The references concerning path integrals have been moved to the [[path integral formulation]] page. --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 18:35, 16 October 2007 (CEST) |
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| The UNED group is engaged in the study of this system and over the years has published a number of research papers dealing with the properties of the quantum hard-sphere system under conditions covering from the fluid to the solid states. The titles of these papers in the list given in "References" are a good summary of the findings and the results obtained. Attention has been focused upon the equilibrium properties, thermodynamic and structural. It is worth realizing that, while there is only one pair radial correlation function in the classical case, the quantum delocalization brings about three different pair radial correlation functions in the quantum case, each of them possessing a definite physical meaning. A great emphasis has been placed on the study of the different structural functions, in both the r-correlation and the k-Fourier spaces that can be determined in a quantum many-body system, as they open an alternative way to carry out computations leading to the fixing of the equation of state. This effort has helped to clarify the role of the path-integral centroids in (equilibrium) quantum statistical mechanics, and also the possibilities of utilizing Ornstein-Zernike classical frameworks in dealing with quantum fluids.
| | == hbar expansion? == |
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| | Circa 2005 I saw a presentation on a quantum "gas in a box". The primary ingredient was an expansion to first-order in hbar. The primary result was that wave functions were space-filling fractals. If one starts at t=0 with all of the gas on one side of the box, then the wave functions interfere destructively on the empty side. Since time evolution is unitary, and the gas is a mix of energies, the result is that at time t>0 the wave functions decohere and start to fill the empty side. Thus, to first order in hbar, the system appears to be ergodic. Did anyone else see this lecture? Anyone have references to this? Search engines cannot find it. [[Special:Contributions/67.198.37.16|67.198.37.16]] 18:55, 1 February 2024 (CET) |
Path integral formulation[edit]
Dear 62.204.197.227,
The references concerning path integrals have been moved to the path integral formulation page. -- Carl McBride (talk) 18:35, 16 October 2007 (CEST)
hbar expansion?[edit]
Circa 2005 I saw a presentation on a quantum "gas in a box". The primary ingredient was an expansion to first-order in hbar. The primary result was that wave functions were space-filling fractals. If one starts at t=0 with all of the gas on one side of the box, then the wave functions interfere destructively on the empty side. Since time evolution is unitary, and the gas is a mix of energies, the result is that at time t>0 the wave functions decohere and start to fill the empty side. Thus, to first order in hbar, the system appears to be ergodic. Did anyone else see this lecture? Anyone have references to this? Search engines cannot find it. 67.198.37.16 18:55, 1 February 2024 (CET)