scholarly journals Minimal Eigenvalue of the Coxeter Laplacian for the Symmetrical Group

1994 ◽  
Vol 167 (2) ◽  
pp. 460-472 ◽  
Author(s):  
R. Bacher

In a recent article F. Hund has treated the problem of the specific heat of the hydrogen molecule on the basis of the wave mechanics. The total number of rotational states are divided due to the homopolar character of the molecule into two groups, to the one of which belong wave functions symmetrical in the two nuclei, and to the other wave functions which are antisymmetrical in the nuclei. Hund has suggested that the presence of both groups in hydrogen may be accounted for by assuming that the nuclei possess a spin, in which case transitions between symmetrical or between antisymmetrical states will have their usual intensity but transitions between symmetrical and antisymmetrical states will be very weak, of the order of the coupling of the nuclear spins. He then writes the following expression for the rotational specific heat, C r /R = σ 2 d 2 / d σ 2 log Q, Q = β [1 + 5 e -6σ + 9 e -20σ + ...] + 3 e -2σ + 7 e -12σ + 11 e -30σ +...., (1) where σ = h 2 /8π 2 I k T and β is the ratio of the weights of the symmetrical group of states to the antisymmetrical group. Hund has found that he obtains a close agreement between (1) and the observed specific heat curve only when β has about the value 2, that is when the symmetrical states have twice the weight of the antisymmetrical. He further obtains for this case I = 1·54 × 10 -41 gm. cm. 2 , the moment of inertia of the H 2 molecule.


Universe ◽  
2018 ◽  
Vol 4 (11) ◽  
pp. 127 ◽  
Author(s):  
Carlo Rovelli ◽  
Francesca Vidotto

We show that the expected lifetime of white holes formed as remnants of evaporated black holes is consistent with their production at reheating. We give a simple quantum description of these objects and argue that a quantum superposition of black and white holes with large interiors is stable, because it is protected by the existence of a minimal eigenvalue of the area, predicted by Loop Quantum Gravity. These two results support the hypothesis that a component of dark matter could be formed by small black hole remnants.


1914 ◽  
Vol 34 ◽  
pp. 76-88
Author(s):  
W. R. Lethaby

The Square Pedestals.—In some notes on the sculpture from the Artemision at the British Museum, printed in the last volume of this Journal (p. 87), I suggested that the fragment No. 1201 most probably belonged to a relief representing either Herakles in the Garden of the Hesperides or Herakles and the Hydra. Subsequent examination and the attempt to make a restoration from the given data have made me sure that the former was the subject of the sculpture. Only this would account for the quiet action of the left hand of Herakles and for the closely associated female figure. If this were indeed the subject, how could its normal elements be arranged so as to suit the conditions of the square pedestal having a vertical joint in the centre, and making proper use of the existing fragment of which Fig. 1 is a rough sketch? This question I have tried to answer. The fragment is now fixed in the side of a built-up pedestal close to its left-hand angle, but there is nothing which settles this position and it is a practically impossible one, for there is not room left in which to complete the figure of Herakles. If, however, we shift the piece to the right hand half of the pedestal, and sketch in the completion of the two figures, we at once see how perfectly the tree and serpent would occupy the centre of the composition and leave the left-hand space for the two other watching maidens—the whole making a symmetrical group.


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