By E. Poisson

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**Extra resources for A Relativist's Toolkit - The Math of Black Hole Mechanics**

**Example text**

1 ; N /H Nˆ Z Z d 1 0 . 2/ ˆ . 2/ d 2 hˆ 0 j O . 1 / O . 2 /hO 12 O . 2 / O . 3). 36), thereby enabling us to simplify the notation considerably. 7 Summary of Two Equivalent Descriptions The preceding considerations have shown that there are at least two equivalent descriptions for many-particle systems in that they yield the same matrix elements and hence the same probabilities for observables quantum mechanically. 39) i D1 j Di C1 Its eigenfunctions are obtained by simultaneously solving the Schrödinger equation: HO ˆ .

We can also see a marked difference between bosons and fermions at low temperatures for each thermodynamic quantity. 18), we may regard the lines of U as the quantum equations of state. 4 High-Temperature Expansions 51 compared with the classical result. This implies that there is an effective attraction (repulsion) between each pair of bosons (fermions) with the same ˛ due to the permutation symmetry. Whereas classical entropy shows unphysical behavior in falling below 0 and diverging to 1, entropy for bosons and fermions appropriately approaches 0 as T !

Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3rd edn. (Butterworth-Heinemann, Oxford, 1991) 4. O. Penrose, Foundations of Statistical Mechanics (Pergamon, Oxford, 1970). The footnote of p. 213 5. J. Sakurai, Modern Quantum Mechanics, rev. edn. (Addison-Wesley, New York, 1994) 6. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948) 7. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955) Chapter 3 Quantum Mechanics of Identical Particles Abstract In general, superconductivity occurs in a system of identical particles; specifically, the conduction electrons in metals.

### A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson

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