On the Canonical Relativistic Kinematics of N-Particle Systems
The ``canonical'' relativistic kinematics is discussed for n-particle systems. Our aim is to derive the formulas in as simple and symmetrical a way as possible and thus to avoid the extra complication...
The ``canonical'' relativistic kinematics is discussed for n-particle systems. Our aim is to derive the formulas in as simple and symmetrical a way as possible and thus to avoid the extra complication...
Correlation Functions and the Critical Region of Simple Fluids
The ``classical'' (e.g. van der Waals) theories of the gas-liquid critical point are reviewed briefly and the predictions concerning the nature of the singularities of the coexistence curve, the speci...
The ``classical'' (e.g. van der Waals) theories of the gas-liquid critical point are reviewed briefly and the predictions concerning the nature of the singularities of the coexistence curve, the speci...
Momentum Distribution in the Ground State of the One-Dimensional System of Impenetrable Bosons
J. Math. Phys. 5, 930 (1964); doi:10.1063/1.1704196
Issue Date: July 1964
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Girardeau has shown that an exact analytical formula may be given for the ground-state wave-function of a system of one-dimensional impenetrable bosons. Starting with this formula, we give a mathematically rigorous analysis leading to the determination of major features of the momentum distribution in the limit of an infinitely large system.
©1964 The American Institute of Physics
| History: | Received 4 February 1964 |
| Permalink: |
http://link.aip.org/link/?JMAPAQ/5/930/1 |
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REFERENCES (20)
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- M. Girardeau, J. Math. Phys. 1, 516 (1960).
- N. N. Bogoliubov, J. Phys. USSR 11, 23 (1947).
- T. D. Schultz, J. Math. Phys. 4, 666 (1963).
- U. Grenander and G. Szegö, Toeplitz Forms and their Applications (University of California Press, Berkeley, California, 1958).
- O. Penrose and L. Onsager,
Phys. Rev. 104, 576 (1956) . - E. H. Lieb and W. Liniger,
Phys. Rev. 130, 1605 (1963) . - E. H. Lieb, Phys. Rev. 130, 2518 (1963), footnote 15.
- Reference 6, footnote 6.
- We put
![[h-bar]](http://scitation.aip.org/stockgif3/planck.gif)
= 1throughout. - Since we are dealing with a single quantum state for the whole system, namely the ground state, this terminology is actually a misnomer. But there is no point in avoiding it as long as its meaning is clearly understood.
- M. Loeve, Probability Theory (D. Van Nostrand Company, New York, 1955), Chap. IV.
- For even N the wave vectors in the fermion problem must be half-odd integral multiples of the basic unit 2
/L, cf. Ref. 8. - This notation will be employed for the limit (8).
- This has been found independently by Professor F. J. Dyson of the Institute for Advanced Study, Princeton, New Jersey.
- See, for instance, the articles by F. J. Dyson, J. Math. Phys. 3, 140, 157, and 166 (1962). The perceptive reader will notice that our ground-state wavefunetion has an interpretation in terms of Dyson's one-dimensional “Coulomb gas on a circle” which, in turn, is related to the eigenvalue distribution of his random matrices.
- In fact, for some purposes the multiple integral seems to offer advantages. Dyson (in an unpublished lecture at the Eastern Theoretical Physics Conference of October 1963) has shown how a not rigorous but very suggestive argument may be based on it, indicating an asymptotic property for large N.
- Communicated privately to the author by Professor G. Szegö.
- Here and in the following we set L = N which corresponds merely to a choice for the unit of length, but for conceptual clarity we occasionally refer to L in the notation.
- In the remainder of this section we suppress the dependence on
which is regarded as a fixed parameter. For the notation used here the reader is advised to consult Appendix 2 or Ref. 4. - To keep the notation uncluttered we suppress the range of indices in the sums and determinants. It is the first N−1 positive integers.
