Approximate Solutions of the Bethe-Salpeter Equation
A modified representation of the Bethe-Salpeter wave function for scalar particles interacting via a massless scalar field is presented and related to Salpeter's approximate wave function. A procedure...
A modified representation of the Bethe-Salpeter wave function for scalar particles interacting via a massless scalar field is presented and related to Salpeter's approximate wave function. A procedure...
Generalization of the ``Edge-of-the-Wedge'' Theorem
It is proved that two analytic functions of several complex variables, having the same boundary values when the imaginary parts of the variables tend to zero inside two arbitrary, but fixed, open cone...
It is proved that two analytic functions of several complex variables, having the same boundary values when the imaginary parts of the variables tend to zero inside two arbitrary, but fixed, open cone...
Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension
J. Math. Phys. 1, 516 (1960); doi:10.1063/1.1703687
Issue Date: November 1960
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A rigorous one-one correspondence is established between one-dimensional systems of bosons and of spinless fermions. This correspondence holds irrespective of the nature of the interparticle interactions, subject only to the restriction that the interaction have an impenetrable core. It is shown that the Bose and Fermi eigenfunctions are related by 
B=FA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one-one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many-boson problem.
Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
B=FA, where A(x1 … xn) is +1 or −1 according as the order pq … r, when the particle coordinates xj are arranged in the order xp<xq< … <xr, is an even or an odd permutation of 1 … n. The energy spectra of the two systems are identical, as are all configurational probability distributions, but the momentum distributions are quite different. The general theory is illustrated by application to the special case of impenetrable point particles; the one-one correspondence between bosons with this particular interaction and completely noninteracting fermions leads to a rigorous solution of this many-boson problem.
Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
| History: | Received March 3, 1960 |
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http://link.aip.org/link/?JMAPAQ/1/516/1 |
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REFERENCES (17)
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- For the case of odd n, the function A(x1
![[centered ellipsis]](http://scitation.aip.org/stockgif3/cellip.gif)
xn) defined by (3) remains well defined if the xj are interpreted modulo L, in which case A satisfies periodic boundary conditions with periodicity length L. On the other hand, for the case of even n the substitution xi
xi±L changes the sign of A. Hence our general theorem on the one-one correspondence is only valid for a system with periodic boundary conditions if n is odd. There are no restrictions on n for boundary conditions of enclosure in a box. - It is an obvious corollary that also all equilibrium thermodynamic properties of the two systems are identical.
- For a proof see O. Penrose and L. Onsager,
Phys. Rev. 104, 576 (1956) , Sec. 5. - We assume that the ground state is nondegenerate, so that 0F can be chosen to be real.
- (a) Note added in proof. This model was treated some time ago in an unpublished work by J. K. Percus and G. J. Yevick (private communication from Professor Yevick);
- (b) B is not required to satisfy the Schrödinger equation on the surfaces Xi = xl, where it vanishes and suffers discontinuities in gradient (but not in value) as a result of the infinite repulsive forces.
- A. C. Aitken, Determinants and Matrices (Oliver and Boyd, Ltd., Edinburgh, 1951), 7th ed., p. 112.
- (a) Note added in proof. It is possible, by a simple change of variables, to extend the results for the ground state of the impenetrable point Bose gas so as to obtain the exact solution for the ground state of a gas of hard-sphere bosons with diameter a>0 enclosed in a one-dimensional periodic box. The resultant expression for the ground-state energy,
, differs only by a “surface” term from a result obtained previously for a Boltzmann system of one-dimensional hard spheres enclosed in a box
- [see R. J. Rubin, J. Chem. Phys. 23, 1183 (1955) for other references]. This agreement is to be expected, since the ground state of a Boltzmann system is identical with that of the corresponding Bose system.
- It is amusing that there exists a soluble one-dimensional problem in the classical statistical mechanics of interacting particles in which all configurational probability distributions are the same as those of the free Fermi gas, hence the same as those of our interacting Bose system; the interparticle interaction in this classical problem is, of course, different from our point interaction.
- It follows from (4) that, for the case of periodic boundary conditions, the momentum of the state B is the same as that of F. Since P =
j(![[h-bar]](http://scitation.aip.org/stockgif3/planck.gif)
/i)![[partial-derivative]](http://scitation.aip.org/stockgif3/part.gif)
/xj is the total linear momentum operator, one has . But it follows from the definition of A [Eq. (3) ff.] that the function PA vanishes except on the planes Xj = xl, where
F vanishes. Hence if PF = kF, then , Q.E.D.
- A. C. Aitken, footnote reference 6, p. 116, prob. 2.
- Here we are labeling the state by its phonon wave number k rather than by the integer j, which is related to k by k = 2
j/L. - Those familiar with Siegert's work on field operators for bosons with impenetrable cores [A. J. F Siegert,
Phys. Rev. 116, 1057 (1959) ] may raise the objection that the free-particle annihilation and creation operator ak and ak† cannot be used in treating particles with hard cores.
Such and objection is based on a misinterpretation of the significance of Eq. (A4) of Siegert's paper: - I am indebted to Professor E. P. Gross for suggesting the following method of finding the ground state vector |0B
. - N. N. Bogolubov, J. Phys. (U.S.S.R.) 11, 23 (1947).
- Note that a−k†(
)![[not-equal]](http://scitation.aip.org/stockgif3/ne.gif)
[a−k()]†. - M. Girardeau and R. Arnowitt,
Phys. Rev. 113, 755 (1959) , Appendix A. - The fact that n0 is not proportional to n is connected with the fact that the ground-state wave function (12) possesses long-range order in view of the very slow rate of change of the factors sin[L−1(xj−xl)];
see O. Penrose and L. Onsager, footnote reference 3. This long-range order does not show up in the pair correlation function (15), or indeed those of any finite order; one has to go to the many-body correlation functions to see it.
, where (x) and †(x), respectively, annihilate and create a boson at point x. If this equation were necessary property of the Bose field operators for particles with hard cores of diameter a, then the objection would be justified. However, Siegert's Eq. (A4) is a sufficient, but not a necessary, condition for the vanishing of the Schrödinger wave function when hard cores overlap. All that is necessary is the much weaker condition
, where | is any state vector descibing particles with impenetrable cores of diameter a (for the case discussed in this Appendix, a = 0). It is more convenient for our purposes to retain field operators (x), †(x) satisfying the usual Bose commutation rules, and hence to interpret the above equation as a subsidiary condition on allowable state vectors |; this subsidiary conditions is merely a transcription of (1) into the language or language of quantized fields.
