Off-Diagonal Long-Range Order and the Momentum Distribution of Electron Pairs in Superconductors, and Helium Atoms in LiquidHe4

The momentum distributions of electron pairs in superconductors and of helium atoms in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are defined by summing the pair or atomic state occupation probability over all internal states of the pair or ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atom corresponding to a given value of total translational momentum. By use of the closure relation for internal wave functions, the momentum distribution of electron pairs is expressed in terms of the two-electron density matrix ${\rho }_2(x_1{x}_2,x_1^{\prime }{x}_2^{\prime })$ of the $2n$-electron system. Similarly, for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atoms, the momentum distribution is expressed in terms of the single-atom density matrix ${\rho }_3(x_1{x}_2\overrightarrow{\mathrm{R}},x_1^{\prime }{x}_2^{\prime }{\overrightarrow{\mathrm{R}}}^{\prime })$ of the $n$-atom system, where $x_1$, $x_2$, and $\overrightarrow{\mathrm{R}}$ are the coordinates of two electrons and an $\alpha$ particle. The equivalence of this definition of the momentum distribution with two other plausible definitions is demonstrated. It is shown that exchange and the exclusion principle produce a strong spreading of the momentum distribution both for a superconductor and for liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, completely suppressing Bose-Einstein condensation in the strong sense of macroscopic occupation of the zero-momentum state. However, a much weaker singularity at zero momentum, characteristic of the off-diagonal long-range order (ODLRO) of ${\rho }_2$ for a superconductor and of ${\rho }_3$ for liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, does occur. By comparison with the hypothetical case of pairs of bosons, it is shown that the exclusion principle (not merely exchange) plays an essential role in broadening the momentum distribution in superconductors and liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$.