A variational method for the ground state of a Bose fluid

A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f 12 ≡ f ( x 1 , x 2 ) of a Bose fluid, such as liquid 4 He, at absolute zero. The assumptions made are: (i) two-body interactions with potential U ij , (ii) trial wave functions of the form f 12 ≡ f ( x 1 , x 2 ) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E = 1 2 n 2 ∫ ∫ d 3 X 1 d 3 X 2 f 12 U 12 − h 2 m − 1 ( ∇ 1 8 ϕ 12 ) + ( ∇ 1 ϕ 12 ) 2 + n ∫ d 3 X 2 f 13 f 23 ϕ 12 ⋅ ∇ 1 ϕ 13 , where n ≡ N/V . It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f . The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.