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• W2126450396 abstract "A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:math> 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 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:math> and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>E</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>∫</mml:mo> <mml:mo>∫</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>d</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant=normal>X</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>d</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:msub> <mml:mrow> <mml:mi mathvariant=normal>X</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mfenced close=} open={> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mfenced close=] open=[> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msubsup> <mml:mi mathvariant=normal>∇</mml:mi> <mml:mn>1</mml:mn> <mml:mn>8</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi mathvariant=normal>∇</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∫</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=normal>d</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:msub> <mml:mrow> <mml:mi mathvariant=normal>X</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>13</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>23</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> </mml:msub> <mml:mo>⋅</mml:mo> <mml:msub> <mml:mi mathvariant=normal>∇</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mn>13</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mstyle> </mml:math> 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." @default.
• W2126450396 created "2016-06-24" @default.
• W2126450396 creator A5054196943 @default.
• W2126450396 date "1960-05-31" @default.
• W2126450396 modified "2023-09-27" @default.
• W2126450396 title "A variational method for the ground state of a Bose fluid" @default.
• W2126450396 cites W1980156840 @default.
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• W2126450396 doi "https://doi.org/10.1098/rspa.1960.0095" @default.
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