# Theory of Quantum Fluids

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Theory of Quantum Fluids is a concise report on the microscopic description of liquid 4He and liquid 3He in the physical density range using simple forms of the potential function between pairs of neutral atoms and the properties of the ground states and limited ranges of low excited states. The monograph covers the properties of the radial distribution function and the three-particle distribution particle; the classical sound field and the correspondence principle; paired phonon states in the free- phonon approximation; the uniform limit and the charged boson system; and the microscopic theory of a single 3He atom in the 4He liquid. Theoretical and experimental physicists will find the book very interesting.

## Table of Contents

Preface

Introduction

Chapter 1. Properties of the Radial Distribution Function

1.1. Definition and General Properties of Distribution Functions

1.2. Radial Distribution Function and Liquid Structure Function

1.3. Sum Rules Involving ρk

1.4. A Fluctuation Theorem for the Ground State

1.5. Examples: Ground States of Noninteracting Boson and Fermion Systems

1.6. Distribution Functions under Equilibrium Conditions

1.7. Necessary Conditions on the Radial Distribution and Liquid Structure Functions

Appendix 1-A. Fourier Analysis of Fluctuations

Appendix 1-B. Families of Radial Distribution Functions

References

Chapter 2. The Three Particle Distribution Function

2.1. Introduction

2.2. Approximate Forms

2.3. The Abe Form

2.4. The Abe Form in the Uniform Limit

2.5. The Hypernetted-Chain Approximation in the Uniform Limit

2.6. A Dynamical Consistency Condition

Appendix 2-A. A Convolution Form for p(4) and Asymmetrical Forms for p(3) and p(4)

Appendix 2-B. Reduction of Eq. (2.42) for Z3 + 1/3F3

References

Chapter 3. The Classical Sound Field and the Correspondence Principle

3.1. Introduction

3.2. Theory of Density Fluctuations

3.3. Quantum Theory of the Continuous Medium

3.4. Consequences of the Long-Range Correlations in the Particle Formulation

Appendix 3-A. Nonlocal Potential Energy in the Theory of Density Fluctuations in the Classical Fluid

Appendix 3-B. Variance and Independence in the |ρk|2 Space

References

Chapter 4. Elementary Excitations in a Boson System-Landau Spectrum

4.1. The Bijl-Feynman Dispersion Formula

4.2. Consequences of Sum Rules

4.3. Improved Theory of the Elementary Excitations

4.4. Phonon-Phonon Interaction

Appendix 4-A. Sum Rules for ω2 and ω3

Appendix 4-B. Evaluation of the Overlap Integraland the Interaction Matrix Element Eqs. (4.39) and (4.40)

References

Chapter 5. Paired Phonon States in the Free-Phonon Approximation

5.1. Introduction

5.2. The Generalized Normalization Integral

5.3. Generalized Distribution Functions

5.4. ρ ± k Model Space and Matrix Elements

5.5. A Normalized Orthogonal Basis in "k" Function Space

5.6. Evaluation of

5.7. Eigenvalues of Η in the "k" Function Space

5.8. Representation of ρk in Terms of Creation and Annihilation Operators

5.9. The Ground-State Eigenfunction

5.10. Evaluation of S'(k)

5.11. Consequence of ω(&) = 0 in the HNC Approximation

Appendix 5-A. Matrix Elements of

Appendix 5-B. Solution of Eq. (5.25)

Appendix 5-C. Contribution of the Three-Phonon Vertex to the Binding Energy of the Boson System

References

Chapter 6. The Boson System at Absolute Zero

6.1. Introduction

6.2. The Extremum Property of the Kinetic Energy

6.3. Iteration-Variation Procedure

6.4. The Virial Theorem

6.5. Numerical Results

6.6. Explicit Trial Functions for g(r) in the Helium Problem

6.7. Corrections to <0|H|0>

References

Chapter 7. The Uniform Limit and the Charged Boson System

7.1. Introduction

7.2. Expectation Value of Η in the Uniform Limit

7.3. Optimum Choice of the Liquid Structure Function

7.4. The Yukawa Potential

7.5. The Charged Boson System

7.6. First-Order Energy Correction for the Boson System

7.7. First-Order Correction to the Liquid Structure Function

7.8. Corrections from the "Paired-Phonon" Analysis

Appendix 7-A. Evaluation of Certain Definite Integrals

References

Chapter 8. Correlated Basis Functions for Fermion Systems. Diagonal Matrix Elements

8.1. Introduction

8.2. The Generalized Normalization Integral

8.3. Additive Approximants and Cluster Integrals—Iwamoto-Yamada (IY) Formalism

8.4. Additive Approximants and Cluster Integrals—Aviles-Hartogh-Tolhoek (AHT) Formalism

8.5. Multiplicative Approximants and Cluster Integrals

8.6. Explicit Formulas for Cluster Integrals and Ground-State-Energy Quantities for Liquid 3He

Appendix 8-A. Exponential Form of the Cluster Expansion for the Generalized Normalization Integrals

Appendix 8-B. Integration Formula for a Spherical Region

References

Chapter 9. Correlated Basis Functions for Fermion Systems. Nondiagonal Matrix Elements and Perturbation Formulas

9.1. Generating Functions

9.2. Nondiagonal Cluster Integrals

9.3. Matrix Elements of the Identity

9.4. Matrix Elements of the Hamiltonian

9.5. Matrix Representations of the Identity and Hamiltonian

9.6. Evaluation of the Ground-State Energy

9.7. Formal Transformation to an Orthonormal Basis System

9.8. Method of Limited Orthogonalization

References

Chapter 10. Low Excited States and Statistical and Transport Properties of Liquid 3He

10.1. Quasiparticle Formulation

10.2. Quasiparticle Energy Spectrum

10.3. Summary of Formulas for Equilibrium and Transport Properties

10.4. Numerical Results

References

Chapter 11. Theory of a 3He Atom in Liquid 4He at Τ = 0

11.1. Statement of the Problem

11.2. Reduction of Matrix Elements

11.3. Numerical Evaluation of Matrix Elements and Energy

11.4. Effective Mass

11.5. The Differential Volume Coefficient

References

Author Index

Subject Index

## Product details

- No. of pages: 280
- Language: English
- Copyright: © Academic Press 1969
- Published: January 1, 1969
- Imprint: Academic Press
- eBook ISBN: 9780323157933