Book contents
- Frontmatter
- Contents
- Preface
- 1 Foundations of quantum statistical mechanics
- 2 Elementary examples
- 3 Quantum statistical master equation
- 4 Quantum kinetic equations
- 5 Quantum irreversibility
- 6 Entropy and dissipation: the microscopic theory
- 7 Global equilibrium: thermostatics and the microcanonical ensemble
- 8 Bose–Einstein ideal gas condensation
- 9 Scaling, renormalization and the Ising model
- 10 Relativistic covariant statistical mechanics of many particles
- 11 Quantum optics and damping
- 12 Entanglements
- 13 Quantum measurement and irreversibility
- 14 Quantum Langevin equation and quantum Brownian motion
- 15 Linear response: fluctuation and dissipation theorems
- 16 Time-dependent quantum Green's functions
- 17 Decay scattering
- 18 Quantum statistical mechanics, extended
- 19 Quantum transport with tunneling and reservoir ballistic transport
- 20 Black hole thermodynamics
- A Problems
- Index
- References
16 - Time-dependent quantum Green's functions
Published online by Cambridge University Press: 05 August 2015
- Frontmatter
- Contents
- Preface
- 1 Foundations of quantum statistical mechanics
- 2 Elementary examples
- 3 Quantum statistical master equation
- 4 Quantum kinetic equations
- 5 Quantum irreversibility
- 6 Entropy and dissipation: the microscopic theory
- 7 Global equilibrium: thermostatics and the microcanonical ensemble
- 8 Bose–Einstein ideal gas condensation
- 9 Scaling, renormalization and the Ising model
- 10 Relativistic covariant statistical mechanics of many particles
- 11 Quantum optics and damping
- 12 Entanglements
- 13 Quantum measurement and irreversibility
- 14 Quantum Langevin equation and quantum Brownian motion
- 15 Linear response: fluctuation and dissipation theorems
- 16 Time-dependent quantum Green's functions
- 17 Decay scattering
- 18 Quantum statistical mechanics, extended
- 19 Quantum transport with tunneling and reservoir ballistic transport
- 20 Black hole thermodynamics
- A Problems
- Index
- References
Summary
Introduction
Mathematically, given a linear differential operator Lx,
one encounters the solution to the inhomogeneous differential equation
Here ρ(x) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let
G(x, y) is the Green's function. This is a function of x with y a parameter. Take G(x, y) to satisfy the same boundary conditions as φ(x). Then
since
An example of Lx is, of course, the Schrödinger operator
We take ρ(x) = V(x)ψ(x, t),ψ(x, t) being the wave function and V(x) the potential operator.
We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al., 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000).
One- and two-time quantum Green's functions and their properties
Let us introduce the creation operator ψ†(r, t) and annihilation operator ψ(r, t) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles:
The Hamiltonian operator for the particles is
and the number density of particles at rt is the operator
Note that r = (r1 … rN) for (1, 2, 3, …, N) and V(|r − r′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.
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- Quantum Statistical Mechanics , pp. 281 - 302Publisher: Cambridge University PressPrint publication year: 2009