Lecture 3-7: General formalism at finite temperature
Reference: Negele & Orland (N&O) Chapter 2
Lecture 3
Introduction
Quantum statistical mechanics (Home reading)
Three ensembles: microcanonical, canonical, grand canonical essemble
Partition function and thermodynamics
Physical response functions and Green’s function
Consider a system subjected to a time-dependent external field,
where the operators and corresponding states are in the Schrödinger picture. We shall study the system through the evolution operator.
Time-ordered operator product
where
Time-ordered exponential
t t where b a and t t n. It may be expanded in a Taylor series as follows, M n a
The evolution operator
Using the time-ordered exponential, the evolution operator may be written
It is easy to verify that it satisfies the equation of motion
and the boundary condition
The response to an infinitesimal perturbation in the external field
The Schrödinger picture and the Heisenberg picture. (Home reading)
The response of a wavefunction to an infinitesimal perturbation by an external field is given by the functional derivative
where the operator and the state in the Heisenberg picture is related to the operator in the Schrödinger picture by
and
Now, consider the response of the expectation value of an operator to an infinitesimal perturbation in the external field.
ˆ ˆ The response of a measurement of O2 (t2 ) to a perturbation couple to O1 is specified by the response function,
The above is one of century results in this chapter.
The n-body real-time Green’s function
The n-body imaginary-time Green’s function
where
Approximation strategies (Home reading)
Asymptotic expansions
Weak coupling and strong coupling Functional integral formulation
A powerful tool for the study of many-body systems
The Feynman path integral for a single particle system
A different formulation to the canonical formalism, the propagator (or the matrix element of the evolution operator) plays the basic role.
From canonical formalism to path integral
The key step is to evaluate the matrix element for the infinitesimal evolution operator
The normal-ordered operator product and exponential
For a single particle in a potential,
The normal-ordered exponential reads,
The normal-ordered exponential is related to the time-ordered exponential through,
The difference is of order 2 .
In the limit 0 , the normal-ordered exponential yields the correct evolution of the wavefunction,
The integral over p is a Gaussian integral yielding
Taking the limit M in the discrete version,
one obtains the Feynman path integral,
where the measure is
An alternative starting point for the formulation of quantum mechanics
The Lagrangian plays important role in quantum mechanics too, especially for the quantization for constraint systems.
So far there is no rigorously mathematical proof for the real time path integral.
A typical trajectory contributing to a path integral
The path integral automatically represent time-ordered products
Imaginary-time path integral and the partition function
Mathematically, it is well defined. Formal unifying for quantum mechanics and statistical mechanics.
Consider the partition function for a single particle system
which may be thought of a sum over close-path propagators in the imaginary time,
Similar to the real time case,
An alternative derivation shows explicitly how the Lagragian in the real time case is transformed into the Hamiltonian in the imaginary time case. (Home reading) Coherent state functional integral
For a general many-particle system expressed in second quantization form, a functional integral representation for the many-particle evolution operator may be obtained using the coherent states.
The closure relation for boson or fermion,
We assume the Hamiltonian is written in normal form, so that
The propagator may be written as
The exponent can be written symbolically
Taking the limit M , we obtain that
where
The partition function for many-particle systems (Home reading)
Homework:
1) Problem 2.3
2) Derive the partition function for non-interacting systems (2.72) using functional integral.
3) Problem 2.6