We examine under this heading the transition amplitude in the Feynman path integral, which we will discuss shortly and will be reduced to an integral over the continuum degrees of freedom, such as a set of p i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_1.jpg"/> and q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_2.jpg"/> . This will involve harmonic (non-interacting) systems that are reduced to the problem of multiple Gaussian integrals over potentially coupled degrees of freedom, q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_3.jpg"/> that we develop appropriate Feynman path integral tools for their solution. Considering that the harmonic oscillator is one of the central objects of theoretical physics then Gaussian integrals are likely to be the key tools of theoretical physics. This will be demonstrated utilizing Feynman path-integral formulation of quantum mechanics where Gaussian integrals will be pivotal for computation in quantum mechanics, quantum statistical mechanics and generally in quantum field theory. It will be interesting to introduce the Maslov correction [ 11 , 12 ] to the wave function which is a jump of π 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline4_4.jpg"/> in the phase when a system passes through a caustic point, a phenomenon related to the second variation as well as to the geometry of paths. This phenomenon is applicable to any system using the quasi-classical approximation. The harmonic oscillator will be one of the objects amongst others to illustrate this.
In quantum statistical mechanics one has the fundamental object ρ(β; x a ,x b ), a matrix entry which defines the density matrix ρl, given in the Feynman formulation by the Euclidean path integral of the preceding chapter: $$\begin{array}{*{20}{c}} {\rho {{{(\beta ;{{x}_{a}},{{x}_{b}})}}^{{\underline{\underline {def}} }}}\int_{{{{x}_{a}} = x(0)}}^{{{{x}_{b}} = x(\beta \hbar )}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]} } \\ { = {{I}_{E}}(0,\beta \hbar ,{{x}_{a}},{{x}_{b}})} \\ \end{array}$$ (14.1.1) where
β=kT1 is the inverse temperature
T1 up to a constant
k1, k being Boltzmann’s constant. From ρ one obtains the all-important partition function Ζ as a trace: $$\begin{array}{*{20}{c}} {Z = \int_{{ - \infty }}^{\infty } {\rho (\beta ;x,x)dx} } \hfill \\ { = \int_{{ - \infty }}^{\infty } {\int_{{x(0) = x}}^{{x(\beta \hbar ) = x}} {\exp ( - {{S}_{E}}(x(t))/\hbar )D[x(t)]dx.} } } \hfill \\ \end{array}$$ (14.1.2)
The transition amplitudes 4.47 and 4.82 of chapter 4 may be generalized for a problem with s degrees of freedom that may be described by a set of operators q ̂ = q ̂ α α = 1 s , p ̂ = p ̂ α α = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqni6_1.jpg"/> : 6.1 Κ q ″ , t ″ ; q ′ , t ′ = ∫ ∏ α = 1 s D p α D q α exp i ℏ ∫ t ′ t ″ ∑ α = 1 s p α q . α − Η p q d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn6_1.jpg"/>
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
It is instructive to note that the principal advantage of linear operators is that their action on any vector is defined purely by their action on a set of basis vectors. Let us consider the linear operator F ̂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqni2_1.jpg"/> transforming the vector a via its components a k : 2.1 F ̂ a i = ∑ k F i k a k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn2_1.jpg"/>
