The flux operator

In Section 5.2, we used an expression for the flux operator. In most quantum mechanics textbooks expressions for the probability current density, or probability flux density, are given in terms of the wave function in the coordinate representation. We need an expression for the flux density operator without reference to any particular representation, and since it is rarely found in the textbooks, let us in the following derive this expression. 

It is related to the probability density for one of the coordinates of a particle to be r. This is seen by forming the expectation value of Pr for the system in the state / ). We find 

We seek an expression for the flux operator by deriving an expression for the time variation of the position projection operator Pr, and compare the resulting equation with the standard continuity equation known in several branches of physics, for example, in fluid dynamics  

Let us therefore determine the rate of change of the position projection operator for the coordinate qi, which in the Heisenberg picture is given by the following commutator equation  

Note that the potential energy term in general cannot be split up into a term only depending on the coordinate qi like the kinetic energy term. The commutator pjy Pr may be written 

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It is readily checked that the matrix elements of the flux operator between two states are... 

We consider the trace at t = 0, Tr[Fe /2kBT9 P)e l2kBT] = Tr[PA], which defines the operator A. It is noted that both the flux operator F and the A operator are Hermitian operators, which implies that Tr[Tk4] = Tr[(PA)t] = Tr pt] = Tr[AP] = Tr[PA], that is, the trace must always be real valued. Furthermore, if the trace is evaluated in a basis of real-valued functions, then the trace must be equal to zero, since the operators contain the momentum operator, which also contains the imaginary unit i = T. Using this result, Eq. (5.108) can be written in the form... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

Let us then derive an expression for the matrix element of the flux operator in the coordinate representation, an expression we need in order to develop the time autocorrelation function of the flux operator in the coordinate representation. We use the axiom for the matrix element of the momentum operator in the coordinate representation, and obtain... 

In Section 5.2, we have seen how the rate constant for a chemical reaction may be determined as a time integral of the auto-time-correlation function of the flux operator... 

In order to use this formal expression in a calculation of the rate constant we need to choose a representation. In the following we will determine the coordinate representation of the correlation function. We use the coordinate representation of the flux operator as derived above. It is introduced in the expression for the time-correlation function by introducing three unit operators like... 

The most elegant way to derive Eqs. (1) and (2) starts from introducing the flux operator Ps through a surface 5. Matrix element of PS between any two functions P and 4> can be defined by the surface integral... 

The reaction path Hamiltonian is particularly useful for evaluating the path integral representation of the trace, Eq. (3.1 ), because the flux operator F does not involve the bath" degrees of freedom (i.e., the transverse vibrational modes Q ), and since they are harmonic oscillators the path integrals over them can be carried out analytically. All that remains to be done numerically is the path integral over only the reaction coordinate degrees of freedom itself. 

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