Quantum Dynamics

Abstract

In classical mechanics, every function F on phase space generates a one-parameter group of diffeomorphisms exp(t LX _F .) (t ∈ R and LX _F is the Lie derivative with respect to the Hamiltonian vector field corresponding to F). Similarly, we learned in §2.4 that in quantum theory every observable a is associated with a one-parameter group of automorphisms b → exp(iat)b exp(—iat). One of the basic postulates quantum theory is that, in units with ħ = 1, the groups generated by the Cartesian position and momentum coordinates x _i and p _j of n particles (j = 1, ....,n) are the same as classically, i.e., displacements respectively in the momenta and positions. Since x _i and p _j do not have bounded spectra, and hence cannot be represented by bounded operators, it is convenient to consider instead the bounded functions$\exp \left( {i\sum\limits_{i = 1}^n {{X_j} \cdot {S_j}} } \right)and\exp \left( {i\sum\limits_{i = 1}^n {{p_j} \cdot {r_j}} } \right),{S_j}{r_j} \in {R^3}$ so as not to have domain questions to worry about. The group of automorphisms can be written in terms of them as follows:

Phase space is the arena of classical mechanics. The algebra of observables in quantum mechanics is likewise constructed with position and momentum, so this section covers the properties of those operators.