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Chapter Summary:

6. Linear Operators in Hilbert Spaces

Having had some experience working with quantum mechanical formalism in various situations,
it is now time to investigate more deeply its mathematical properties. In particular, we are going to describe the time evolution and, more generally, symmetry transformations and their generators.

Time evolution according to the Schr\"odinger equation has the property that it conserves the norm of wave packets. In mathematical terms, the relation between the initial state (at time t=0) and the state at time t can be described as the action of a unitary operator U(t). The set of operators U(t) is called the unitary group generated by the Hamiltonian operator H. In the same way all self-adjoint operators generate unitary groups. For example, the momentum operator generates the unitary group of translations and the angular momentum operator generates rotations. For self-adjoint operators satisfying the canonical commutation relations, the corresponding unitary groups satisfy the Weyl relations. We shall have occasion to use the Weyl relations for calculating the time evolution of harmonic oscillator states in the next chapter.

A system is said to be symmetric with respect to a certain group of transformations, if these transformations commute with the time evolution. Equivalently, the generator of the symmetry transformation commutes with the Hamiltonian of the system. An immediate consequence is Noether's theorem. If the Schrödinger equation has a symmetry, then any physical observable corresponding to the generator of the symmetry is a conserved quantity (a constant of motion). For example, momentum is conserved in a system with translational symmetry.