Whether classical or quantum, solving the equations of motion analytically exactly is often a formidable task. Some properties of the motion may, however, be inferred without solving the equations of motion. One such important property is the symmetry of the Hamiltonian under some transformation. The interesting and practically useful aspect of symmetry is that it characterizes the constants of the motion. The symmetry may be under some transformation of space and time, called the geometrical symmetry, or it may be in some abstract space resulting from particular functional form of the Hamiltonian, called the dynamical symmetry. This chapter investigates the consequences of such symmetries on the quantum mechanical description of the motion of a particle.

Symmetry Transformation

Consider a system whose state for some observer is. If the observer performs the measurement of an observable on the system, we know that the probability of observing an eigenvalue corresponding to an eigenstate as an outcome is. Now, let the same experiment be described in a coordinate system transformed with respect to the first one. Since the result of a measurement is not expected to depend on the coordinate system used, the probability of observing certain eigenvalue as the outcome should not change. Generalizing the arguments above, we expect the transition probability between the states to be independent of the coordinate system. This chapter is concerned with investigating the consequences of the invariance of the probabilities under linear transformation of the coordinate system.